Interval linear algebra

Transcription

1 Charles University in Prague Faculty of Mathematics and Physics HABILITATION THESIS Milan Hladík Interval linear algebra Department of Applied Mathematics Prague 204

2 Contents Preface 3 Interval linear algebra 5. Introduction Systems of interval linear equations Enclosure methods Linear dependencies AE solution set The interval eigenvalue problem The symmetric case The general case Further topics Bibliography 6 Reprints of papers 20 M. Hladík. New operator and method for solving real preconditioned interval linear equations. SIAM J. Numer. Anal., M. Hladík. Solution set characterization of linear interval systems with a specific dependence structure. Reliab. Comput., M. Hladík. Description of symmetric and skew-symmetric solution set. SIAM J. Matrix Anal. Appl., M. Hladík. Enclosures for the solution set of parametric interval linear systems. Int. J. Appl. Math. Comput. Sci., E. D. Popova & M. Hladík. Outer enclosures to the parametric AE solution set. Soft Comput., M. Hladík, D. Daney & E. Tsigaridas. Bounds on real eigenvalues and singular values of interval matrices. SIAM J. Matrix Anal. Appl., M. Hladík, D. Daney & E. Tsigaridas. An algorithm for addressing the real interval eigenvalue problem. J. Comput. Appl. Math., M. Hladík, D. Daney & E. Tsigaridas. Characterizing and approximating eigenvalue sets of symmetric interval matrices. Comput. Math. Appl., M. Hladík, D. Daney & E. Tsigaridas. A filtering method for the interval eigenvalue problem. Appl. Math. Comput., M. Hladík. Bounds on eigenvalues of real and complex interval matrices. Appl. Math. Comput.,

3 Preface This habilitation thesis consists of 0 journal papers, supplemented by a commentary. The papers are (co-)authored by M. Hladík, and their subject belongs to interval linear algebra. The thesis is structured as follows. In the first part, we introduce the reader to the topic with particular emphasis on author s contribution. More specifically, we focus on interval linear systems of equations (Section.2) and the interval eigenvalue problem (Section.3). We briefly mention also further related areas the author works in (Section.4). Generally, most of the author s contribution is of algorithmic essence developing polynomial methods for approximating NPhard problems. Some of his results are more theoretical he extended the list of known NP-hard problems by posting several new ones, and also derived explicit description of some complicated sets. In the second part, we attach reprints of those 0 papers the thesis is based on. Below, we give a short summary accompanied with a brief description. For interval linear systems of equations, we attach the papers: Milan Hladík. New operator and method for solving real preconditioned interval linear equations. SIAM J. Numer. Anal., 52():94 206, 204. In this paper, we develop a new method for solving interval linear equations. It outperforms some classical methods with respect to both time and sharpness of enclosures. Milan Hladík. Solution set characterization of linear interval systems with a specific dependence structure. Reliab. Comput., 3(4):36 374, 2007 We extend characterization of the solution set of interval linear equations for the case when there is a simple dependence structure between the interval coefficients. Milan Hladík. Description of symmetric and skew-symmetric solution set. SIAM J. Matrix Anal. Appl., 30(2):509 52, We derive an explicit characterization of the solution set of interval linear equations under the restriction that the constraint matrix must be symmetric resp. skew-symmetric. Milan Hladík. Enclosures for the solution set of parametric interval linear systems. Int. J. Appl. Math. Comput. Sci., 22(3):56 574, 202. We generalize some enclosure methods for interval linear equations to the case when the matrix and the right-hand side entries depend linearly on interval parameters. Evgenija D. Popova and Milan Hladík. Outer enclosures to the parametric AE solution set. Soft Comput., 7(8):403 44, 203. We consider systems of linear equations, where the elements of the matrix and of the right-hand side vector are linear functions of interval parameters. We study parametric AE solution sets, which are defined by 3

4 -quantification of parameters. We generalize classical methods to obtain polynomially computable outer bounds for parametric AE solution sets. The interval eigenvalue problem is accompanied by reprints of the following papers: Milan Hladík, David Daney, and Elias Tsigaridas. Bounds on real eigenvalues and singular values of interval matrices. SIAM J. Matrix Anal. Appl., 3(4):26 229, 200. We propose computationally cheap bounds on the eigenvalues for general and for symmetric interval matrices, and also for singular values of (nonsquare) interval matrices. Milan Hladík, David Daney, and Elias P. Tsigaridas. An algorithm for addressing the real interval eigenvalue problem. J. Comput. Appl. Math., 235(8): , 20. We develop an algorithm for approximation of the real eigenvalues of interval matrices with a given precision. Milan Hladík, David Daney, and Elias P. Tsigaridas. Characterizing and approximating eigenvalue sets of symmetric interval matrices. Comput. Math. Appl., 62(8): , 20. We present a characterization of some of the boundary eigenvalues of symmetric interval matrices. Based on this result, we introduce an inner approximation algorithm that in many cases finds exact bounds. Milan Hladík, David Daney, and Elias P. Tsigaridas. A filtering method for the interval eigenvalue problem. Appl. Math. Comput., 27(2): , 20. We propose a filtering method that iteratively improves a given outer approximation of eigenvalues of an interval matrix. Milan Hladík. Bounds on eigenvalues of real and complex interval matrices. Appl. Math. Comput., 29(0): , 203. We present a computationally cheap and tight formula for bounding real and imaginary parts of eigenvalues of real or complex interval matrices. 4

5 . Interval linear algebra Interval computation, roughly speaking, studies problems with interval input data. Intervals naturally appear in many situations: Rounding errors A real number cannot be represented exactly in the floating point arithmetic. In order to get reliable results, it is a natural approach to enclose the result of each operation in as small as possible interval covering the true value. For example, [ , ], 3 2 [ , ], π [ , ]. Computer-assisted proofs Some mathematical proofs, such as Kepler conjecture or the Double bubble problem, were carried out with aid of computers. Employing interval methods here was necessary to obtain verified results. CSP and global optimization In the Constraint satisfaction problem or in Global optimization, we have to find a solution (or the best solution) subject to some nonlinear constraints. Interval computation enables in a numerically reliable way to remove parts of the initial domain that contain no solution, and by an extensive space search it encloses all solutions with arbitrary precision. Modelling uncertainty Intervals are also used to model uncertain, inexact or incomplete data. Results of measurements are often expressed in the form of v± v, meaning that the true value, which is not observable, lies in the interval [v v, v + v]. Continuous variables are sometimes discretized into a finite set of values typically, we split time into time slots (days, years,... ). Incomplete data may arise due to lack of knowledge or data protection personal information (salary, age,... ) is often categorized in interval ranges. Interval linear algebra gives a necessary mathematical background for interval computation. The same role the standard linear algebra plays in the modern science, the interval linear algebra plays for studying interval problems. Fundamental problems of interval linear algebra are analogous to those of traditional linear algebra; examples include solving interval linear equations, testing nonsingularity or computing eigenvalues of interval matrices. Thus, interval linear algebra is a basis for solving more complex interval-valued problems, such as interval mathematical programming, interval least squares and statistics on interval data, for instance. 5

6 . Introduction Notation. We begin with the key notion of an interval matrix, which is the family of matrices A := {A R m n ; A A A}, where A, A R m n, A A, are given. The inequality for matrices and vectors is understood entrywise throughout the thesis. The center and radius matrices of A are defined as A c := (A + A), 2 A := (A A). 2 The set of all interval matrices of size m n is denoted by IR m n. Interval vectors and intervals are considered as special cases of interval matrices. The magnitude of A IR m n is defined as mag(a) := max { A ; A A} = max( A, A ), where the maximum and absolute value functions are understood entrywise. Let a set S R n be given. The interval hull of S, denoted by S, is the smallest interval vector containing S, that is S := v. v IR n : S v As we will see later, to determine the interval hull is a computationally hard problem in general. So we often weaken the goal to compute an enclosure instead. An enclosure of a set S R n is any interval vector v IR n of such that S v. Naturally, one seeks for as small as possible enclosures. The computation of interval hulls or enclosures of various sets (described explicitly or implicitly) belongs to the basic problems studied in interval computation. Interval arithmetic. Interval arithmetic is the basic tool to compute enclosures for the images of real functions defined by arithmetic expressions. Let be a basic operation addition, subtraction, multiplication or division. For a, b IR we define a b := {a b; a a, b b}, with 0 b in case of division. It is not hard to see that for particular operations the interval arithmetic reads a + b = [a + b, a + b], a b = [a b, a b], ab = [min(ab, ab, ab, ab), max(ab, ab, ab, ab)], a/b = [min(a/b, a/b, a/b, a/b), max(a/b, a/b, a/b, a/b)]. Looking at real numbers as zero-radius intervals, interval arithmetic generalizes the classical one, and we can use mixed expressions like 2[3, 4]+5 with the meaning [2, 2][3, 4] + [5, 5]. Interval addition and multiplication is known to be commutative, associative and sub-distributive. More details on interval arithmetic and interval computation in general can be found in books [, 33, 34, 35]. 6

7 Interval extensions of functions. One of the fundamental problems in interval analysis is computation of the range of a function over intervals. Let f : R n R be a real function and x IR n. The range of f over x is defined as f(x) := {f(x); x x}. For continuous and monotone functions, the range is easy to determine. If f is non-decreasing on x IR, then f(x) = [f(x), f(x)] and likewise for nonincreasing functions. For some simple functions, such as the sine or square, the range is easily determined, too. In general, the range f(x) need not be an interval. In fact, it may be neither closed nor connected. Moreover, checking whether 0 f(x) is an undecidable problem [5]. Thus one aims at determining its interval hull f(x) instead. Besides numerical aspects of exact calculation, computing f(x) is still a very difficult problem in general. That is why we usually cannot hope for exact computation of f(x), and we have to rely on more-or-less tight enclosures. Bad news are that even calculating a sufficiently tight enclosure may be still a computationally hard problem. Consider a mapping f : IR n IR, called an interval function. We say that it is inclusion isotonic if for every x, y IR n : x y f(x) f(y). Further, the interval function f : IR n IR is an interval extension of f : R n R if for every x R n : f(x) = f(x). These two properties, inclusion isotonicity and interval extension, are enough to get a proper enclosure for the range of f(x). The following theorem is already by Moore [33]. Theorem. (Fundamental theorem of interval analysis). If f : IR n IR is inclusion isotonic and is an interval extension of f : R n R, then for every x IR n : f(x) f(x). Interval arithmetic is one example of an inclusion isotonic interval extension. Suppose that we have an arithmetic expression for f : R n R using only a finite number of arithmetic operations. The corresponding natural interval extension f of f is defined by that expression when replacing real arithmetic by the interval one. Theorem.2. Natural interval extension of an arithmetic expression is both an interval extension and inclusion isotonic. Dependency problem. When evaluating arithmetic expressions by interval arithmetic, we face the so called dependency problem. As an example, consider f(x) = x 2 2x and x x = [, 2]. Direct evaluation yields the enclosure f(x) x 2 2x = [ 3, 2], but an equivalent formulation f(x) = (x ) 2 results in the exact image f(x) = (x ) 2 = [, 0]. The reason is that in the latter expression, the parameter appears only once, and so the evaluation by interval arithmetic is exact, while in the former the parameter appears twice. Unfortunately, not every function can be expressed in this way, and that is why direct evaluations can suffer from high overestimation of the true image. 7

8 .2 Systems of interval linear equations Let A IR m n and b IR m. The system of interval linear equations is a family of linear systems We denote this family shortly as Ax = b, A A, b b. Ax = b, but the aim is not to find an (interval) vector x that satisfies these equations. A solution is defined as a solution to a system Ax = b for some A A and b b. The solution set is defined as a set of all solutions and denoted Σ := {x R n ; A A, b b : Ax = b}. (.) Comprehensive works on interval system include [, 6, 33, 34, 35]. Characterization. The well-known characterization of Σ comes from [38]. Theorem.3 (Oettli Prager, 964). The solution set Σ is described by the inequality system A c x b c A x + b. From the description of Σ we see that it represents a non-convex polyhedral set, which is however convex in each orthant. The problem of checking Σ is NP-hard; see [32]. This is true even in the class of problems with m = n; cf. [6]..2. Enclosure methods Here we restrict ourselves to the most typical case m = n, and we want to compute an enclosure of Σ. Many methods for computing enclosures of Σ are just extensions of the classical methods for the real case. This involves Gaussian elimination or Jacobi and Gauss Seidel iterations. There are also some specific algorithms developed directly for the interval case, for example the Krawczyk method (see [30, 34, 35]), the Hansen Bliek Rohn method (see [3, 6, 8, 36, 37, 43]) or the Magnitude method [25]. Preconditioning. Most of the methods use preconditioning, which was first introduced for interval systems in Hansen [7]. Preconditioning by a matrix C R n n means that we multiply both sides of Ax = b by C in interval arithmetic to obtain a new interval system (CA)x = Cb. Due to properties of interval arithmetic, the solution set to the new system contains the original one. Even though the new solution set is larger than the original one, the overestimation is usually small. The main argument for preconditioning is that many methods perform better with preconditioning. 8

9 Mostly, we precondition by a numerically computed (A c ), which has good both theoretical properties and practical performance. If we precondition by (A c ), then the interval hull can be computed in polynomial time by using the Hansen Bliek Rohn method. However, other methods are useful as well since they are faster and the overestimation is low. We briefly describe the Gauss Seidel and Magnitude methods. Gauss Seidel method. It is an iterative method that start with an initial enclosure x of Σ, and then iteratively improves ( x i := b i ) a ij x j x i, i =,..., n. a ii j i Magnitude method. It is a very recent method from Hladík [25] using a kind of generalization of the Gauss Seidel method. We assume that Ax = b is already preconditioned. Next, we assume that A c = I n, which is easily done by suitable inflation of interval entries of A. Eventually, we assume that for the spectral radius of A, denoted by ρ(a ), we have ρ(a ) < ; this is a necessary and sufficient condition for A to contain nonsingular matrices only. Now, the method works as follows:. Compute u, an enclosure of the solution of the real system Au = mag(b). 2. Calculate d i := a ii /( ((A ) 2 ) ii ), i =,..., n. 3. Evaluate where γ i := a ii /d i. x i := b i + ( j i a iju j γ i u i )[, ], i =,..., n, a ii + γ i [, ] As a result, we have Σ x. The third step of the method is one Gauss Seidellike iteration applied to the enclosure [ u, u] Σ. It was proved in [25] that the computed enclosure x si always as tight as that calculated by the interval Gauss Seidel method. Moreover, if d i are replaced by (A ) ii, i =,..., n, then Σ = x, up to numerical precision..2.2 Linear dependencies So far, we considered the case when parameters can attain any value from their interval domains independently of each other. This assumption is very restrictive and hardly satisfied in practice. Mostly, there are some dependencies between the interval parameters. Thus, to take dependencies into account is an important and challenging problem. As a rich class of systems with dependencies, consider problems with linear parametric structure A(p)x = b(p), 9

10 where A(p) = K k= A kp k, b(p) = K k= b kp k and p p for some given interval vector p IR K, matrices A,..., A K R n n and vectors b,..., b n R n. This linear parametric case covers a wide area of interval systems with dependencies. For instance, the interval systems Ax = b, where the constraint matrix is supposed to be symmetric, skew-symmetric, Toeplitz or Hankel. The solution set. The solution set of a parametric interval system is defined as Σ p = {x R n ; A(p)x = b(p) for some p p}. Many researcher tried to generalize the Oettli Prager Theorem.3 for the parametric case, but no simple characterization appeared. Hladík [] derived an Oettli Prager-like characterization of interval linear equations and inequalities with a simple dependence structure given by multiple appearance of a submatrix in the constraints. This was utilized by Hladík [6] to characterize solutions of a system of complex interval equations, where complex intervals have a rectangular form. Fourier Motzkin type elimination characterizing the solution set by a system of nonlinear inequalities was utilized in [2, 39], but the number of inequalities may be doubly exponential. The symmetric solution set. Symmetry of the constraint matrix is a special type of the linear parametric form. Herein, the symmetric solution set reads {x R n ; Ax = b for some symmetric A A}. Explicit description of this set was derived in Hladík [2], and it uses only exponentially many inequalities a ij x i x j (p i q j ) + i,j= A x + b r, b i x i (p i + q i ) r i x i (p i q i ) i= i= for all vectors p, q {0, } n \ {0, } such that p lex q and (p = q i : p i = q i = 0), where r := A c x + b c and lex denotes the lexicographical ordering. It remains open whether there is a polynomial characterization by means of nonlinear inequalities. Anyway, checking whether x Σ p for a given x R n is polynomially decidable via linear programming even for a general linear parametric solution set. Enclosures. To find an enclosure of Σ p, we can simply forget the dependencies and enclose by standard methods the relaxed system Ax = b, where A := A(p) and b := b(p) are evaluated by interval arithmetic. Utilizing the dependencies, however, leads to tighter enclosures. Various extensions of standard methods to the parametric case were presented in [42, 47, 48, 50], among others. In particular, Hladík [20] generalized the Bauer Skeel and the Hansen Bliek Rohn bounds for this case and combined them together, yielding a more efficient algorithm. 0

11 .2.3 AE solution set In the definition (.) of the standard solution set, interval parameters are associated with existential quantifiers. In some problems, universal quantifiers may appear, too. Thus, a natural generalization of the solution concept is to associate each parameter p k i,j {a ij, b i } with a quantification Q k {, }. Now, a solution is any vector x R n satisfying the quantified formula Q p, Q 2 p 2,..., Q n 2 +np n 2 +n : Ax = b. To treat such solutions is a tempting problem. The known results are concerned with the special case of π 2 -quantification, called AE quantification in interval community (AE for All-Exists ). The interval quantities that are universally quantified are denoted by A, b, and the existential ones by A, b. Thus, the interval system Ax = b can be written as (A + A )x = b + b, and the so called AE solution set is defined Σ AE := { x R n ; A A b b A A b b : (A + A )x = b + b }. Special cases. There are some important special cases of the general AEsolution concept. For example, tolerable solutions are defined by the condition A A, b b : Ax = b, and controllable solutions are defined by b b, A A : Ax = b. Characterization. Surprisingly, AE solution set can be described in the same manner as Oettli Prager; see [49] and references therein. Theorem.4 (Shary, 995). We have Theorem.5 (Rohn, 996). We have Σ AE = { x R n ; A x b b A x }. Σ AE = { x R n ; A c x b c ( rad A rad A ) x + rad b rad b }. Combining linear dependencies with AE solutions, we come up with a general model, which is in the scope of current research. Popova [40] utilized Fourier Motzkin type elimination to characterize the solution set by a system of nonlinear inequalities. Concerning enclosures of the corresponding solution set, Popova and Hladík [4] recently generalized the single-step Bauer Skeel method, and for the tolerable solution set, they proposed a linear programming based method that yields optimal enclosures under some assumptions.

12 .3 The interval eigenvalue problem.3. The symmetric case If A R n n is symmetric, then it has real eigenvalues, and thus we may suppose they are sorted non-increasingly λ (A) λ n (A). Given an interval matrix A IR n n with A and A symmetric, the corresponding symmetric interval matrix is defined as Its eigenvalue sets are defined as A S := {A A; A = A T }. λ i (A S ) := {λ i (A); A A S }, i =,..., n. Each eigenvalue set λ i (A S ) consists of ith eigenvalues of all symmetric matrices in A. By the continuity of eigenvalues and compactness and convexity of A S it is easy to see that λ i (A S ), i =,..., n are compact intervals. They may be disjoint or they may overlap, but one interval can never lie in the interior of another one. Exact bounds. The largest and smallest eigenvalues of A S can be calculated by an exponential time formula by Hertz [9]. These two extremal eigenvalues are attained at matrices of special form. Theorem.6 (Hertz, 992). We have λ (A S ) = max λ ( z {±} n A c + diag(z)a diag(z) ), λ n (A S ) = min λ ( z {±} n n A c diag(z)a diag(z) ). The other boundary points of the eigenvalue sets need not be attained at these matrices, moreover, they need not be attained at vertex matrices (matrices with entries a ij {a ij, a ij } i, j). Hladík et al. [28] extended the Hertz theorem for boundary eigenvalues of λ i (A S ), i =,..., n, that lie in no other eigenvalue set. Based on these results, they also developed an algorithm (the so called submatrix vertex enumeration) that computes exact bounds (up to the numerical precision) under some additional assumptions. In the general case, it returns inner and outer approximations of the eigenvalues sets. A complete characterization of the eigenvalue sets still remains a challenging open problem. Enclosures. Calculation of the eigenvalue sets is computationally intractable. Even checking whether 0 λ i (A S ) for some i =,..., n is NP-hard; for some more NP-hardness results see [3]. That is why we will again turn our attention to tight enclosures of the eigenvalue sets. A simple enclosure of the eigenvalue set is obtained by the Weyl theorem; cf. [26, 46]. Recall that ρ(a) stands for the spectral radius of A. 2

13 Theorem.7. We have λ i (A S ) [λ i (A c ) ρ(a ), λ i (A c ) + ρ(a )], i =,..., n. Hladík et al. [26] also employed the Cauchy interlacing property in two different ways to enclose the eigenvalue sets by other means, and proposed several other computationally cheap bounds. Combining these approaches together leads to an efficient enclosing algorithm with respect to both time and tightness. Contracting method. Hladík et al. [29] developed an iterative contractor. It starts with an enclosure of the eigenvalue sets and iteratively makes them tighter. The method works only for disjoint enclosures and does not converge to the optimal bounds in general, but the formula is computationally cheap, converges in a low number of steps, and often reduces the overestimation significantly. The method is based on the following theorem. For a given λ 0 that is an eigenvalue of no matrix in A S it determines a neighbourhood interval containing no eigenvalues as well. Theorem.8. Let λ 0 n i=λ i (A S ) and define M := A λ 0 I. Then (λ 0 + λ) n i=λ i (A S ) for all real λ satisfying λ < 2 ρ ( I QM c + I QM c T + Q M + (M ) T Q ) T, (.2) ρ ( Q + Q T ) where Q R n n, Q 0, is an arbitrary matrix. Even though the theorem is valid for any Q 0, an appropriate choice of Q influences effectivity of the formula (.2). The authors recommended the choice of (M c ) or its numerical approximation..3.2 The general case We have mentioned that calculation of eigenvalue bounds for symmetric interval matrices is a computationally hard problem. Bounding (complex) eigenvalues for general interval matrix A IR n n is even much more difficult. The following enclosure for all eigenvalues of all A A is by Hladík [22]. Originally developed for eigenvalues of complex interval matrices, we state it in the real form for the sake of simplicity. Theorem.9. For any A A and its eigenvalue ν = λ + iµ we have λ n ( (A + 2 AT ) S ) λ λ ( (A + 2 AT ) S ), ( 0 λ (A ) S ( 2 AT ) 0 n µ λ (A ) S 2 AT ) 2 (AT A) 0. 2 (AT A) 0 The theorem generalizes and improves some older formulae [0, 45]. The main idea behind is to reduce the general case to the symmetric one. In particular, the reduction uses the right end-point of the largest eigenvalue set and the left end-point of the smallest eigenvalue set. Thus, the more efficient method for computing the extremal eigenvalues of symmetric interval matrices is employed, the more efficient are these formulae. 3

14 Enclosing eigenvalues of A A by circles was presented by Hladík et al. [26]. They adapted the well known Bauer Fike theorem and its generalization by Chu [5] from the perturbation theory of eigenvalues of diagonalizable and not necessarily diagonalizable matrices, respectively. Upper bounds on the maximal spectral radius, max{ρ(a); A A}, were dealt with in Hladík [7]. He proposed two computationally cheap and tight formulae and adapted the contracting method mentioned above to refine the computed values. Real eigenvalues. Now, we focus on real eigenvalues of general matrices A A. The set of real eigenvalues is defined as Λ(A) := {λ R; Ax = λx, x 0, A A}. The iterative contractor from [29] discussed above is applicable for this situation, too, as well as the circle enclosures. A more thorough investigation was presented by Hladík et al. [27]. Adapting sufficient or necessary conditions for regularity of interval matrices and enhancing various techniques from interval computations, they developed an algorithm approximating Λ(A) by means of inner and outer enclosures. Moreover, by using eigenvalue theorems from Rohn [44], they managed to achieve exact boundary points (up to numerical precision) of Λ(A) under some mild assumptions only..4 Further topics Interval linear algebra is a basis for solving many interval-valued problems. Interval linear inequalities. Generalization of the Oettli Prager Theorem.3 to interval mixed linear equations and inequalities is due to Hladík [23]. He also considered strong solvability (i.e., AE solutions with universal quantifiers only) for such systems. Interval linear programming. Linear programming with coefficients varying in given intervals was surveyed in Hladík [2]. Hladík [3] proposed a general scheme for computing the range of optimal values subject to variations of parameters in given intervals; the scheme involves not only basic linear programming formulations using equations or inequalities, but it ables to handle dependencies between the parameters as well. Further, Hladík [24] studied the conditions under which an optimal basis remains optimal under any perturbation of parameters in intervals, and proposed a sufficient condition for such basis stability. NP-hardness results in the area of multiobjective interval linear programming were presented by Hladík [9]. He showed, for instance, that checking whether a given solution remains Pareto optimal for any perturbation of the objective value coefficients in given intervals is a co-np-complete problem. 4

15 Interval nonlinear programming. Hladík [8] proposed a general framework for determining bounds of the optimal values of nonlinear programming problems when input data vary in given intervals. He applied the approach in two classes of optimization problems: Convex quadratic programming and posynomial geometric programming. Hladík [4] considered a similar problem for interval-valued linear fractional programming problems. Interval matrix games. Suppose that payoffs of bimatrix games are subject to interval uncertainties. Hladík [5] discussed the problem of existence of an equilibrium being common for all instances of interval values. He also characterized the set of all possible equilibria by means of a linear mixed integer system. Interval linear regression. Enclosing least square solutions for interval-valued overdetermined linear equations, with straightforward applications in statistics, was considered in Černý, Antoch and Hladík [4]. 5

16 Bibliography [] G. Alefeld and J. Herzberger. Introduction to Interval Computations. Computer Science and Applied Mathematics. Academic Press, New York, 983. [2] G. Alefeld, V. Kreinovich, and G. Mayer. On the solution sets of particular classes of linear interval systems. J. Comput. Appl. Math., 52(-2): 5, [3] C. Bliek. Computer Methods for Design Automation. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, July 992. [4] M. Černý, J. Antoch, and M. Hladík. On the possibilistic approach to linear regression models involving uncertain, indeterminate or interval data. Inf. Sci., 244:26 47, 203. [5] K.-w. E. Chu. Generalization of the Bauer-Fike theorem. Numer. Math., 49(6):685 69, 986. [6] M. Fiedler, J. Nedoma, J. Ramík, J. Rohn, and K. Zimmermann. Linear Optimization Problems with Inexact Data. Springer, New York, [7] E. Hansen. Interval arithmetic in matrix computations, Part I. J. Soc. Ind. Appl. Math., Ser. B, Numer. Anal., 2(2): , 965. [8] E. R. Hansen. Bounding the solution of interval linear equations. SIAM J. Numer. Anal., 29(5): , 992. [9] D. Hertz. The extreme eigenvalues and stability of real symmetric interval matrices. IEEE Trans. Autom. Control, 37(4): , 992. [0] D. Hertz. Interval analysis: Eigenvalue bounds of interval matrices. In C. A. Floudas and P. M. Pardalos, editors, Encyclopedia of optimization, pages Springer, New York, [] M. Hladík. Solution set characterization of linear interval systems with a specific dependence structure. Reliab. Comput., 3(4):36 374, [2] M. Hladík. Description of symmetric and skew-symmetric solution set. SIAM J. Matrix Anal. Appl., 30(2):509 52, [3] M. Hladík. Optimal value range in interval linear programming. Fuzzy Optim. Decis. Mak., 8(3): , [4] M. Hladík. Generalized linear fractional programming under interval uncertainty. Eur. J. Oper. Res., 205():42 46, 200. [5] M. Hladík. Interval valued bimatrix games. Kybernetika, 46(3): , 200. [6] M. Hladík. Solution sets of complex linear interval systems of equations. Reliab. Comput., 4:78 87,

17 [7] M. Hladík. Error bounds on the spectral radius of uncertain matrices. In T. Simos, editor, Proceedings of the International Conference on Numerical Analysis and Applied Mathematics 20 (ICNAAM-20), G-Hotels, Halkidiki, Greece, 9-25 September, volume 389 of AIP Conference Proceedings, pages , Melville, New York, 20. American Institute of Physics (AIP). [8] M. Hladík. Optimal value bounds in nonlinear programming with interval data. TOP, 9():93 06, 20. [9] M. Hladík. Complexity of necessary efficiency in interval linear programming and multiobjective linear programming. Optim. Lett., 6(5): , 202. [20] M. Hladík. Enclosures for the solution set of parametric interval linear systems. Int. J. Appl. Math. Comput. Sci., 22(3):56 574, 202. [2] M. Hladík. Interval linear programming: A survey. In Z. A. Mann, editor, Linear Programming New Frontiers in Theory and Applications, chapter 2, pages Nova Science Publishers, New York, 202. [22] M. Hladík. Bounds on eigenvalues of real and complex interval matrices. Appl. Math. Comput., 29(0): , 203. [23] M. Hladík. Weak and strong solvability of interval linear systems of equations and inequalities. Linear Algebra Appl., 438(): , 203. [24] M. Hladík. How to determine basis stability in interval linear programming. Optim. Lett., 8(): , 204. [25] M. Hladík. New operator and method for solving real preconditioned interval linear equations. SIAM J. Numer. Anal., 52():94 206, 204. [26] M. Hladík, D. Daney, and E. Tsigaridas. Bounds on real eigenvalues and singular values of interval matrices. SIAM J. Matrix Anal. Appl., 3(4):26 229, 200. [27] M. Hladík, D. Daney, and E. P. Tsigaridas. An algorithm for addressing the real interval eigenvalue problem. J. Comput. Appl. Math., 235(8): , 20. [28] M. Hladík, D. Daney, and E. P. Tsigaridas. Characterizing and approximating eigenvalue sets of symmetric interval matrices. Comput. Math. Appl., 62(8): , 20. [29] M. Hladík, D. Daney, and E. P. Tsigaridas. A filtering method for the interval eigenvalue problem. Appl. Math. Comput., 27(2): , 20. [30] R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Comput., 4:87 20, 969. [3] V. Kreinovich, A. Lakeyev, J. Rohn, and P. Kahl. Computational Complexity and Feasibility of Data Processing and Interval Computations. Kluwer,

18 [32] A. V. Lakeev and S. I. Noskov. On the solution set of a linear equation with the right-hand side and operator given by intervals. Sib. Math. J., 35(5): , 994. [33] R. E. Moore. Interval Analysis. Prentice-Hall, Englewood Cliffs, NJ, 966. [34] R. E. Moore, R. B. Kearfott, and M. J. Cloud. Introduction to Interval Analysis. SIAM, Philadelphia, PA, [35] A. Neumaier. Interval Methods for Systems of Equations. Cambridge University Press, Cambridge, 990. [36] A. Neumaier. A simple derivation of the Hansen-Bliek-Rohn-Ning-Kearfott enclosure for linear interval equations. Reliab. Comput., 5(2):3 36, 999. [37] S. Ning and R. B. Kearfott. A comparison of some methods for solving linear interval equations. SIAM J. Numer. Anal., 34(4): , 997. [38] W. Oettli and W. Prager. Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides. Numer. Math., 6: , 964. [39] E. D. Popova. Explicit characterization of a class of parametric solution sets. Comptes Rendus de L Academie Bulgare des Sciences, 62(0):207 26, [40] E. D. Popova. Explicit description of AE solution sets for parametric linear systems. SIAM J. Matrix Anal. Appl., 33(4):72 89, 202. [4] E. D. Popova and M. Hladík. Outer enclosures to the parametric AE solution set. Soft Comput., 7(8):403 44, 203. [42] E. D. Popova and W. Krämer. Inner and outer bounds for the solution set of parametric linear systems. J. Comput. Appl. Math., 99(2):30 36, [43] J. Rohn. Cheap and tight bounds: The recent result by E. Hansen can be made more efficient. Interval Comput., 993(4):3 2, 993. [44] J. Rohn. Interval matrices: Singularity and real eigenvalues. SIAM J. Matrix Anal. Appl., 4():82 9, 993. [45] J. Rohn. Bounds on eigenvalues of interval matrices. ZAMM, Z. Angew. Math. Mech., 78(Suppl. 3):S049 S050, 998. [46] J. Rohn. A handbook of results on interval linear problems. Technical Report 63, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, 202. [47] S. M. Rump. Verification methods for dense and sparse systems of equations. In J. Herzberger, editor, Topics in Validated Computations, Studies in Computational Mathematics, pages 63 36, Amsterdam, 994. Elsevier. Proceedings of the IMACS-GAMM International Workshop on Validated Computations, University of Oldenburg. 8

19 [48] S. M. Rump. Verification methods: Rigorous results using floating-point arithmetic. Acta Numer., 9: , 200. [49] S. P. Shary. A new technique in systems analysis under interval uncertainty and ambiguity. Reliab. Comput., 8(5):32 48, [50] I. Skalna. A method for outer interval solution of systems of linear equations depending linearly on interval parameters. Reliab. Comput., 2(2):07 20, [5] W. Zhu. Unsolvability of some optimization problems. Appl. Math. Comput., 74(2):92 926,

20 SIAM J. NUMER. ANAL. Vol. 52, No., pp c 204 Society for Industrial and Applied Mathematics Downloaded 02/03/4 to Redistribution subject to SIAM license or copyright; see NEW OPERATOR AND METHOD FOR SOLVING REAL PRECONDITIONED INTERVAL LINEAR EQUATIONS MILAN HLADÍK Abstract. We deal with real preconditioned interval linear systems of equations. We present a new operator, which generalizes the interval Gauss Seidel operator. Also, based on the new operator and properties of well-known methods, we propose a new algorithm, called the magnitude method. We illustrate by numerical examples that our approach outperforms some classical methods with respect to both time and sharpness of enclosures. Key words. linear interval systems, solution set, interval matrix AMS subject classifications. 65G40, 5A06 DOI. 0.37/ Introduction. We consider a real system of linear equation with coefficients varying inside given intervals, and we want to find a guaranteed enclosure for all emerging solutions. Since determining the best enclosure to the solution set is an NPhard problem [2], the approaches to calculating it may be computationally expensive [6, 5, 20] in the worst case. That is why the field was driven to develop cheap methods for enclosing the solution set, not necessarily optimally. There are many methods known; see, e.g., [, 2, 3, 4, 8, 9, 0, 7, 9]. Extensions to parametric interval systems were studied in [5, 3, 9], among others, and quantified solutions were investigated, e.g., in [2, 3, 2]. We will use the following interval notation. An interval matrix A is defined as A := [A, A] ={A R m n ; A A A}, where A, A R m n are given. The center and radius of A are, respectively, defined as A c := (A + A), 2 AΔ := (A A). 2 The set of all m-by-n interval matrices is denoted by IR m n. Interval vectors and intervals can be regarded as special interval matrices of sizes m-by- and -by-, respectively. For a definition of interval arithmetic, see, e.g., [8, 9]. Extended interval arithmetic with improper intervals of type [a, a], a>a, was discussed, e.g., in [7, 2]. We will use improper intervals only for the simplicity of exposition of interval expressions. For example, a +[b, b], where b>0, is a shorthand for the interval [a + b, a b]. The magnitude of an A IR m n is defined as mag(a) :=max( A, A ), where max( ) is understood entrywise. The comparison matrix of A IR n n is the matrix A R n n with entries A ii := min{ a ; a a ii }, i =,...,n, A ij := mag(a ij ), i j. Received by the editors March 25, 203; accepted for publication (in revised form) October 23, 203; published electronically January 28, 204. This work was supported by CE-ITI (GAP202/2/G06) of the Czech Science Foundation. Faculty of Mathematics and Physics, Department of Applied Mathematics, Charles University, 8 00 Prague, Czech Republic (milan.hladik@matfyz.cz). 94 Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

21 OPERATOR AND METHOD FOR INTERVAL LINEAR EQUATIONS 95 Let A IR n n, b IR n, and consider a set of systems of linear equations Downloaded 02/03/4 to Redistribution subject to SIAM license or copyright; see Ax = b, A A, b b, commonly called a system of interval linear equations. The corresponding solution set is defined as Σ := {x R n ; A A b b : Ax = b}. The aim is to compute as tight as possible an enclosure of Σ by an interval vector x IR n, meaning that Σ x. ByΣ := Σ we denote the interval hull of Σ, i.e., the smallest interval enclosure of Σ with respect to inclusion. Thus, enclosing Σ or Σ by interval vectors is the same objective. Throughout the paper, we assume that A c = I n ; that is, the midpoint of A is the identity matrix. This assumption is not without loss of generality, but most of the solvers utilize a preconditioning that results in interval linear equations A x = b, A RA, b Rb, where R is the numerically computed inverse of A c. Thus, the midpoint of RA is nearly the identity matrix. To be numerically safe, we then relax the interval system to A x = b, A [I n mag(i n RA),I n +mag(i n RA)], b Rb. Even though preconditioning causes an enlargement of the solution set, it is easier to handle. Since we do not miss any old solution, any enclosure to the preconditioned system is a valid enclosure for the original one as well. The assumption A c = I n has many consequences. The solution set of such an interval linear system is bounded (i.e., A contains no singular matrix) if and only if ρ(a Δ ) <, where ρ(a Δ ) stands for the spectral radius of A Δ ; the sufficiency follows from [4] and the necessity from [9, Prop. 4..7]. So in the rest of the paper we assume that this is satisfied. Another nice property of the interval system in question is that the interval hull of the solution set can be determined exactly (up to numerical accuracy) by calling the Hansen Bliek Rohn method [2, 6]. Ning and Kearfott [] (see also [0]) proposed an alternative formula for computing Σ. We state it below and use the following notation: u := A mag(b), d i := ( A ) ii, i =,...,n, α i := a ii /d i, i =,...,n. Notice also that the comparison matrix A can now be expressed as A = I n A Δ. Theorem. (Ning Kearfott, 997). We have (.) Σ i = b i +(u i /d i mag(b i ))[, ], i =,...,n. a ii + α i [, ] The disadvantage of the Hansen Bliek Rohn method is that we have to safely compute the inverse of A. There are other procedures to compute a verified enclosure of Σ; see [8, 9]. They are usually faster, on account of tightness of the Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

22 96 MILAN HLADÍK Downloaded 02/03/4 to Redistribution subject to SIAM license or copyright; see resulting enclosures. We briefly recall three of them, the well-known interval Jacobi, Gauss Seidel, and Krawczyk iteration methods. They must be initiated by a starting enclosure x 0 Σ. In our examples we employ the formula from [8], which in our case draws Σ mag(b) (.2) A Δ [, ] n, provided A Δ <. Iteration methods can usually be expressed by an operator P : IR n IR n having the property (.3) (x Σ) P(x). Thus, the operator removes no solution included in x. As a consequence, if P(x) =, then x contains no solution. Basically, iterations then can have the plain form x P(x), or the form with intersections x P(x) x. The Krawczyk method is based on the operator x b +(I n A)x. Denote by D the interval diagonal matrix, whose diagonal is the same as that of A, and A is used for the interval matrix A with zero diagonal. The interval Jacobi operator reads x D (b A x), where D is the diagonal matrix with entries /d,...,/d nn. The interval Gauss Seidel operator proceeds by evaluating the above expression row by row and using the already updated entries of x in the subsequent rows. That is, the interval Gauss Seidel iteration reads for i =ton do : x i := a ii b i j i a ij x j. In the remainder of the paper, we will be concerned with the Krawczyk method, and with the Gauss Seidel iterations in particular. By x GS and x K we denote the limit enclosures computed by the interval Gauss Seidel and Krawczyk methods, respectively. The theorem below is adapted from [9] and gives an explicit formula for the enclosures. Theorem.2. We have Moreover, (.4) x GS = D (b +mag(a )u[, ]), x K = b + A Δ u[, ]. u = mag(σ) =mag(x GS ) = mag(x K ). Property (.4), not stressed enough in the literature, shows an interesting relation between the mentioned methods. In each coordinate, all corresponding enclosures have one endpoint in common (that one with the larger absolute value). Thus, the enclosures differ from one side only (but the difference may be large). Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

23 OPERATOR AND METHOD FOR INTERVAL LINEAR EQUATIONS 97 Downloaded 02/03/4 to Redistribution subject to SIAM license or copyright; see 2. New interval operator. Theorem 2.. Let Σ x IR n.then Σ i b i j i a ijx j +[γ i, γ i ]u i (2.) a ii + γ i [, ] for every γ i [0,α i ] and i =,...,n. Proof. Let i {,...,n}. First, we prove the statement for γ i = α i. By Theorem., Σ i = b i +(u i /d i mag(b i ))[, ]. a ii + α i [, ] The denominator is the same as in (2.), and it is a positive interval. Thus, it is sufficient to compare the numerators only. We have b i +(u i /d i mag(b i ))[, ] = b i +(u i /d i ( A u) i )[, ] = b i + a Δ iju j ( a ii /d i )u i [, ] j i b i + a Δ ij mag(x j ) γ i u i [, ] j i = b i j i a ij x j +[γ i, γ i ]u i. For γ i = 0, equation (2.) reduces to the interval Gauss-Seidel operator. Now, we suppose that 0 <γ i <α i. Defining v i := b i j i aδ ij u j[, ], we have to show the inclusion v i + α i u i [, ] a ii + α i [, ] v i + γ i u i [, ] a ii + γ i [, ]. We show it by comparing the left endpoints only; the right endpoints are compared accordingly. We distinguish three cases: () Let v i + γ i u i 0. Then we want to show that which is simplified to or v i + γ i u i a ii + γ i v i + α iu i a ii + α i, v i (α i γ i ) a ii u i (α i γ i ) v i a ii u i. This is always true since for any x Σ and the corresponding A A and b b we have 0=(Ax b) i = a ij x j b i a ij u j b i = a ii u i v i. j= j= Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

24 98 MILAN HLADÍK Downloaded 02/03/4 to Redistribution subject to SIAM license or copyright; see (2) Let v i + γ i u i < 0andv i + α i u i 0. Then the statement is obvious. (3) Let v i + α i u i < 0. Then we want to show that Simplifying, we obtain or v i + γ i u i a ii γ i v i + α iu i a ii α i. v i (α i γ i ) a ii u i (α i γ i ) v i a ii u i, which holds true. The proposed operator is based on the inclusion (2.) and reads (2.2) for i =ton do : x i := b i j i a ijx j +[γ i, γ i ]u i. a ii + γ i [, ] This is the sequential form as in the Gauss Seidel iterations. We can also formulate a Jacobi-like form with independent evaluations for each coordinate. Denote by D the diagonal matrix with entries (a + γ [, ]),...,(a nn + γ n [, ]),andby b the interval vector with entries b +[γ, γ ]u,...,b n +[γ n, γ n ]u n. Then the second version of the operator becomes x D (b A x), where A is the same as in the Jacobi operator. For simplicity of exposition, we will work with the former formulation of the operator. Obviously, for γ = 0 we get the interval Gauss Seidel operator, so our operator can be viewed as its generalization. The proof also shows that the best choice for γ is γ = α. In order to make the operator applicable, we have to compute u and d or some lower bounds of them. Notice that replacing the exact values of u and d by lower bounds causes a slight overestimation, and one gets a superset in the right-hand sides of (2.) and (2.2). Thus, the operator will still satisfy the fundamental property (.3). The tighter the bounds on u and d, the better; however, if we spend too much time calculating almost exact u and d, then it makes no sense to use the operator when we can call the Ning Kearfott formula directly. So, it is preferable to derive cheap and possibly tight lower bounds on u and d. We suggest the following ones. Proposition 2.2. We have u mag(b)+a Δ (mag(b)+a Δ mag(b))), d i d i := a ii /( ((A Δ ) 2 ) ii ), i =,...,n. Proof. The first part follows from ( ) u = A mag(b) =(I n A Δ ) mag(b) = (A Δ ) k mag(b) (I n + A Δ +(A Δ ) 2 ) mag(b) = mag(b)+a Δ (mag(b)+a Δ mag(b)). k=0 Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

25 OPERATOR AND METHOD FOR INTERVAL LINEAR EQUATIONS 99 The second part follows from Downloaded 02/03/4 to Redistribution subject to SIAM license or copyright; see whence d i = ((A Δ ) k ) ii k=0 ( d =diag( A )=diag (A Δ ) ), k a ii +((A Δ ) 2 ) ii ( + a Δ ii +((AΔ ) 2 ) ii +((A Δ ) 2 ) ii a Δ ii +((AΔ ) 2 ) 2 ii + ) = a ii +(+a Δ ii )((A Δ ) 2 ) ii (+((A Δ ) 2 ) ii +((A Δ ) 2 ) 2 ii + ) = a ii + a ii ((A Δ ) 2 a ii ) ii ((A Δ = ) 2 ) ii ((A Δ. ) 2 ) ii Notice that both bounds require computational time O(n 2 ). In particular, the diagonal of (A Δ ) 2 is computable in square time, but the exact diagonal of (A Δ ) 3 would be too costly. The following result shows that the above estimation of d is tight enough to ensure that γ 0. Notice that this would not be satisfied in general if we used the simpler estimation d diag(a +(A Δ ) 2 ). Proposition 2.3. We have γ i := a ii /d i 0, i =,...,n. Proof. We can write k=0 γ i = a ii /d i = a ii ((AΔ ) 2 ) ii a ii a ii (aδ ii )2 +a Δ ii = a Δ ii ( aδ ii )= Comparison to the interval Gauss Seidel method. Since our operator is a generalization of the interval Gauss Seidel iteration, it is natural to compare them. Let x be an enclosure of Σ, let i {,...,n}, and denote by û alower bound estimation on u. We compare the results of ours and the interval Gauss Seidel operators, that is, b i j i a ijx j +[γ i, γ i ]û i b i j i and a ijx j. a ii + γ i [, ] a ii If γ i = 0, then both intervals coincide, so let us assume that γ i > 0. Denote v i := b i j i a ijx j. We compare the left endpoints of the intervals v i +[γ i, γ i ]û i a ii + γ i [, ] and v i a ii ; the right endpoints are compared accordingly. We distinguish three cases: () Let v i 0. Then we want to show that This is simplified to v i a ii v i + γ iû i a ii + γ i. v i γ i a ii û i γ i Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

26 200 MILAN HLADÍK or Downloaded 02/03/4 to Redistribution subject to SIAM license or copyright; see v i a ii û i. If û i = u i or û i is not far from u i, then the inequality holds true. (2) Let v i < 0andv i + γ i û i 0. Then the inequality is obviously satisfied. (3) Let v i + γ i û i < 0. Then we want to show that This is simplified to or v i a ii v i + γ i û i a ii γ i. v i γ i a ii û i γ i v i a ii û i. This is true, provided that both û i and x are sufficiently good approximations of u i and Σ, respectively. The above discussion indicates that our operator with γ i > 0 is effective only if x is sufficiently tight and the reduction of the enclosure is valid from the smaller side (in the absolute value sense) only. Since a ij, i j, are symmetric intervals, the reduction in the smaller sides of x i s makes no improvement in the next iterations. The only influence is by the size of mag(x) since a ij x j = a ij mag(x) j. j i j i Therefore, the following incorporation of our operator seems the most effective: Compute x Σ by the interval Gauss Seidel method, and then call one iteration of our operator. Example 2.4. Let [8, 0] [3, 5] [8, 0] [3, 5] A = [5, 7] [0, 2] [6, 8], b = [6, 8], [4, 6] [7, 9] [5, 7] [5, 7] and consider the interval linear system Ax = b, A A, b b, preconditioned by the numerically computed inverse of A c. As in the subsequent examples, the computations were done in MATLAB. Interval arithmetics and some basic interval functions were provided by the interval toolbox INTLAB v6 [8]. The resulting intervals displayed below are exact within outward rounding to four digits. We start with the initial enclosure computed by (.2), x 0 =.8065([, ], [, ], [, ]) T. The interval Gauss Seidel method then is terminated after four iterations, yielding the enclosure x =([.2820, 0.074], [0.847,.564], [.0822, ]) T. Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

27 OPERATOR AND METHOD FOR INTERVAL LINEAR EQUATIONS 20 This is not yet equal to the limit enclosure Downloaded 02/03/4 to Redistribution subject to SIAM license or copyright; see x GS =([.283, 0.067], [0.849,.5637], [.082, ]) T, due to the finite number of iterations; we terminate the iterations when the absolute improvement in each coordinate from both above and from below is less than min i,j (A Δ ij)/00 = 0.0. By Proposition 2.2, we obtain the following lower bounds: whence we calculate u (.633,.4367, ) T, d (.2343,.2536,.2030) T, γ := (0.0387, , ) T. These values are quite conservative since the optimal values would be for γ = α, where α =(0.0632, , ) T. Nevertheless, the computed γ is sufficient to reduce the overestimation of x. One iteration of our operator results in the tighter enclosure x 2 =([.2820, ], [0.226,.564], [.0822, ]) T. For completeness, notice that the interval hull of the preconditioned system is Σ =([.283, ], [0.257,.5637], [.082, 0.044]) T. 3. Magnitude method. Property (.4) and the analysis at the end of section 2. motivate us to compute enclosure to Σ along the following lines. First, we compute the magnitude of Σ, thatis,u = A mag(b), and then we apply one iteration of the presented operator on the initial box x =[ u, u], producing b i j i a iju j +[γ i, γ i ]u i, i =,...,n. a ii + γ i [, ] Herein, the lower bound on d is computed by Proposition 2.2. In view of the proof of Theorem 2., we can express the result equivalently as (.), but in that formula, an upper bound on d is required, so we do not consider it here. Instead, we reformulate it in the slightly simpler form omitting improper intervals: b i +( j i aδ ij u j γ i u i )[, ], i =,...,n. a ii + γ i [, ] Algorithm 3. gives a detailed and numerically reliable description of the method. Algorithm 3... Compute u, an enclosure for the solution of A u = mag(b). 2. Calculate d, a lower bound on d by Proposition Evaluate x i := b i +( j i aδ ij u j γ i u i )[, ], i =,...,n, a ii + γ i [, ] where γ i := a ii /d i. Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

28 202 MILAN HLADÍK Downloaded 02/03/4 to Redistribution subject to SIAM license or copyright; see Properties. First, notice that the computations of u and d in steps and 2 are independent, so they may be parallelized. Now, let us compare the magnitude method with the Hansen Bliek Rohn and the interval Gauss Seidel method. The propositions below show that the magnitude method is superior to the interval Gauss Seidel method, and it gives the best possible enclosure as long as u and d are determined exactly. Since u is computed tightly, the possible deficiency is caused only by an underestimation of d. Proposition 3.2. If u and d are calculated exactly, then x = Σ. Proof. This follows from the proof of Theorem 2.. Proposition 3.3. We have x x GS.Ifγ =0, then equality holds. Proof. Let i {,...,n}, and without loss of generality assume that Σ c i 0. Then x i = b i j i a iju j +[γ i, γ i ]u i, a ii + γ i [, ] = b i (A [ u, u]) i = b i a ii x GS i j i a iju j a ii. Denoting v i := b i j i a iju j, we can rewrite the above expressions as x i = v i +[γ i, γ i ]u i a ii + γ i [, ], x GS i = v i a ii. By the assumption, x i = xgs i = u i, so we have to compare the left endpoints of x i and xgs only. We distinguish three cases: () Let v i 0. Then we want to show that This is simplified to v i a ii v i + γ iu i a ii + γ i. v i γ i a ii u i γ i. If γ i = 0, then the above inequality holds as equation; otherwise for any γ i > 0itis true as well. (2) Let v i < 0andv i + γ i u i 0. Then the statement is obvious. (3) Let v i + γ i u i < 0. Then we want to show that This is simplified to v i a ii v i + γ iu i a ii γ i. v i γ i a ii u i γ i, which is true for any γ i 0, too. Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

29 OPERATOR AND METHOD FOR INTERVAL LINEAR EQUATIONS 203 x 2 Downloaded 02/03/4 to Redistribution subject to SIAM license or copyright; see Fig. 3.. Example 3.4: The solution set in gray, the preconditioned system in light gray, and three enclosures for verifylss, the interval Gauss Seidel, and the magnitude method (from largest to smallest) Numerical examples. Example 3.4. Consider the interval linear system Ax = b, A A, b b, with ( ) ( ) [2, 4] [8, 0] [4, 6] A =, b =. [2, 4] [4, 6] [8, 0] Figure 3. depicts the solution set to Ax = b, A A, b b, indarkergrayand the preconditioned system by (A c ) in light gray. We compare three methods for enclosing the solution set. The function verifylss from the package INTLAB [8] yields the enclosure 0 2 x =([ , 0.838], [.9279, 0.072]) T. The interval Gauss Seidel method initiated with the enclosure x 0 =([ 5, 5], [ 5, 5]) T gives in four iterations a tighter enclosure than verifylss, x 2 =([ , ], [.9093, 0.380]) T, but it requires almost double computational time. In contrast, our magnitude method produces a bit tighter enclosure, x =([ , ], [.909, 0.374]) T, but with less computational effort than the other methods. The enclosure is also very close to the optimal one (for the preconditioned system), Σ =([ , ], [.909, 0.47]) T. Enclosures x, x 2, x are illustrated in Figure 3. in a nested way. In the example below, we present a limited computational study. Example 3.5. We considered randomly generated examples for various dimensions and interval radii. The entries of A c and b c were generated randomly in [ 0, 0] with uniform distribution. All radii of A and b were equal to the parameter δ>0. x Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

30 204 MILAN HLADÍK Table 3. Example 3.5: Computational time for randomly generated data. Downloaded 02/03/4 to Redistribution subject to SIAM license or copyright; see n δ verifylss Gauss Seidel Magnitude Magnitude (γ = 0) Table 3.2 Example 3.5: Tightness of enclosures for randomly generated data. n δ verifylss Gauss Seidel Magnitude Magnitude (γ = 0) The computations were carried out in MATLAB (R200b) on a sixprocessor machine AMD Phenom(tm) II X6 090T Processor, CPU 800 MHz, with 5579 MB RAM. We compared four methods with respect to computational time and tightness of resulting enclosures, namely, the verifylss function from INTLAB, the interval Gauss Seidel method, the proposed magnitude method (Algorithm 3.), and eventually the magnitude method with γ = 0. The last one yields the limit Gauss Seidel enclosure, and it is faster than the magnitude method since we need not compute a lower bound on d. Table 3. shows the running times in seconds, and Table 3.2 shows the tightness for the same data. Each record is an average of 00 runs. The tightness was measured by the sum of the resulting interval radii with respect to the optimal interval hull Σ Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

31 OPERATOR AND METHOD FOR INTERVAL LINEAR EQUATIONS 205 computed by the Ning Kearfott formula (.). Precisely, we display Downloaded 02/03/4 to Redistribution subject to SIAM license or copyright; see n i= xδ i n i= ΣΔ i where x is the calculated enclosure. Thus, the closer to, the sharper the enclosure. The results of our experiments show that the magnitude method with γ =0saves some time (about 0% to 20%), but the loss in tightness may be larger. Compared to the interval Gauss Seidel method, the magnitude method wins significantly both in time and tightness. Compared to verifylss, our approach produces tighter enclosures. Provided interval entries of the equation system are wide, the magnitude method is also cheaper; for narrow enough intervals, the situation is changed and verifylss needs less computational effort. For both variants of the magnitude method, we used verifylss for computing a verified enclosure of u = A mag(b) (step of Algorithm 3.). So it might seem curious that (for wide input intervals) verifylss beats itself. 4. Conclusion. We have proposed a new operator for tightening solution set enclosures of interval linear equations. Based on this operator and a property of limit enclosures of classical methods, we came up with a new algorithm, called the magnitude method. It provably always outperforms the interval Gauss Seidel method with respect to the quality of approximation. Numerical experiments indicate that it is efficient in both computational time and tightness of enclosures, particularly for wide interval entries. In future research, we would like to extend our approach to parametric interval systems. Also, overcoming the assumption A c = I n and considering nonpreconditioned systems is a challenging problem. Very recently, a new version of INTLAB was released (unfortunately, no longer free of charge), so numerical studies utilizing enhanced INTLAB functions would be of interest, too. REFERENCES [] O. Beaumont, Solving interval linear systems with linear programming techniques, Linear Algebra Appl., 28 (998), pp [2] M. Fiedler, J. Nedoma, J. Ramík, J. Rohn, and K. Zimmermann, Linear Optimization Problems with Inexact Data, Springer, New York, [3] E. Hansen and S. Sengupta, Bounding solutions of systems of equations using interval analysis, BIT, 2 (98), pp [4] E. R. Hansen, Bounding the solution of interval linear equations, SIAMJ.Numer.Anal.,29 (992), pp [5] M. Hladík, Enclosures for the solution set of parametric interval linear systems, Int. J. Appl. Math. Comput. Sci., 22 (202), pp [6] C. Jansson, Calculation of exact bounds for the solution set of linear interval systems, Linear Algebra Appl., 25 (997), pp [7] E. Kaucher, Interval analysis in the extended interval space IR, in Fundamentals of Numerical Computation (Computer-Oriented Numerical Analysis), Comput. Suppl. 2, Springer, Vienna, 980, pp [8] R. E. Moore, R. Baker Kearfott, and M. J. Cloud, Introduction to Interval Analysis, SIAM, Philadelphia, [9] A. Neumaier, Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, UK, 990. [0] A. Neumaier, A simple derivation of the Hansen-Bliek-Rohn-Ning-Kearfott enclosure for linear interval equations, Reliab. Comput., 5 (999), pp [] S. Ning and R. B. Kearfott, A comparison of some methods for solving linear interval equations, SIAM J. Numer. Anal., 34 (997), pp , Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

32 206 MILAN HLADÍK Downloaded 02/03/4 to Redistribution subject to SIAM license or copyright; see [2] E. D. Popova, Explicit description of AE solution sets for parametric linear systems, SIAM J. Matrix Anal. Appl., 33 (202), pp [3] E. D. Popova and M. Hladík, Outer enclosures to the parametric AE solution set, Soft Comput., 7 (203), pp [4] G. Rex and J. Rohn, Sufficient conditions for regularity and singularity of interval matrices, SIAM J. Matrix Anal. Appl., 20 (998), pp [5] J. Rohn, Systems of linear interval equations, Linear Algebra Appl., 26 (989), pp [6] J. Rohn, Cheap and tight bounds: The recent result by E. Hansen can be made more efficient, Interval Comput., 993 (993), pp [7] J. Rohn and G. Rex, Enclosing solutions of linear equations, SIAM J. Numer. Anal., 35 (998), pp [8] S. M. Rump, INTLAB INTerval LABoratory, in Developments in Reliable Computing, T. Csendes, ed., Kluwer Academic Publishers, Dordrecht, The Netherlands, 999, pp [9] S. M. Rump, Verification methods: Rigorous results using floating-point arithmetic, ActaNumer., 9 (200), pp [20] S. P. Shary, On optimal solution of interval linear equations, SIAM J. Numer. Anal., 32 (995), pp [2] S. P. Shary, A new technique in systems analysis under interval uncertainty and ambiguity, Reliab. Comput., 8 (2002), pp Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

33 Reliable Computing (2007) 3: DOI: 0.007/s x Springer 2007 Solution Set Characterization of Linear Interval Systems with a Specific Dependence Structure MILAN HLADÍK Charles University, Faculty of Mathematics and Physics, Department of Applied Mathematics, Malostranské nám. 25, 8 00, Prague, Czech Republic, milan.hladik@matfyz.cz (Received: 29 June 2006; accepted: December 2006) Abstract. This is a contribution to solvability of linear interval equations and inequalities. In interval analysis we usually suppose that values from different intervals are mutually independent. This assumption can be sometimes too restrictive. In this article we derive extensions of Oettli-Prager theorem and Gerlach theorem for the case where there is a simple dependence structure between coefficients of an interval system. The dependence is given by equality of two submatrices of the constraint matrix.. Introduction Coefficients and right-hand sides of systems of linear equalities and inequalities are rarely known exactly. In interval analysis we suppose that these values vary in some real intervals independently. But in practical applications (for instance electrical circuit problem [5], [7]) they are sometimes related. General case of parametric dependences has been considered e.g. in [6], [9], where various algorithms for finding inner and outer solutions were proposed. Linear interval systems with more specific dependencies were studied e.g. in [], [2]. There were derived basic characteristics (shape, enclosures, etc.), especially for cases where the constraint matrix is supposed to be symmetric or skew-symmetric. But any explicit condition such as Oettli-Prager theorem [8] (for linear interval equations) or Gerlach theorem [4] (for linear interval inequalities) has never appeared. In this paper we focus on weak solvability of linear interval systems with a simple dependence structure and derive explicit (generally nonlinear) conditions for such a solvability. Let us introduce some notation. The ith row of a matrix A is denoted by A i,,the jth column by A, j. The vector e =(,,) T is the vector of all ones. An interval matrix is defined as A I =[A, A] ={A R m n A A A}, where A A are fixed matrices. By A c 2 (A + A), AΔ (A A) 2

34 362 MILAN HLADÍK we denote the midpoint and radius of A I, respectively. The interval matrix addition and subtraction is defined as follows A I + B I A I B I =[A + B, A + B], =[A B, A B]. A vector x R n is called a weak solution of a linear interval system A I x = b I, if Ax = b holds for some A A I, b b I. Analogously we define a term weak solution for other types of interval systems (cf. [3]). 2. Generalization of Oettli-Prager Theorem In this section we generalize the Oettli-Prager [8] characterization of weak solutions of linear interval equations to the case where there is a specific dependence between some coefficients of the constraint matrix. LEMMA 2.. Given s, s 2, p i, q i R,i=,,n. Let us denote the function ƒ(u, u 2 ) s u + s 2 u 2 + n p i u + q i u 2. Then the problem i= min {ƒ(u, u 2 ); (u, u 2 ) R 2 } (2.) has an optimal solution (equal to zero) if and only if holds. q i s 2, i= q k p i q i p k q k s p k s 2 k =,,n i= Proof. The objective function ƒ(u, u 2 ) is positive homogeneous and hence the problem (2. ) has an optimal solution iff ƒ(u, u 2 ) 0 holds for u = ±, u 2 R and for u =0,u 2 = ±. Let us consider the following cases: (i) Letu =. Then the function ƒ(, u 2 )=s + s 2 u 2 + n p i + q i u 2 of one i= parameter represents a broken line. It is sufficient to check nonnegativity of this function in the breaks and nonnegativity of the limits in ±. The breaks are p k, q k q k = 0,k =,,n. Hence we derive k =,,n, q k = 0: s p k q k s 2 + p i p k q i q 0. (2.2) k i=

35 SOLUTION SET CHARACTERIZATION OF LINEAR INTERVAL SYSTEMS To be lim u 2 ƒ(, u 2) 0, it must the inequality q i s 2 hold and to be i= lim u 2 ƒ(, u 2) 0, it must q i s 2 hold. We obtain next condition i= q i s 2. (2.3) i= (ii) Letu =. Then analogously as in first paragraph we obtain for the function ƒ(, u 2 )= s + s 2 u 2 + n p i + q i u 2 the condition i= k =,,n, q k = 0: s + p k q k s 2 + i= All the conditions (2. 2), (2. 4) can we written in one n p i + p k q i q 0, (2.4) k q i s 2. (2.5) i= k =,,n : p i q k p k q i s q k p k s 2. (2.6) i= The assumption q k = 0 is not necessary, for in the case q k = 0 the inequality (2. 6) is included in (2. 3). (iii) Letu = 0. Then the condition ƒ(0, ±) 0 is included in the condition (2. 3). THEOREM 2.. Let A I R m n, B I, C I R m h, b I, c I R m. Then for certain A A I, B B I, C C I, b b I, c c I vectors x, y R n, z R h form a solution of the system Ax + Bz = b, (2.7) Ay + Cz = c (2.8) if and only if they satisfy the following system of inequalities B Δ z y T + C Δ z x T + b Δ y T + c Δ x T A Δ x + B Δ z + b Δ r, (2.9) where r A c x B c z + b c, r 2 A c y C c z + c c. A Δ y + C Δ z + c Δ r 2, (2.0) +A Δ xy T yx T r y T r 2 x T, (2.)

36 364 MILAN HLADÍK Proof. Denote a I A I l,, b I Bl,, I c I Cl,, I β I bl I, γ I cl I. Consider the lth equations in systems (2. 7) (2. 8) and denote them by ax + bz = β, ay + cz = γ, (2.2) where a a I, b b I, c c I, β β I, γ γ I. Suppose that the vector a a I in demand has the ith component in the form a i ai c + α i ai Δ for α i,. The condition (2. 2) holds iff for a certain α, n relations a c x + a c y + α i ai Δ x i + b c z β c b Δ z β Δ, β c + b Δ z + β Δ, i= α i ai Δ y i + c c z γ c c Δ z γ Δ, γ c + c Δ z + γ Δ i= hold. Equivalently, iff the following problem { max 0 T α; α i ai Δ x i r + β, α i ai Δ x i r + β, i= α i ai Δ y i r 2 + β 2, i= i= i= } α i ai Δ y i r 2 + β 2, α e, α e has an optimal solution for r a c x b c z+β c, r 2 a c y c c z+γ c, β β Δ +b Δ z, β 2 γ Δ + c Δ z. From duality theory in linear programming this problem has an optimal solution iff the problem { min ( r + β )u +(r + β )u 2 +( r 2 + β 2 )u 3 +(r 2 + β 2 )u 4 + (v i + w i ); ai Δ x iu + ai Δ x iu 2 ai Δ y iu 3 + ai Δ y iu 4 + v i w i =0 i =,,n, } u, u 2, u 3, u 4, v i, w i 0 i =,,n has an optimal solution. After substitution ũ u 2 u, ũ 3 u 4 u 3 we can rewrite this problem as { min (r + β )ũ +2β u +(r 2 + β 2 )ũ 3 +2β 2 u 3 + (v i + w i ); i= ai Δ x i ũ + ai Δ y i ũ 3 + v i w i =0 i =,,n, } u ũ, u 3 ũ 3, u, u 3, v i, w i 0 i =,,n. For optimal v i, w i, u, u 3 we have u = ( ũ ) +, u 3 = ( ũ 3 ) +,andv i + w i = a Δ i x iũ + a Δ i y iũ 3 (since one of v i, w i is equal to zero). Hence the problem can be i=

37 SOLUTION SET CHARACTERIZATION OF LINEAR INTERVAL SYSTEMS reformulated as { min (r + β )ũ +2β ( ũ ) + +(r 2 + β 2 )ũ 3 +2β 2 ( ũ 3 ) + + ai Δ x iũ + ai Δ y iũ 3 ; ũ, ũ 3 R i= The positive part of real number p is equal to p + = ( p + p ) and the problem 2 comes in the form { min r ũ + β ũ + r 2 ũ 3 + β 2 ũ 3 + }. ai Δ x i ũ + ai Δ y i ũ 3 ; ũ, ũ 3 R i= }. (2.3) Now we use Lemma 2. with u replaced by ũ, u 2 replaced by ũ 3, n by n +2, and next s r, s 2 r 2, p i a Δ i x i (i =,,n), p n+ β, p n+2 0, q i a Δ i y i (i =,,n), q n+ 0, q n+2 β 2. Hence the problem (2. 3) has an optimum iff β y k + β 2 x k + ai Δ x i + β r, i= ai Δ y i + β 2 r 2, i= ai Δ y kx i x k y i y k r x k r 2 i= k =,,n holds, or, equivalently iff a Δ x + b Δ z + β Δ r, a Δ y + c Δ z + γ Δ r 2, b Δ z y T + β Δ y T + c Δ z x T + γ Δ x T + a Δ xy T yx T r y T r 2 x T holds. These inequalities represents the lth inequalities from systems (2. 9) (2. ), which proves the statement. In the case that x = y we immediately have the following corollary. COROLLARY 2.. Let A I R m n, B I, C I R m h, b I, c I R m. Then for certain A A I, B B I, C C I, b b I, c c I vectors x R n, z R h form a solution of the system Ax + Bz = b, (2.4) Ax + Cz = c (2.5)

38 366 MILAN HLADÍK if and only if they represent a weak solution of the linear interval system A I x + B I z = b I, (2.6) A I x + C I z = c I, (2.7) B I z C I z = b I c I. (2.8) 3. Generalization of Gerlach Theorem Now we generalize the Gerlach [4] characterization of weak solutions of linear interval inequalities to the case where there is a specific dependence between some coefficients of the constraint matrix. THEOREM 3.. Let A I R m n, B I, C I R m h, b I, c I R m. Then for certain A A I, B B I, C C I, b b I, c c I vectors x, y R n, z R h form a solution of the system Ax + Bz b, (3.) Ay + Cz c (3.2) if and only if they satisfy the system of inequalities A Δ x + B Δ z + b A c x + B c z, (3.3) A Δ y + C Δ z + c A c y + C c z, (3.4) r y k + r 2 x k + A Δ y k x x k y 0 k =,,n : x k y k < 0, (3.5) where r b A c x B c z + B Δ z, r 2 c A c y C c z + C Δ z. Proof. Denote a I A I l,, b I Bl,, I c I Cl,, I β I bl I, γ I cl I. Let us consider the lth inequalities in systems (3. ) (3. 2) and denote them by ax + bz β, ay + cz γ, (3.6) where a a I, b b I, c c I, β β I, γ γ I. Let us consider the vector in demand a a I in the form with the ith component a i a c i +α i a Δ i, α i,. The condition (3. 6) holds iff for a certain α, n we have a c x + a c y + α i ai Δ x i + b c z β + b Δ z, i= α i ai Δ y i + c c z γ + c Δ z i= or, equivalently, iff the following problem { } max 0 T α; α i ai Δ x i r, α i ai Δ y i r 2, α e, α e, i= i=

39 SOLUTION SET CHARACTERIZATION OF LINEAR INTERVAL SYSTEMS where r β a c x b c z+b Δ z, r 2 γ a c y c c z+c Δ z, has an optimal solution. From the duality theory in linear programming this problem has an optimal solution iff the same holds for the problem { min r u + r 2 u 2 + (v i + w i ); i= } ai Δ x i u + ai Δ y i u 2 + v i w i =0, u, u 2, v i, w i 0 i =,,n. For optimal solution v i, w i the relation v i + w i = ai Δx iu + ai Δy iu 2 holds. Hence we can the linear programming problem rewrite as { } min r u + r 2 u 2 + ai Δ x iu + ai Δ y iu 2 ; u, u 2 0. i= Since the objective function ƒ(u, u 2 )=r u +r 2 u 2 + n ai Δ x i u +ai Δ y i u 2 is positive i= homogeneous, it is sufficient (similarly as in the proof of Lemma 2.) to check its nonnegativity only for special points: (i) Ifu =0, u 2 =,thenƒ(0, ) 0 is equal to r 2 + a Δ y 0, which is the lth inequality from the system (3. 4). (ii) Letu =, u 2 0. The function ƒ(, u 2 ) represents a broken line with breaks in u 2 =0andinu 2 = x k 0, y y k = 0. For the first case the condition k ƒ(, 0) 0 is equal to r + a Δ x 0, which is the lth inequality from the system (3. 3). In the second case, for the breaks of the objective function ƒ(, u 2 ) we obtain the following inequality r + r 2 x k y k + ai Δ i= x i x k y k y i 0 k =,,n : xk y k 0, y k = 0. Since x k = x k, y k y the inequality is equal to (w.l.o.g. assume xk, y k = 0, for k otherwise we get redundant condition) r y k + r 2 x k + a Δ y k x x k y 0, k =,,n : x k y k < 0, which is the lth inequality from the system (3. 5). Note that the system (3. 5) from Theorem 3. is empty if x = y, orifx, y 0. Hence from Theorem 3. two corollaries directly follow. COROLLARY 3.. Let A I R m n, B I, C I R m h, b I, c I R m. Then for certain A A I, B B I, C C I, b b I, c c I vectors x R n, z R h form a solution of the system Ax + Bz b, Ax + Cz c

40 368 MILAN HLADÍK if and only if x is a weak solution of the interval system A I x + B I z b I, A I x + C I z c I. COROLLARY 3.2. Let A I R m n, B I, C I R m h, b I, c I R m. Then for certain A A I, B B I, C C I, b b I, c c I vectors x, y R n, z R h form a nonnegative solution of the system Ax + Bz b, Ay + Cz c if and only if x is a solution of the system Ax + Bz b, Ay + Cz c. Remark 3.. Contrary to the situation in common analysis, in interval analysis it is not generally possible to transform an interval system of equations A I x = b I to the interval system of inequalities A I x b I, A I x b I.However,ifthe interval system of inequalities is integrated with certain dependence structure, such a transformation is possible. The interval system A I x = b I is weakly solvable iff there exist A A I, b b I such that the system Ax + bx n+ 0, A( x)+b( x n+ ) 0, x n+ = is solvable. From Theorem 3. (with assignment y = x, y n+ = x n+ ) it follows the solvability of the system A Δ x + b Δ x n+ A c x + b c x n+, A Δ x + b Δ x n+ A c x b c x n+, ( A c x b c x n+ ) x k +(A c x + b c x n+ ) x k + A Δ 0 0 k =,,n +, equivalently, x n+ =, A Δ x + b Δ A c x b c, (3.7) which is the statement of Oettli-Prager theorem on solvability of A I x = b I. 4. Mixed Equalities and Inequalities In previous sections we studied dependence structure for linear interval equations and inequalities, respectively. Now we turn our attention to mixed linear interval equations and inequalities with a dependence structure.

41 SOLUTION SET CHARACTERIZATION OF LINEAR INTERVAL SYSTEMS THEOREM 4.. Let A I R m n, B I, C I R m h, b I, c I R m. Then for certain A A I, B B I, C C I, b b I, c c I vectors x, y R n, z R h form a solution of the system Ax + Bz = b, (4.) Ay + Cz c (4.2) if and only if they satisfy the system of inequalities A Δ x + B Δ z + b Δ r, (4.3) A Δ y + r 2 0, (4.4) r y T diag(sgn x)+r 2 x T +(b Δ + B Δ z ) y T + A Δ xy T yx T 0, (4.5) where r b c A c x B c z, r 2 c A c y C c z + C Δ z. Proof. Denote a I A I l,, b I Bl,, I c I Cl,, I β I bl I, γ I cl I. Let us consider the lth equality and inequality in the systems (4. ) and (4. 2) and denote them by ax + bz = β, ay + cz γ, (4.6) where a a I, b b I, c c I, β β I, γ γ I. Let the vector a a I in demand have its ith component in the form a i a c i + α i a Δ i,whereα i,. The condition (3. 6) holds iff for a certain α, n we have a c x + a c y + α i ai Δ x i + b c z β c b Δ z β Δ, β c + b Δ z + β Δ, i= α i ai Δ y i + c c z γ + c Δ z i= or, equivalently iff the following problem { max 0 T α; α i ai Δ x i r + β, i= α i ai Δ x i r + β, i= } α i ai Δ y i r 2, α e, α e i= has an optimal solution (for r β c a c x b c z, β β Δ + b Δ z, r 2 γ a c y c c z + c Δ z ). From the duality theory in linear programming this problem has an optimal solution iff the problem { min (r β )u +(r + β )u 2 + r 2 u 3 + (v i + w i ); i= ai Δ x i u + ai Δ x i u 2 + ai Δ y i u 3 + v i w i =0, } u, u 2, u 3, v i, w i 0 i =,,n

42 370 MILAN HLADÍK has an optimal solution. After substitution ũ u 2 u we can rewrite this problem as { min (r + β )ũ +2β u + r 2 u 3 + (v i + w i ); i= ai Δ x i ũ + ai Δ y i u 3 + v i w i =0, } u ũ, u, u 3, v i, w i 0 i =,,n. For optimal solution v i, w i, u it must v i + w i = ai Δ x i ũ + ai Δ y i u 3, u =( ũ ) + hold. Hence the problem is simplified to { min (r + β )ũ +2β ( ũ ) + + r 2 u 3 + ai Δ x i ũ + ai Δ y i u 3 ; i= } ũ R, u 3 0. Since the positive part of a real number p is equal to p + = ( p + p ), the linear 2 programming problem has the final form { min r ũ + r 2 u 3 + β ũ + i= } ai Δ x iũ + ai Δ y iu 3 ; ũ R, u 3 0. The objective function ƒ(u, u 2 )=r ũ + r 2 u 3 + β ũ ) + n ai Δx iũ + ai Δy iu 3 of i= this problem is positive homogeneous, thus it is sufficient (similar as in the proof of Lemma 2.) to check the nonnegativity only for some special points: (i) Ifũ = ±, u 3 =0,thenƒ(±, 0) 0 is equal to ± r + β + a Δ x 0, or equivalently β + a Δ x r, which is the lth inequality from the system (4. 3). (ii) Letu 3 =. The function ƒ(ũ, ) represents a broken line with breaks in ũ =0andinũ = y k 0, x x k = 0. For the first case the function ƒ(0, ) 0is k equal to r 2 + a Δ y 0, which is the lth inequality from the system (4. 4). In the second case, for each nonzero break of the objective function ƒ(ũ, ) we obtain inequality y k r + r 2 + β y k x k x k + ai Δ i= y k x i + y i x 0. k This inequality can be expressed (since for x k = 0 we get a redundant condition) as r y k sgn(x k )+r 2 x k + β y k + a Δ y k x x k y 0, k =,,n,

43 SOLUTION SET CHARACTERIZATION OF LINEAR INTERVAL SYSTEMS or in the vector form r y T diag(sgn x)+r 2 x T + β y T + a Δ xy T yx T 0, which corresponds to the lth inequality from the system (4. 5). Putting x = y we immediately have he following corollary. COROLLARY 4.. Let A I R m n, B I, C I R m h, b I, c I R m. Then for certain A A I, B B I, C C I, b b I, c c I vectors x R n, z R h form a solution of the system Ax + Bz = b, (4.7) Ax + Cz c (4.8) if and only if they are a weak solution of the interval system A I x + B I z = b I, (4.9) A I x + C I z c I, (4.0) C I z B I z c I b I. (4.) Proof. According to Theorem 4. vectors x R n, z R h form a solution of the system (4. 7) (4. 8) iff they satisfy the system A Δ x + B Δ z + b Δ r, A Δ x + r 2 0, r x T diag(sgn x)+r 2 x T +(b Δ + B Δ z ) x T + A Δ xx T xx T 0. From [3, Theorems 2.9 and 2.9] it follows that the first and second inequalities of this system are equivalent to (4. 9) and (4. 0), respectively. The third inequality can be rewritten as (b c A c x B c z) x T +(c A c x C c z + C Δ z ) x T +(b Δ + B Δ z ) x T 0. (4.2) If x = 0, then the statement holds. Assume that x = 0. Then the inequality (4. 2) can be simplified to (b c A c x B c z)+(c A c x C c z + C Δ z )+(b Δ + B Δ z ) 0, and consequently to C Δ z + B Δ z + c b C c z B c z. According to [3, Theorem 2.9] this inequality is equivalent to (4. ). THEOREM 4.2. Let A I R m n, B I i R m h, C I j R m h, b I i R m, c I j R m, i =,,p, j =,,q. Then for certain matrices A A I, B i B I i, C j C I j, b i b I i,

44 372 MILAN HLADÍK c j c I j,i=,,p, j =,,q vectors x R n, z R h form a solution of the system Ax + B i z = b i, i =,,p, (4.3) Ax + C j z c j, j =,,q (4.4) if and only if they are a weak solution of the interval system A I x + Bi I z = bi I, i =,,p (4.5) A I x + Cj I z cj I, j =,,q (4.6) ( B I i Bk I ) z = b I i bk I, i, k : i<k. (4.7) ( C I j Bi I ) z c I j bi I, i, j. (4.8) Proof. One implication is obvious. If for certain A A I, B i Bi I, C j Cj I, b i bi I, c j cj I vectors x R n, y R h satisfy the system (4. 3) (4. 4), then they represent a weak solution of the interval system (4. 5) (4. 8) as well. To prove the second implication, denote a I A I l,, bi i (Bi I ) l,, cj I (Cj I ) l,, βi I (bi I ) l, γj I (cj I) l, i =,,p, j =,,q. Let us consider the lth inequalities in systems (4. 3) (4. 4) and denote them by ax + b i z = β i, i =,,p, (4.9) ax + c j z γ j, j =,,q, (4.20) where a a I, b i bi I, c j cj I, β i βi I, γ j βj I. Denote r i a c x + bi c z βi c, i =,,p and s j a c x + cj cz γ j c, j =,,q. Vectors x, z satisfy the system (4. 9) (4. 20) iff there exists α a Δ x, a Δ x for which we have r i + α b Δ i z + β Δ i, s j + α cj Δ z + γj Δ, or equivalently i =,,p, j =,,q α a Δ x, α a Δ x, α r i b Δ i z + β Δ i, α r i b Δ i z β Δ α s j + c Δ j z + γ Δ j, i, i =,,p, i =,,p, j =,,q. Such a number α exists iff the following four conditions hold. (i) First condition: a Δ x r i + bi Δ z + β i Δ, a Δ x r i bi Δ z β i Δ, i =,,p, i =,,p,

45 SOLUTION SET CHARACTERIZATION OF LINEAR INTERVAL SYSTEMS or, equivalently r i b Δ i z + β Δ i, i =,,p. According to [3, Theorem 2.9] the first condition is equivalent to the condition that vectors x, z represent a weak solution of the interval equation a I x + b I i z = β I i, which corresponds to the lth equation in the system (4. 5). (ii) Second condition: a Δ x s j + c Δ j z + γ Δ j, j =,,q. According to [3, Theorem 2.9] the second condition is equivalent to the condition that vectors x, z represent a weak solution of the interval inequality a I x + c I j z γ I j, which corresponds to the lth inequality in the system (4. 6). (iii) Third condition: r k bk Δ z βk Δ r i + bi Δ z + βi Δ, i, k =,,p, i<k, r i bi Δ z βi Δ r k + bk Δ z + βk Δ, i, k =,,p, i<k, or, equivalently r i r j (bi Δ + bk Δ ) z + βi Δ + βk Δ, i, k =,,p, i<k. According to [3, Theorem 2.9] the third condition is equivalent to the condition that vectors x, z represent a weak solution of the interval equation (b I i bi k )z = β I i β I k, i, k =,,p, i<k, which corresponds to the lth equation in the system (4. 7). (iv) Fourth condition: r i b Δ i z β Δ i or, equivalently s j r i (b Δ i s j + c Δ j z + γ Δ j, + c Δ j ) z + β Δ i + γ Δ j, i =,,p, j =,,q, i =,,p, j =,,q. According to [3, Theorem 2.9] the fourth condition is equivalent to the condition that vectors x, z represent a weak solution of the interval inequality (c I j bi i )z γ I j β I i, i =,,p, j =,,q, which corresponds to the lth inequality in the system (4. 8).

46 374 MILAN HLADÍK References. Alefeld, G., Kreinovich, V., and Mayer, G.: On Symmetric Solution Sets, in: Herzberger, J. (ed.), Inclusion Methods for Nonlinear Problems. Proceedings of the International GAMM-Workshop, Munich and Oberschleissheim, 2000, Wien, Springer, Comput. Suppl. 6 (2003), pp Alefeld, G., Kreinovich, V., and Mayer, G.: The Shape of the Solution Set for Systems of Interval Linear Equations with Dependent Coefficients, Math. Nachr. 92 (998), pp Fiedler, M., Nedoma, J., Ramík, J., Rohn, J., and Zimmermann, K.: Linear Optimization Problems with Inexact Data, Springer-Verlag, New York, Gerlach, W.: Zur Lösung linearer Ungleichungssysteme bei Störung der rechten Seite und der Koeffizientenmatrix, Math. Operationsforsch. Stat., Ser. Optimization 2 (98), pp Kolev, L. V.: Interval Methods for Circuit Analysis, Word Scientific, Singapore, Kolev, L. V.: Solving Linear Systems Whose Elements Are Nonlinear Functions of Interval Parameters, Numerical Algorithms 37 (2004), pp Kolev, L. V. and Vladov, S. S.: Linear Circuit Tolerance Analysis via Systems of Linear Interval Equations, ISYNT 89 6th International Symposium on Networks, Systems and Signal Processing, June 28 July, Zagreb, Yugoslavia, 989, pp Oettli, W. and Prager, W.: Compatibility of Approximate Solution of Linear Equations with Given Error Bounds for Coefficients and Right-Hand Sides, Numer. Math. 6 (964), pp Popova, E.: On the Solution of Parameterised Linear Systems, in: Kraemer, W. and Wolff von Gudenberg, J. (eds), Scientific Computing, Validated Numerics, Interval Methods, Kluwer Academic Publishers, Boston, 200, pp

47 SIAM J. MATRIX ANAL. APPL. Vol. 30, No. 2, pp c 2008 Society for Industrial and Applied Mathematics DESCRIPTION OF SYMMETRIC AND SKEW SYMMETRIC SOLUTION SET MILAN HLADÍK Abstract. We consider a linear system Ax = b, where A is varying inside a given interval matrix A, and b is varying inside a given interval vector b. The solution set of such a system is described by the well-known Oettli Prager Theorem. But if we are restricted only on symmetric/skew symmetric matrices A A, the problem is much more complicated. So far, the symmetric/skew symmetric solution set description could be obtained only by a lengthy Fourier Motzkin elimination applied on each orthant. We present an explicit necessary and sufficient characterization of the symmetric and skew symmetric solution set by means of nonlinear inequalities. The number of the inequalities is, however, still exponential w.r.t. the problem dimension. Key words. linear interval systems, solution set, interval matrix, symmetric matrix AMS subject classifications. 65G40, 5A06, 5A57 DOI. 0.37/ Notation IR m n the set of all m-by-n interval matrices IR n the set of all n-dimensional interval vectors S interval hull of a set S R n, i.e., the smallest box [a,b ] [a n,b n ] that contains all the elements of S lex strict lexicographic ordering of vectors, i.e., u lex v if for some k we have u i = v i, i<k, and u k <v k lex lexicographic ordering of vectors, i.e., u lex v if u lex v or u = v v absolute value of a vector v, i.e., the vector with components v i = v i A i,. the ith row of a matrix A e k the kth basis vector (with convenient dimension), i.e., the kth column of the identity matrix r + positive part of a real number r, i.e., r + = max(0,r). Introduction. Real-life problems are often subject to uncertainties in data measurements. Such uncertainties can be dealt with by methods of interval analysis [] instead of exact values we compute with compact real intervals. An interval matrix is defined as A =[A, A] ={A R m n A A A}, where A A are fixed matrices (n-dimensional interval vectors can be regarded as interval matrices n-by-). By A c 2 (A + A), AΔ (A A) 2 we denote the midpoint and radius of A, respectively. Received by the editors January 23, 2007; accepted for publication (in revised form) by A. Frommer February 2, 2008; published electronically May 9, Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské nám. 25, 8 00, Prague, Czech Republic (milan.hladik@matfyz.cz). 509 Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

48 50 MILAN HLADÍK Let us consider a system of linear interval equations The solution set Ax = b. Σ {x R n Ax = b, A A, b b} is described by the well-known Oettli Prager condition [] x Σ A Δ x + b Δ A c x b c. In interval analysis, we usually suppose, that values vary in given intervals independently. But in some applications, dependencies can occur (cf. [5], [9]). Especially, we focus on some types of the matrix A. The symmetric solution set is defined as Σ sym {x R n Ax = b, A = A T,A A, b b}, and the skew symmetric solution set as Σ skew {x R n Ax = b, A = A T,A A, b b}. These sets have been exhaustively studied in recent years (see [2], [3], [4], [5], [6], and [7]). Applications involve Markov chains [8] and truss mechanics [0], for instance. Descriptions of Σ sym and Σ skew can be obtained by a Fourier Motzkin elimination applied on each of 2 n orthants. Contrary to Σ, the symmetric solution set Σ sym is not polyhedral, its shape is described by quadrics (see [3], [4], [5], and [6]), and it is not convex in general, even if intersected with an orthant. The paper is organized as follows. In section 2 we derive a solution set characterization for a system of linear interval equations, where specific dependences occur. As consequences, we obtain a description of the symmetric solution set Σ sym (section 3), and a description of the skew symmetric solution set Σ skew (section 4). The basic properties of Σ sym, which were mentioned above, simply follow from the proposed Theorem 3. in section 3 (illustrated by Figures 3. and 3.2). 2. Linear interval equations with particular dependences. This section provides a characterization of the linear interval system equipped with a certain dependency (Theorem 2.2); the matrix A occurs twice in the system in (2.3) and transposed in (2.4). We will see later in sections 3 and 4 that the description of the symmetric/skew symmetric solution set is a simple consequence of Theorem 2.2. Another reason for dealing with such a dependency is that similar relations (occurrence of a matrix and its transposition in a system) can appear in some applications, e.g., optimality conditions in linear programming. First we state an auxiliary result. Lemma 2.. Let a,b,d R m, a 2,b 2,d 2 R n, and C R m n. The function m (2.) f(u, v) (a ) T u +(b ) T u +(a 2 ) T v +(b 2 ) T v + c ij d 2 ju i + d i v j i= j= is nonnegative for all u R m and v R n iff it is nonnegative for all u, v satisfying at least one of the following conditions: (i) u i {0,d i } i =,...,m, and v j {0, d 2 j } j =,...,n; (ii) u i {0, d i } i =,...,m, and v j {0,d 2 j } j =,...,n; (iii) (u T,v T ) T = ±e k for some k {,...,m+ n}. Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

49 DESCRIPTION OF SYMMETRIC SOLUTION SET 5 Proof. One implication is obvious: If f(u, v) is nonnegative for all u R m and v R n, then it is nonnegative in particular points. The converse will be proven by induction w.r.t. the dimension m + n. If m + n =, then without loss of generality (w.l.o.g.) assume m =,n =0. The function f(u) =a u + b u is nonnegative for all real u iff it is nonnegative for u = ±, which is managed by the third condition of Lemma 2.. The induction step will be proven by contradiction. Let us assume that f(u, v) < 0 for some vectors u, v. (a) Suppose there are some vectors u, v such that f(u, v) < 0 and u i = 0 for some index i. Delete the ith component in u and denote the resulting vector by ũ. Replace b 2 by b 2, where b 2 j = b2 j + c ij d i, j =,...,n, and apply the induction hypothesis to ũ, v. Hence f(ũ,v ) 0 for some vectors ũ R m and v R n satisfying one of the conditions (i) (iii). Canonical embedding of ũ to the space R m yields the pair of vectors u R m and v R n such that f(u,v ) 0, and one of the conditions (i) (iii) is true. Thus, a contradiction. (b) Suppose there are some vectors u, v such that f(u, v) < 0 and v j = 0 for some index j. Here the assertion follows analogously to case (a). (c) Assume as the remaining case that no component of u, v is zero for all vectors u, v with f(u, v) < 0. First we show that d i 0 for every i =,...,m, and d2 j 0 for every j =,...,n. If w.l.o.g. d i = 0 for some i, then we have f(u, v) =f(u,...,u i, 0,u i+,...,u m,v)+f(0,...,0,u i, 0,...,0, 0) < 0. That is, one of the two summands is negative, which contradicts our assumption. Now, choose vectors ũ, ṽ with f(ũ, ṽ) < 0 such that the number of absolute values in (2.) that are zero is maximal. Define the graph G =(V,E), where the vertex set V consist of ũ i, i =,...,m, and ṽ j, j =,...,n. The edge set E contains such pairs {ũ i, ṽ j } for which d 2 jũi + d i ṽj = 0. We distinguish three cases and show that each of them contradicts some assumption.. The graph G is connected. Choose (ũ i, ṽ j ) E and define z ṽj d 2 j 0. Then ũ i = d i z, and ṽ j = d 2 j z. Due to the connectivity of G, we can extend this property by induction to all i, j: If(ũ i, ṽ j ) E and ũ i = d i z, then ṽ j = d 2 j z. If (ũ i, ṽ j ) E and ṽ j = d 2 j z, then ũ i = d i z. Hence (2.2) ũ i = d i z i =,...,m, ṽ j = d 2 jz j =,...,n. Define u z ũ, v z ṽ. Vectors u,v satisfy the first or the second condition of Lemma 2. (depending on the sign of z ), but f(u,v )= z f(u, v) < 0. Thus, a contradiction. 2. The graph G is not connected and E. We will construct vectors u,v with f(u,v ) < 0 and at least one component of u or v to be zero, which contradicts the assumption of case (c). Take a connected component G =(V,E )ofg such that E. Then the property (2.2) holds when restricted on G : ũ i = d i z i :ũ i V, ṽ j = d 2 jz j :ṽ j V. Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

50 52 MILAN HLADÍK Consider the function g(z) f(u(z),v(z)) as a function of variable z, where u i (z) = { d i z ũ i V, ũ i otherwise, v j (z) = { d 2 j z ṽ j V, ṽ j otherwise. Then g(z) is a piecewise linear function (broken line) on R. Moreover, it is linear on a neighborhood N(z )ofz, that is, g(z) =pz + q, z N(z ) for some p, q R. W.l.o.g. assume that z > 0 and consider two possibilities. Let g(z) be nondecreasing on N(z ). In this case g(z) is nondecreasing on the interval [0,z ], since otherwise there is a break point in (0,z ) contradicting our assumption on the maximal number of zero absolute values. From g(0) g(z )= f(ũ, ṽ) < 0wegetg(0) = f(u(0),v(0)) < 0 with u(0) i = 0 for all indices i such that ũ i V (at least one exists due to E ). This contradicts the assumption of case (c). Let g(z) be decreasing on N(z ). Then g(z) is decreasing on [z, ) (otherwise we are in contradiction with our assumption on the maximal number of zero absolute values). Moreover, for sufficiently large z we have f(u,v ) < 0, where u i = { d i z ũ i V, 0 otherwise, v j = { d 2 j z ṽ j V, 0 otherwise. As V V, the vectors u,v contradict the assumption of case (c). 3. The graph G is not connected and E =. Define the function g(z) f(ũ z,ũ 2,...,ũ m, ṽ,...,ṽ n ). This function is linear on a neighborhood N(z )of z. If g(z) is nondecreasing on N(z ), then it is nondecreasing on [0,z ] (otherwise we are in contradiction with our assumption on the maximal number of zero absolute values). From g(0) g(z )=f(ũ, ṽ) < 0wegetg(0) = f(0, ũ 2,...,ũ m, ṽ,...,ṽ n ) < 0. This contradicts the assumption of case (c). If g(z) is decreasing on N(z ), then it is decreasing on [z, ) (otherwise we are in contradiction with our assumption on the maximal number of zero absolute values). Moreover, for sufficiently large z we have f(ũ z,0,...,0, 0,...,0) < 0. This also contradicts the assumption of case (c). Theorem 2.2. Let A IR n n, b IR n, and d IR n. Then vectors x, y R n form a solution of the system (2.3) (2.4) Ax = b, A T y = d for some A A, b b, and d d iff they satisfy the following system of inequalities: (2.5) (2.6) (2.7) a Δ ij y i x j (p i q j ) + i,j= A Δ x + b Δ r, A Δ y + d Δ r 2, (b Δ i y i p i + d Δ i x i q i ) (ri y i p i ri 2 x i q i ) i= i= p, q {0, } n, where r A c x + b c, r 2 (A c ) T y + d c. Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

51 DESCRIPTION OF SYMMETRIC SOLUTION SET 53 Proof. Let x, y R n. Then x, y satisfy (2.3) (2.4) iff for a certain α [, ] n n the following relations hold: A c i,.x + α ik a Δ ikx k [b c i b Δ i,b c i + b Δ i ] i =,...,n, k= ( T Ac.,j) n y + α kj a Δ kjy k [d c j d Δ j,d c j + d Δ j ] j =,...,n. k= Equivalently, iff the following linear programming problem max 0 α ij subject to i,j= α ik a Δ ikx k ri + b Δ i k= α ik a Δ ikx k ri + b Δ i k= α kj a Δ kjy k rj 2 + d Δ j k= α kj a Δ kjy k rj 2 + d Δ j k= i =,...,n, i =,...,n, j =,...,n, j =,...,n, α ij i, j =,...,n, α ij i, j =,...,n has an optimal solution. Recall duality in linear programming [2], [3]. The linear programs max b T ỹ subject to ÃT ỹ c and min c T x subject to à x = b, x 0 are dual to each other. Moreover, their optimal values are equal as long as at least one of the problems is feasible (i.e., the constraints are satisfiable). Thus our linear programming problem has an optimal solution iff the dual problem { min ( r + b Δ ) T w +(r + b Δ ) T w 2 +( r 2 + d Δ ) T w 3 } +(r 2 + d Δ ) T w 4 + (wij 5 + wij) 6 i,j= subject to a Δ ijx j wi + a Δ ijx j wi 2 a Δ ijy i wj 3 + a Δ ijy i wj 4 + wij 5 wij 6 =0 i, j =,...,n, w,w 2,w 3,w 4,w 5,w 6 0 has an optimal solution. The dual problem is feasible as its constraints are fulfilled when all the variables are equal to zero, for instance. After substitution u w 2 w, Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

52 54 MILAN HLADÍK v w 4 w 3 we can rewrite this problem: min (r + b Δ ) T u +2(b Δ ) T w +(r 2 + d Δ ) T v +2(d Δ ) T w 3 + subject to (wij 5 + wij) 6 i,j= a Δ ijx j u i + a Δ ijy i v j + wij 5 wij 6 =0 i, j =,...,n, w u, w 3 v, w,w 3,w 5,w 6 0. For w,w 3,w 5, and w 6 some necessary optimality conditions can be given. For each i, j at least one of wij 5,w6 ij is zero (otherwise subtract from them a sufficiently small ε>0 and obtain a better solution). If wij 5 = 0, then w6 ij = aδ ij x ju i + a Δ ij y iv j 0, and hence wij 5 + w6 ij = aδ ij x ju i + a Δ ij y iv j. Similarly, if wij 6 = 0, then wij 5 = (aδ ij x ju i + a Δ ij y iv j ) 0, and hence wij 5 + w6 ij = aδ ij x ju i + a Δ ij y iv j. Therefore w 5 ij + w 6 ij = a Δ ijx j u i + a Δ ijy i v j holds in any case. Next, the only constraints involving the variable wi, i {,...,n} are w i u i and wi 0. Since the objective function coefficient by w i is nonnegative, the optimal wi should be as small as possible. That is, w i = max ( u i, 0)=( u i ) +. Hence we have w =( u) +, and the equation w 3 =( v) + follows analogously. Using these necessary optimality conditions, the optimization problem can be reformulated as an unconstrained optimization problem: { min (r + b Δ ) T u +2(b Δ ) T ( u) + +(r 2 + d Δ ) T v u,v R n +2(d Δ ) T ( v) + + i,j= } a Δ ij x j u i + y i v j. The positive part of a real number p is equal to p + = 2 (p + p ), and the problem comes in the form min u,v R n (r ) T u +(b Δ ) T u +(r 2 ) T v +(d Δ ) T v + a Δ ijx j u i + a Δ ijy i v j. i,j= As a Δ ij is nonnegative (because it is the radius of an interval), the objective function can be written (2.8) f(u, v) (r ) T u +(b Δ ) T u +(r 2 ) T v +(d Δ ) T v + a Δ ij x j u i + y i v j. Note that, it is positive homogeneous, that is, f(λu, λv) =λf(u, v) λ 0. i,j= If f(ũ, ṽ) < 0 for some vectors ũ, ṽ R n, then f(λũ, λṽ) tends to for λ, and the problem does not attain an optimum. On the other hand, if f(u, v) 0 for Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

53 DESCRIPTION OF SYMMETRIC SOLUTION SET 55 all u, v R n, then the optimal solution is u = v = 0. Thus the optimization problem has an optimal solution iff the objective function is nonnegative for all u, v R n. We now use Lemma 2. with a r, a 2 r 2, b b Δ, b 2 d Δ, C A Δ, d y, d 2 x, and m = n. It follows that it is sufficient to test nonnegativity of f(u, v) for three cases:. u i {0,y i } i =,...,n, and v j {0, x j } j =,...,n; 2. u i {0, y i } i =,...,n, and v j {0,x j } j =,...,n; 3. (u T,v T ) T = ±e k for some k {,...,2n}. The first and second cases yield ± ri y i p i + b Δ i y i p i ri 2 x i q i + d Δ i x i q i + a Δ ij y i x j (p i q j ) 0 or i= i= a Δ ij y i x j (p i q j ) + i,j= i= i= i= i,j= (b Δ i y i p i + d Δ i x i q i ) (ri y i p i ri 2 x i q i ), where p, q {0, } n. In the third case when u = ±e k and v =0,weget ±rk + b Δ k + a Δ kj x j 0, j= which is the kth Oettli Prager inequality in (2.5). Likewise u =0,v = ±e k yields the kth Oettli Prager inequality in (2.6). 3. Symmetric solution set. In this section, we suppose w.l.o.g. that A = A T, i.e., matrices A c,a Δ are symmetric. Otherwise we restrict our considerations on the interval matrix (a ij a ji ) n i,j=. Theorem 3., which is a simple corollary of Theorem 2.2, enables us to obtain an explicit description of the symmetric solution set Σ sym. Nevertheless, the number of inequalities in the description is still exponential. Therefore when checking x Σ sym for only one vector x, it is better from the theoretical viewpoint to use the linear programming problem (from the proof of Theorem 2.2), which is polynomially solvable [3]. The question whether Σ sym can be described by a polynomial number of inequalities is still open. Theorem 3.. Let r A c x+b c. The symmetric solution set Σ sym is described by the following system of inequalities: (3.) A Δ x + b Δ r, a Δ ij x i x j (p i q j ) + b Δ (3.2) i x i (p i + q i ) r i x i (p i q i ) i,j= for all vectors p, q {0, } n \{0, } such that (3.3) p lex q and (p = q i : p i = q i =0). i= Proof. For every A A, the matrix 2 (A + AT ) A is symmetric, and for every b,b 2 b we have 2 (b + b 2 ) b. Thus, Σ sym can be equivalently described as the set of all x R n satisfying (3.4) Ax = b, (3.5) A T x = b 2 i= i= Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

54 56 MILAN HLADÍK for some A A, b,b 2 b. Put y x, d b and apply Theorem 2.2 on system (3.4) (3.5). We obtain that Σ sym is described by (3.) (3.2) for all p, q {0, } n. To reduce the number of inequalities in (3.2), it is sufficient due to symmetry to consider only vectors p, q {0, } n for which p lex q. Obviously, the case p = q is also redundant. The inequality (3.2) corresponding to p = 0 and any q {0, } n can be omitted for the following reason. Multiplying the Oettli Prager system (3.) by the vector ( x q,..., x n q n ) we obtain a Δ ij x i x j q i + b Δ i x i q i r i x i q i r i x i q i. i,j= i= Due to the symmetry of A Δ the first sum is equal to n i,j= aδ ij x ix j q j, and hence the inequality a Δ ij x i x j q j + b Δ i x i q i r i x i q i i,j= i= is a consequence of the Oettli Prager system. The inequality (3.2) corresponding to any p {0, } n and q = is redundant as it is a consequence of the inequality (3.2) with p p, q 0 (which is redundant for the same reason as before); the right-hand sides of the inequalities are the same, and the left-hand side of the former inequality includes all of the left-hand side terms of the latter inequality and possibly some more positive terms. Finally, we prove redundancy for all inequalities (3.2) with p, q {0, } n \{0, }, p lex q, and i= i= i= (3.6) p q and i :(p i = q i =). Clearly, (3.6) is equivalent to (3.7) i :(p i = q i = ) and i : p i = q i =. Such an inequality is a consequence of the inequality (3.2) with p q, q p. The vectors p,q satisfy the condition (3.3). We compute the number of inequalities for system (3.2). Proposition 3.2. The system (3.2) consists of 2 (4n 3 n 2 2 n +3) inequalities. Proof. There are (2 n 2) 2 pairs of vectors p, q satisfying p, q {0, } n \{0, }. Since for each pair p, q just one of the conditions p lex q, p = q, orq lex p is true, the number of the vectors p, q satisfying p, q {0, } n \{0, }, p lex q, is equal to 2( (2 n 2) 2 (2 n 2) ) = 2 (2n 2)(2 n 3). Now we focus on condition (3.7) which determines the bad cases. For every p, q define I p,q {i =,...,n p i = q i =}, J p,q {i =,...,n p i + q i =}. ( Vectors p, q {0, } n satisfy (3.7) iff I p,q and I p,q + J p,q = n. The value n ) k 2 n k identifies the number of p, q {0, } n for which I p,q = k and J p,q = n k. Summing up for all k =,...,n and using binomial expansion of ( + 2) n we obtain the number of pairs p, q {0, } n with property (3.7) is equal to ( n ) 2 n + ( n 2 ) 2 n ( n n ) 2 0 =3 n 2 n. Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

55 DESCRIPTION OF SYMMETRIC SOLUTION SET 57 From this amount we have to exclude the cases when p =orq =: 3 n 2 n 2 (2 n )+. Exactly half of them satisfy p lex q. Eventually, we obtain the number in question: 2 (2n 2)(2 n 3) ( 3 n 2 n 2 (2 n )+ ) = 2 2 (4n 3 n 2 2 n +3). The number of inequalities in (3.2) is exponential, but not as tremendous as by using Fourier Motzkin elimination (no better upper bound is known than the double exponential one). Moreover, system (3.2) is characterized explicitly and is much more easy to handle. Concretely, for n = 2 we have only one additional inequality (in comparison to two inequalities obtained by Fourier Motzkin elimination [4]), for n = 3 this number rises up to 2 (cf. [3], [4], [6]; Fourier Motzkin elimination leads to 44 inequalities). Example 3.3. For the two-dimensional case, the symmetric solution set is described by the system consisting of the Oettli Prager inequalities (3.) a Δ x + a Δ 2 x 2 + b Δ a c x a c 2x 2 + b c a Δ 2 x + a Δ 22 x 2 + b Δ 2 a c 2x a c 22x 2 + b c 2 supplemented by only one inequality (3.2) a Δ x 2 + a Δ 22x b Δ x + b Δ 2 x 2 a c x 2 + a c 22x b c x b c 2x 2. In the list below we mention some particular examples. Figures 3. and 3.2 illustrate a solution set (light gray color) and a symmetric solution set (gray color):. (Figure 3.) A = ( [,2] [0,a] ) ( [0,a], b = 22 ) ; here the interval hull Σ can be arbitrarily larger than Σ sym, depending on the real parameter a>0. 2. (Figure 3.2) A = ( [ 5,5] ) ( [ 5,5], b = ) [,3] ; here Σ is unbounded, but Σsym is bounded. Fig. 3.. Solution set arbitrarily larger than symmetric solution set, a =4. Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

56 58 MILAN HLADÍK Fig Unbounded solution set and bounded symmetric solution set. 3. For A = ( [0,] [,2] ) ( [,2] [,0], b = [,] ) [,] we have Σ = Σsym and both are bounded. ) (, b = [0,] ) [0,] we have Σ = Σsym and both are un- 4. For A = ( [,] [0,2] [0,2] [,] bounded. 4. Skew symmetric solution set. In this section, let us suppose w.l.o.g. that A = A T and the diagonal of A is zero. Therefore A c is skew symmetric and A Δ is a symmetric matrix. The description of the skew symmetric solution set Σ skew is a consequence of Theorem 2.2. Proposition 4.. Let r A c x + b c. The skew symmetric solution set Σ skew is described by the following system of inequalities: (4.) (4.2) a Δ ij x i x j (p i q j ) + i,j= A Δ x + b Δ r, b Δ i x i (p i + q i ) r i x i (p i + q i ) i= i= p, q {0, } n \{0}, p lex q. Proof. For all A A and b,b 2 b we have that 2 (A AT ) A is a skew symmetric matrix and 2 (b + b 2 ) b. Thus, Σ skew can be equivalently described as the set of all x R n satisfying (4.3) (4.4) Ax = b, A T ( x) =b 2 for some A A, b,b 2 b. Put y x, d b. Then r = A c x + b c = ( A c )( x)+b c = (A c ) T y + d c = r 2 r. Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

57 DESCRIPTION OF SYMMETRIC SOLUTION SET 59 Apply Theorem 2.2 on system (4.3) (4.4). We obtain that Σ skew is described by (4.) (4.2). To reduce the number of inequalities in (4.2), it is sufficient due to symmetry to consider only vectors p, q {0, } n \{0} for which p lex q. The number of inequalities in (4.2) is 2 n (2 n ) and can be furthermore decreased to the number 2 n n ; see Theorem 4.2 where we claim that it is sufficient to consider only such inequalities for which p = q, the others being redundant. Theorem 4.2. Let r A c x + b c. The skew symmetric solution set Σ skew is described by the following system of inequalities: (4.5) A Δ x + b Δ r, (4.6) a Δ ij x i x j (p i p j ) + b Δ i x i p i r i x i p i p {0, } n \{0}, p e k. i<j i= i= Proof. For given vectors p, q {0, } n denote the inequality (4.2) corresponding to p, q by Ineq(p, q). Let p, q be fixed, and define vectors s, t {0, } n componentwise by s i = { p i = q i =, 0 otherwise, t i = { (p i =) (q i =), 0 otherwise. We prove that Ineq(p, q) is a consequence of the inequality (4.7) ( ) Ineq(s, s)+ineq(t, t) 2 and hence can be omitted. The right-hand side of the inequality Ineq(p, q) is p i= r ix i + q i= r ix i = s i= r ix i + t i= r ix i, which is not greater than s i= r ix i + t i= r ix i, the right-hand side of (4.7). The second sum in Ineq(p, q) is equal to p i= bδ i x i + q i= bδ i x i, which is equal to s i= bδ i x i + t i= bδ i x i, the second sum in (4.7). To prove the similar relations for the corresponding first sums let us note that diagonal terms (i.e., when i = j) in Ineq(p, q) are nonnegative, while diagonal terms are zero in (4.7). We gather the remaining terms into symmetric pairs and show that for each i<jone has a Δ ij x i x j (p i q j ) + a Δ ij x j x i (p j q i ) 2 In fact, we prove a stronger inequality + 2 ( ) a Δ ij x i x j (s i s j ) + a Δ ij x i x j (t i t j ) ( ) a Δ ij x j x i (s j s i ) + a Δ ij x j x i (t j t i ) = a Δ ij x i x j (s i s j ) + a Δ ij x i x j (t i t j ). p i q j + p j q i s i s j + t i t j. Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

58 520 MILAN HLADÍK This can be shown simply by the enumeration of all possible values of p i,p j,q i, and q j, which is done in the following: p i p j q i q j p i q j + p j q i s i s j t i t j s i s j + t i t j Now we have that the right-hand side of Ineq(p, q) is less or equal to the righthand side of (4.7), and the left-hand side of Ineq(p, q) is greater or equal to the left-hand side of (4.7). Therefore, Ineq(p, q) is redundant, and (4.2) can be replaced by the system (4.8) a Δ ij x i x j (p i p j ) + i<j b Δ i x i p i r i x i p i p {0, } n \{0}. i= The last reduction follows from the fact that for each unit vector p e k the corresponding inequality in (4.8) represents an x k -multiple of the kth Oettli Prager inequality (4.5). The resulting number of inequalities in the description is again exponential. But in comparison with the upper bound 8 ( ) 3 2 κ+ 2, κ = 2n(n + ), for the final number of inequalities obtained by Fourier Motzkin elimination (see [4]), the improvement is significant. For n = 2, system (4.6) comprises one inequality, and for n = 3 we get four inequalities. In these cases, Fourier Motzkin elimination yields two and eight inequalities, respectively. Example 4.3. For n = 2, system (4.6) is composed of only one inequality i= b Δ x + b Δ 2 x 2 b c x + b c 2x 2. In this two-dimensional case the set Σ skew represents a polyhedral set, which is convex in each orthant (cf. [4]). The following are some particular examples:. For A = ( 0 [,2] ) ( [ 2, ] 0, b = [0,2] ) [ 2,2] we have Σ = Σskew and both are bounded. 2. For A = ( 0 [,] [,] 0 unbounded. ), b = ( [0,2] [ 2,2] ) we have Σ = Σskew and both are Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

59 DESCRIPTION OF SYMMETRIC SOLUTION SET 52 REFERENCES [] G. Alefeld and J. Herzberger, Introduction to Interval Computations, Academic Press, London, 983. [2] G. Alefeld and G. Mayer, On the symmetric and unsymmetric solution set of interval systems, SIAM J. Matrix Anal. Appl., 6 (995), pp [3] G. Alefeld, V. Kreinovich, and G. Mayer, The shape of the symmetric solution set, in Proceedings of the International Workshop on Applications of Interval Computations, El Paso, 995, B. Kearfott and V. Kreinovich, eds., Kluwer Academic Publishers, Dordrecht, 996. [4] G. Alefeld, V. Kreinovich, and G. Mayer, On the shape of the symmetric, persymmetric, and skew-symmetric solution set, SIAM J. Matrix Anal. Appl., 8 (997), pp [5] G. Alefeld, V. Kreinovich, and G. Mayer, The shape of the solution set for systems of interval linear equations with dependent coefficients, Math. Nachr., 92 (998), pp [6] G. Alefeld, V. Kreinovich, and G. Mayer, On symmetric solution sets, in Inclusion Methods for Nonlinear Problems: With Applications in Engineering, Economics and Physics. Proceedings of the International GAMM-Workshop, Munich and Oberschleissheim, 2000, Comput. Suppl. 6, J. Herzberger, ed., Springer, Wien, 2003, pp. 22. [7] G. Alefeld, V. Kreinovich, and G. Mayer, On the solution sets of particular classes of linear interval systems, J. Comput. Appl. Math., 52 (2003), pp. 5. [8] R. Araiza, G. Xiang, O. Kosheleva, and D. Škulj, Under interval and fuzzy uncertainty, symmetric Markov chains are more difficult to predict, in Proceedings of the 26th International Conference of the North American Fuzzy Information Processing Society NAFIPS 2007, M. Reformat and M. R. Berthold, eds., San Diego, CA, 2007, pp [9] M. Hladík, Solution set characterization of linear interval systems with a specific dependence structure, Reliab. Comput., 3 (2007), pp [0] Z. Kulpa, A. Pownuk, and I. Skalna, Analysis of linear mechanical structures with uncertainties by means of interval methods, Comput. Assist. Mech. Eng. Sci., 5 (998), pp [] W. Oettli and W. Prager, Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides, Numer. Math., 6 (964), pp [2] M. Padberg, Linear Optimization and Extension, Springer, Berlin, 999. [3] A. Schrijver, Theory of Linear and Integer Programming, John Wiley & Sons Ltd., Chichester, UK, 998. Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

60 Int. J. Appl. Math. Comput. Sci., 202, Vol. 22, No. 3, DOI: /v ENCLOSURES FOR THE SOLUTION SET OF PARAMETRIC INTERVAL LINEAR SYSTEMS MILAN HLADÍK Department of Applied Mathematics, Faculty of Mathematics and Physics Charles University, Malostranské nám. 25, 8 00, Prague, Czech Republic milan.hladik@matfyz.cz Faculty of Informatics and Statistics University of Economics, nám. W. Churchilla 4, 3067, Prague, Czech Republic We investigate parametric interval linear systems of equations. The main result is a generalization of the Bauer Skeel and the Hansen Bliek Rohn bounds for this case, comparing and refinement of both. We show that the latter bounds are not provable better, and that they are also sometimes too pessimistic. The presented form of both methods is suitable for combining them into one to get a more efficient algorithm. Some numerical experiments are carried out to illustrate performances of the methods. Keywords: linear interval systems, solution set, interval matrix.. Introduction Solving systems of interval linear equations is a fundamental problem in interval computing (Fiedler et al., 2006; Neumaier, 990). Therein, one assumes that the matrix entries and the right-hand side components perturb independently and simultaneously within given intervals. However, this assumption is hardly true in practical problems. Very often various correlations between input quantities appear, e.g., in robotics (Merlet, 2009) or in dynamic systems (Busłowicz, 200). Linear dependences were investigated by several authors. The first paper on parametric interval systems (with a special structure) is that by Jansson (99). For a special class of parametric systems, Neumaier and Pownuk (2007) proposed an effective method. The general problem of interval parameter dependent linear systems was first treated by Rump (994). Theoretical papers involve, e.g., characterization of the boundary of the solution set (Popova and Krämer, 2008), the quality of the solution set (Popova, 2002), or an explicit characterization of a class of parametric interval systems (Hladík, 2008; Popova, 2009). Shapes of the particular solution sets were first analyzed by Alefeld et al. (997; 2003). Kolev (2006) proposed a direct method and an iterative one (Kolev, 2004) for computing an enclosure of the solution set. ParametrizedGauss Seidel iteration was employed by Popova (200). A direct method was given by Skalna (2006), and a monotonicity approach by Popova (2006a), Rohn (2004), and Skalna (2008). Inner and outer approximations by a fixed-point method were developed by Rump (994; 200), and implemented by Popova and Krämer (2007). A Mathematica package for solving parametric interval systems is introduced by Popova (2004a). Let p := [p, p] ={p R K p p p} be an interval vector. By p c := 2 (p + p) and pδ := 2 (p p) we denote the corresponding center and the radius vector. Analogous notation is used for interval matrices. We suppose that the reader is familiar with the basic interval arithmetic. In this paper, we consider a general parametric system of interval linear equations in the form where A(p)x = b(p), p p, () K K A(p) = p k A k, b(p) = p k b k. k= k=

61 562 M. Hladík Herein, p is the interval vector representing K interval parameters, and A k R n n and b k R n, k =,...,K, are given matrices and vectors. Notice that this linear parametric form comprises affine linear parametric matrices and vectors, K K A 0 + p k A k, b 0 + p k b k, k= k= since one can simply introduce an idle parameter p 0 := [, ]. In our approach, no better results are obtained explicitly for the affine linear parametric structure. The solution set is defined as Σ := {x R n A(p)x = b(p), p p}. We use the following notation: ρ(a) stands for the spectral radius of a matrix A, A i. for the i-th row of A, I for the identity matrix and e i for its ith column. The diagonal matrix with entries z,...,z n is denoted by diag(z), and A(p) is a short form for a family A(p), p p. We write interval quantities in boldface. The paper is structured as follows. In Section 2 we discuss the regularity of a parametric interval matrix, and in Section 3 enclosures of a parametric interval linear system. We generalize the Bauer Skeel and the Hansen Bliek Rohn bounds, which were developed for a standard interval linear system; for the reader s convenience, we recall the original formulae in Appendix. Moreover, we propose efficient refinements of both methods. 2. Regularity of parametric interval matrices In order to develop an enclosure for the parametric interval system we have to discuss the regularity of the parametric interval matrix A(p) first. The parametric interval matrix is called regular if A(p) is nonsingular for every p p. Preconditioning and relaxing the parametric interval matrix, we obtain an interval matrix i.e., A = K p k (RA k ), k= [ K A ij = min ( p k (RA k ) ij, p k (RA k ) ) ij, k= K max ( p k (RA k ) ij, p k (RA k ) ] ) ij. k= Clearly, if A is regular, then so is A(p). Thus we can employ the well-known Beeck Rump sufficient condition for the regularity of interval matrices (Beeck, 975; Rump, 983; Rex and Rohn, 998). Theorem. Let R R n n be such that ( ) K ρ I RA(p c ) + p Δ k RAk <. (2) k= Then A(p) is regular. Usually, the best choice for the matrix R is the numerically computed inverse of A(p c ). In the following, we consider the case R = A(p c ). For this special case, the sufficient condition was already stated by Popova (2004b). How strong is the sufficient condition presented in Theorem? The following result shows a class of problems where the condition is not only sufficient, but also necessary. It is a generalization of Rohn s result (Rohn, 989, Corollary 5..(ii)). Proposition. Suppose that A(p c ) is nonsingular and there are z {±} n and y {±} K such that for every k {,...,K} we have y k diag(z) A(p c ) A k diag(z) 0. Then A(p) is regular if and only if ( K ) ρ p Δ k A(pc ) A k <. k= Proof. One implication is obvious in view of Theorem. We prove the converse by contradiction. Denote A := = K p Δ k A(pc ) A k k= K p Δ k y k diag(z) A(p c ) A k diag(z), k= and suppose for contradiction that ρ(a ). SinceA is non-negative, according to the Perron Frobenius theorem (Horn and Johnson, 985; Meyer, 2000) there is some non-zero vector x such that A x = ρ(a )x or, equivalently, ( I ) ρ(a ) A x =0. Premultiplying by A(p c ) diag(z),weget ( A(p c ) diag(z) ) ρ(a ) A(pc )diag(z) A x =0 or ( K k= ) ( p c k y ) k ρ(a ) pδ k A k (diag(z) x) =0.

62 Enclosures for the solution set of parametric interval linear systems 563 The vector diag(z) x is non-zero, and the constraint matrix belongs to A(p) since p c k y k ρ(a ) pδ k p k, k =,...,K. Thus we found a singular matrix in A(p), which is a contradiction. 3. Enclosures for parametric interval linear systems The main problem studied within this paper is to find a tight enclosure for the solution set Σ, where an enclosure is any interval vector containing Σ. A simple enclosure can be acquired by relaxing the system () to an interval linear system Ax = b, where (by using interval arithmetic) K K A := p k A k, b := p k b k. k= k= Since many efficient solvers of interval linear systems use preconditioning, we should note that instead of preconditioning the system Ax = b by a matrix R it is better to precondition the original data. That is, consider A x = b, where K K A := p k (RA k ), b := p k (Rb k ). (3) k= k= Proposition 2. We have A RA and b Rb. Proof. Let i, j {,...,n}. Due to the sub-distributivity of interval arithmetic, we can write K ( A ij = K n ) p k (RA k ) ij = p k R il A k lj = k= K k= l= k= p k R il A k lj = n l= k= ( K ) R il p k A k lj = l= k= l= K R il (p k A k lj ) R il A lj =(RA) ij. l= We proceed similarly for b Rb. To obtain tighter enclosures, we have to inspect parametric systems more carefully. Recently, Popova (2009) proved that the inequality system given below in (4) is an explicit description of a parametric interval linear system of the so-called zero or first class; in this class, for each k =,...,K, the nonzero entries of (A k b k ) are situated in one row only. First we show this is a necessary (but not sufficient in general) characterization for any parametric interval linear system. Theorem 2. If x R n solves () for some p p, thenit solves A(p c )x b(p c ) K p Δ k Ak x b k. (4) k= Proof. Let x R n be a solution to A(p)x = b(p) for some p p. Then, in a similar way as for the well known Oettli Prager theorem, we derive A(p c )x b(p c ) K = p c k (Ak x b k ) k= K K = p c k (Ak x b k ) p k (A k x b k ) k= k= K = (p c k p k)(a k x b k ) k= K p c k p k A k x b k k= K p Δ k A k x b k. k= A sufficient and necessary characterization of Σ is given below in terms of infinite systems of inequalities. From another viewpoint, the system is composed of a union of systems (4) over all possible preconditionings of (). An open question arises whether or not particular extremal points of Σ can be achieved by an appropriate preconditioning of (). Theorem 3. We have that x Σ if and only if it solves y T (A(p c )x b(p c )) for every y R n. K p Δ k y T (A k x b k ) (5) k= Proof. Let x R n.thenx Σ if and only if there is a vector q [, ] K such that A(p c )x b(p c )= K q k p Δ k (A k x b k ). k= Set d := A(p c )x b(p c ),andletd R n K be a matrix whose k-th column is equal to p Δ k (Ak x b k ), k =,...,K. Then x Σ if and only if there is an optimal solution to the linear system Dq = d, q,

63 564 M. Hladík or, in other words, if and only if the linear program max 0 T q subject to Dq = d, q has an optimal solution. Consider the corresponding dual problem min d T y + T (u + v) subject to D T y + u v =0,u,v 0, which is always feasible. According to the theory of duality in linear programming (Padberg, 999; Schrijver, 998), the existence of an optimal solution to one problem implies the same for the second one and the optimal values are equal. For an optimal solution of the dual problem and every i {,...,K} either u i =0or v i =0.Otherwise,we can subtract a small positive amount from both u i and v i and decrease the optimal value. If u i =0,then(u+v) i = v i =(D T y) i 0. Similarly, v i =0implies (u + v) i = u i = (D T y) i 0. Hence we can derive u+v = D T y, and the dual problem takes the form min d T y + T D T y subject to y R n. Since the objective function is positive homogeneous, the problem has an optimal solution (equal to zero) if and only if the objective function is non-negative, i.e., d T y + T D T y 0, y R n or, substituting y := y, y T d y T D, y R n. In the setting of D and d,weget(5). Based on Theorem 2 we develop a generalization of the Bauer Skeel bounds (Rohn, 200; Stewart, 998) to parametric interval systems. Note that the generalized Bauer Skeel bounds yield the same enclosure as the direct method by Skalna (2006). However, the following form is more convenient for combining it with the Hansen Bliek Rohn method and for refinements. Theorem 4. Suppose that A(p c ) is nonsingular. Write M := K p Δ k A(p c ) A k, k= x := A(p c ) b(p c ). If ρ(m) <,then [ K x (I M) p Δ k A(p c ) (A k x b k ), x +(I M) k= K k= is an interval enclosure to Σ. ] p Δ k A(p c ) (A k x b k ) Proof. Preconditioning the system A(p)x = b(p) by the matrix A(p c ), we obtain an equivalent system A(p c ) A(p)x = A(p c ) b(p), or K K p k A(p c ) A k x = p k A(p c ) b k, p p. k= k= According to Theorem 2 each solution to this system satisfies A(p c ) A(p c )x A(p c ) b(p c ) K p Δ k A(pc ) (A k x b k ). k= Rearranging the system, we get x x = Equivalently, ( I K k= K p Δ k A(pc ) (A k x b k ) (6) k= K p Δ k A(pc ) (A k (x x + x ) b k ) k= K p Δ k A(p c ) A k (x x ) k= + K p Δ k A(p c ) (A k x b k ) k= K p Δ k A(p c ) A k x x k= + K p Δ k A(pc ) (A k x b k ). k= ) p Δ k A(p c ) A k x x K p Δ k A(pc ) (A k x b k ). k= From ρ(m) <, it follows (Fiedler et al., 2006; Meyer, 2000, Theorem.3) that (I M) = M j. j=0 Since the matrix M is non-negative, so is (I M). Thus we may multiply the system by (I M) to obtain K x x (I M) p Δ k A(pc ) (A k x b k ). k=

64 Enclosures for the solution set of parametric interval linear systems 565 This means, that x x (I M) x x +(I M) K k= K k= p Δ k A(p c ) (A k x b k ), p Δ k A(p c ) (A k x b k ). The Hansen Bliek Rohn method (Fiedler et al., 2006; Rohn, 993, Theorem 2.39) gives an enclosure for the solution set of an interval linear system. The following is a generalization to parametric interval linear systems; however, the result is the same as the Hansen Bliek Rohn bounds applied on the preconditioned system (3) by R := A(p c ). For the reader s convenience, we present a detailed proof, which will be followed up in the next section for a refinement. Note that an alternative form of the enclosure was developed by Neumaier (999) as well as Ning and Kearfott (997). Theorem 5. Suppose that A(p c ) is nonsingular. Using the notation from Theorem 4, write M := (I M), K x 0 := M x + p Δ k M A(p c ) b k. k= If ρ(m) <, then any solution x to () satisfies { x i max x 0 i +(x i x i )m ii, and x i min ( x 0 2m ii i +(x i x i )m ) } ii, { x 0 i +(x i + x i )m ii, ( x 0 2m ii i +(x i + x i )m ) } ii. Proof. From the proof of Theorem 4 we know that each solution to () satisfies x x K p Δ k A(pc ) (A k x b k ) (7) k= K K p Δ k A(pc ) A k x + p Δ k A(pc ) b k. k= This inequality system implies k= k= x x K K p Δ k A(pc ) A k x + p Δ k A(pc ) b k, (8) k= and x x K K p Δ k A(p c ) A k x + p Δ k A(p c ) b k. (9) k= k= Let i {,...,n}. Consider the system (9) in which the i-th inequality is replaced by the i-th inequality from (8), x x +(x i x i x i + x i )e i K K p Δ k A(p c ) A k x + p Δ k A(p c ) b k. k= This can be rewritten as ( I K k= k= ) p Δ k A(p c ) A k x +(x i x i )e i x +(x i x i )e i + K p Δ k A(p c ) b k. k= From ρ(m) <, it follows (Fiedler et al., 2006; Meyer, 2000, Theorem.3) that (I M) = M j. j=0 Since the matrix M is non-negative, M =(I M) I. Thus we may multiply the system by M 0 to obtain x +(x i x i )M e i Setting M x +(x i x i )M e i K + p Δ k M A(p c ) b k. k= x 0 = M x + the system reads K p Δ k M A(p c ) b k, k= x +(x i x i )M e i x 0 +(x i x i )M e i. The i-th inequality becomes x i +(x i x i )m ii x0 i +(x i x i )m ii. Distinguish two cases. If x i 0,then x i x 0 i +(x i x i )m ii.

65 566 M. Hladík If x i < 0,then x i +2x i m ii x0 i +(x i x i )m ii. Using the fact that M I, we get that 2m ii 2 > and ( x i x 0 2m ii i +(x i x i )m ) ii. Summing up, we have an upper bound on x i as follows: { x i max x 0 i +(x i x i )m ii, ( ) } x 0 2m ii i +(x i x i )m ii. To obtain a lower bound on x i,werealizethatax = b if and only if A( x) = b. Thus, we apply the previous result to the parametric interval system A(p)( x) = b(p). That is, the sign of b c and x will be changed and { x i max x 0 i +( x i x i )m ii, ( ) } x 0 2m ii i +( x i x i )m ii, or, { x i min x 0 i +(x i + x i )m ii, ( x 2m 0 ii i +(x i + ) } x i )m ii. 4. Refinement of enclosures Now we show that the enclosures discussed in the previous section can be made tighter. The idea is to use those enclosures to check some sign invariances, and if they hold true, then the process of deriving the enclosures can be refined. Note that the proposed refinements run always in polynomial time. Let x be the enclosure obtained by Theorems 4 or 5 or by any other method, and let k {,...,K}. Write a k := A(p c ) (A k x b k ). We will employ notations from Theorems 4 and 5, too. For the refinements, we assume ρ(m) <. 4.. Refinement of the Bauer Skeel bounds. First, we consider the Bauer Skeel bounds. If a k 0, then for every x Σ one has A(p c ) (A k x b k ) = A(p c ) A k (x x )+A(p c ) (A k x b k ). (0) Otherwise, if a k 0,then A(p c ) (A k x b k ) = A(p c ) A k (x x ) A(p c ) (A k x b k ). () Otherwise, we estimate the term from above as in the proof Remark. The Bauer Skeel and Hansen Bliek Rohn methods are similar to each other since they are derived from the same basis. Nevertheless, as we will see in Section 6, both methods are incomparable, that is, sometimes the former is better and sometimes the latter. Thus, to obtain enclosure as tight as possible we propose to compute both and take their intersection. The overall computational cost is low since we calculate the inverses A(p c ), M = (I M) and other intermediate expressions only once. Using notations of Theorems 4 and 5, we compute the upper endpoints of the resulting enclosure as the minima of x i + M i. K k= p Δ k A(p c ) (A k x b k ), and { max x 0 i +(x i x i )m ii, ( ) } x 0 2m ii i +(x i x i )m ii, i =,...,n. We proceed similarly for the lower endpoints. A(p c ) (A k x b k ) A(p c ) A k x x + A(p c ) (A k x b k ). (2) Anyway, the inequality (6) can be written as x x K p Δ k A(pc ) (A k x b k ) k= Y (x x )+y + Z x x + z for some Y,Z R n n, y, z R n,andz 0. Here, Y and y are summed up from (0) and (), whereas Z and z come from (2). Now, we proceed as follows: whence and x x Y x x + y + Z x x + z, (I Y Z) x x y + z, x x +(I Y Z) (y + z), x x (I Y Z) (y + z).

66 Enclosures for the solution set of parametric interval linear systems 567 Algorithm (Refinement of the Bauer Skeel method) : Y := 0; y := 0; Z := 0; z := 0; 2: x := A(p c ) b(p c ); 3: Let x be an initial enclosure to Σ; 4: for k =,...,K do 5: a k := A(p c ) (A k x b k ); 6: for j =,...,ndo 7: if a k j 0 then 8: Y j. := Y j. + p Δ k A(pc ) j. A k ; y j := y j + p Δ k A(pc ) 9: else if a k j 0 then 0: Y j. := Y j. p Δ k A(pc ) : else 2: Z j. := Z j. + p Δ k A(pc ) j. A k ; y j := y j p Δ k A(pc ) j. (A k x b k ); j. (A k x b k ); j. A k ; z j := z j + p Δ k A(pc ) j. (A k x b k ) ; 3: end if 4: end for 5: end for 6: return [ x (I Y Z) (y + z), x +(I Y Z) (y + z) ], an enclosure to Σ. Since Y + Z is non-negative and Y + Z M,the inverse matrix (I Y Z) exists and is non-negative. Notice that even tighter bounds can be calculated by splitting the terms of (6) componentwise. That is, we check the signs of ai k and ak i for every i =,...,n,and use the i-th estimate either in (0), () or (2) accordingly. The method is described in Algorithm. In the following we claim that the resulting enclosure is always as good as the initial Bauer Skeel bounds. Proposition 3. Let x be the enclosure obtained by Theorem 4, and x the enclosure obtained by Algorithm. Then x x. Proof. Recall that and x = x +(I M) From = x + j= K k= K M j k= p Δ k A(p c ) (A k x b k ) p Δ k A(pc ) (A k x b k ), x = x +(I Y Z) (y + z) = x + ( Y + Z) j (y + z). y + z j= K p Δ k A(pc ) (A k x b k ) k= and 0 Y + Z M,we obtain x x. We proceed Similarly for x x Refinement of the Hansen Bliek Rohn bounds. We will refine the Hansen Bliek Rohn bounds in the same manner as the Bauer Skeel ones. If a k 0,then A(p c ) (A k x b k ) = A(p c ) A k x A(p c ) b k. (3) Otherwise, if a k 0,then A(p c ) (A k x b k ) = A(p c ) A k x + A(p c ) b k. (4) Otherwise, we use the standard estimation for the Hansen Bliek Rohn method, A(p c ) (A k x b k ) A(p c ) A k x + A(p c ) b k. (5) Thus (7) takes the form of x x K p Δ k A(pc ) (A k x b k ) k= Yx y + Z x + z ( Y + Z) x y + z, where Y,Z R n n, y, z R n,andz 0. Next, we proceed as in the proof of Theorem 5. The method is summarized in Algorithm 2. We show that the refinement of the Hansen Bliek Rohn method is in each component at least as tight as the original Hansen Bliek Rohn bounds. Proposition 4. Let x be the enclosure obtained by Theorem 5, and x the enclosure obtained by Algorithm 2. Then x x.

67 568 M. Hladík Algorithm 2 (Refinement of the Hansen Bliek Rohn method) : Y := 0; y := 0; Z := 0; z := 0; 2: x := A(p c ) b(p c ); 3: Let x be an initial an enclosure to Σ; 4: for k =,...,K do 5: a k := A(p c ) (A k x b k ); 6: for j =,...,ndo 7: if a k j 0 then 8: Y j. := Y j. + p Δ k A(pc ) j. A k ; y j := y j + p Δ k A(pc ) j. b k ; 9: else if a k j 0 then 0: Y j. := Y j. p Δ k A(pc ) j. A k ; y j := y j p Δ k A(pc ) j. b k ; : else 2: Z j. := Z j. + p Δ k A(pc ) j. A k ; z j := z j + p Δ k A(pc ) j. b k ; 3: end if 4: end for 5: end for 6: M := (I Y Z) ; x 0 := M ( x y + z); 7: for i =,...,ndo 8: x i {x := max 0 i +(x i x i )m ii, ( 2m x 0 ii i +(x i x i ii) } )m ; 9: x i := min { x 0 i +(x i + x i )m ii, 2m ii ( x 0 i +(x i + x i )m ii) } ; 20: end for 2: return x, an enclosure to Σ. Proof. Let i {,...,n}. We prove x i x i.thelower case is done accordingly. Write M := (I Y Z), x 0 := M ( x y + z). Clearly, M M and x 0 x 0. Thus x 0 i +(x i + x i )m ii x 0 i +(x i + x i )m ii. Since m ii,wehave and the term 2m ii, ( x 0 2m ii i +(x ) i x i )m ii is the maximizer in Step 8 of Algorithm 2 if and only if it is non-positive. In this case, ( x 0 2m ii i +(x i x i )m ) ii ( x 0 2m ii i +(x ) i x i )m ii, which completes the proof. 5. Time complexity Let us analyse the theoretical time complexity of the proposed methods. Both Bauer Skeel and Hansen Bliek Rohn methods have the same asymptotic time complexities. The most computationally expensive is to calculate the matrix M. It costs O(n 3 K) operations by using a naive implementation. However, the matrices A k, k =,...,K, are usually sparse, in which case the complexity is lower. Denote by P the maximum number of non-zero entries in some A k, k =,...,K, that is, the maximum number of appearances of some parameter p k. Then, computation of M can be implemented in O(nK(n+P )),the matrix inverse is in O(n 3 ) and the remaining calculation is negligible with respect to the worst case time complexity. Thus the algorithms are in O(n 3 + n 2 K + np K). For instance, for symmetric interval systems, we have P =2, K = 2 n(n ), so the total cost is O(n4 ). For Toeplitz systems we have P = O(n), K = O(n), so the time complexity is O(n 3 ). Concerning the refinements discussed in Section 4 it turns out that their asymptotic time complexity is the same as that of the original methods, that is, O(n 3 + n 2 K + np K). Of course, the multiplicative terms are greater, which causes the higher computational time presented in Section 6. The iterative methods by Rump or Popova and Krämer require O(n 3 +n 2 KI) operations, where I stands for the number of iterations. Thus our approach is not

68 Enclosures for the solution set of parametric interval linear systems 569 asymptotically worse provided that P = O(nI). 6. Examples and numerical experiments In his paper, Rohn (200) claims that for the standard system of interval linear equations the Hansen Bliek Rohn bounds are never worse than the Bauer Skeel ones. In the following examples we show that this is not the case for (more general) parametric systems. Surprisingly, the Bauer Skeel bounds are sometimes notably better (Example 2). Example. Consider Okumura s problem of a linear resistive network (Popova and Krämer, 2008, Example 5.3.). It obeys the form of () with A(p) = p + p 6 p 6 0 p 6 p 2 + p 6 + p 7 p 7 0 p 7 p 3 + p 7 + p p p 8 0 p 4 + p 8 + p 9 p 9, p 9 p 5 + p 9 b(p) = (0, 0, 0, 0, 0) T,andp i [0.99,.0], i =,...,9. The Bauer Skeel bounds computed according to Theorem 4 are ([7.048, 7.67], [4.73, ], [5.3933, 5.558], [2.377, ], [.060,.27]) T, and the refinement by Algorithm yields ([7.05, 7.667], [4.80, ], [5.3938, 5.553], [2.382, ], [.0605,.23]) T. The Hansen Bliek Rohn method (Theorem 5) results in a not-as-tight enclosure, ([6.9693, 7.250], [4.0689, 4.297], [5.350, 5.562], [2.083, ], [.0397,.43]) T. The refinement by Algorithm 2 gives ([6.9925, 7.93], [4.34, ], [5.3799, ], [2.324, 2.237], [.0576,.244]) T. Notice that for this example the exact interval hull of the solution set Σ is known (Popovaand Krämer, 2008), ([7.070, 7.663], [4.93, ], [5.3952, 5.550], [2.392, ], [.064,.2]) T. Example 2. From Example 3.4 of Popova and Krämer (2008) we have ( ) ( ) p p 2 p2 + x = 3, p 2 p p 2 where p [ 2, ] and p 2 [3, 5]. Here, the Bauer Skeel enclosure gives ([0.282,.2052], [.403, ]) T, whereas the Hansen Bliek Rohn method yields ([ , ], [ 4.878, ]) T. No refinement for this very low dimensional example was successful. Example 3. The last example is devoted to numerical experiments with randomly generated data. Even though the real-life data are not random, such experiments reveal something on the performance of algorithms. The computations were carried out in MATLAB (R2008b) on a machine with an AMD Athlon 64 X2 Dual Core Processor 4400+, 2.2 GHz, CPU with 004 MB RAM. Interval arithmetics and some basic interval functions were provided by the interval toolbox INTLAB v5.3 (Rump, 2006). We used just a simple implementation of the methods presented. Notice, for large-scale problems in particular, that a more subtle implementation utilizing the sparsity of matrices A k, k =,...,K, could be used. First, we consider systems with symmetric matrices that were generated in the following way. First, entries of A c were chosen randomly and independently in [ 0, 0] with uniform distribution, and then we set A c := A c +(A c ) T +0nI. The entries of the radius matrix A Δ are equal to R, wherer>0is a parameter. The right-hand side interval vector was chosen to be degenerate (zero width) with entries chosen randomly from [ 0, 0]. In diverse settings of the dimension n and the radius R we carried out sequences of 0 runs. The results are summarized in Table. We compare the resulting enclosures by relative sums of radii with respect to the Bauer Skeel bounds. That is, for a given enclosure w and the Bauer Skeel enclosure v,wedisplay wi Δ i= vi Δ i= On the average, the Bauer Skeel (BS) method gives tighter enclosures than the Hansen Bliek Rohn (HBR) one. The refinement is more conclusive for the latter than for the former..

69 570 M. Hladík Table. Symmetric systems with random data. n R relative sum of radii average execution time (in sec.) BS refined BS HBR refined HBR BS refined BS HBR refined HBR Second, we consider Toeplitz systems, i.e, systems with matrices A satisfying a i,j = a i+,j+, i, j =,...,n. Herein, A c i, and Ac,i, i =2,...,n,were chosen randomly in [ 0, 0], whereas A c, in [ 0 + 0n, 0 + 0n]. The entries of A Δ are equal to R. The right-hand side vector was again degenerate with entries selected randomly in [ 0, 0]. The results are displayed in Table 2. Third, we consider symmetric systems again generated in the same way as above. We compare the combination of the Bauer Skeel and Hansen Bliek Rohn methods (Remark ) with the interval Cholesky method (Alefeld and Mayer, 993; 2008). We implemented the basic version of the interval Cholesky method since the more sophisticated algorithm based on pivot tightening (Garloff, 200) is intractable, having the exponential complexity. Table 3 demonstrates that the proposed method is much more efficient than the interval Cholesky one. Even though the computing time is slightly better for the latter, the former yields a significantly tighter enclosure. Finally, we did some comparisons with the parametric solver by Popova (2004a; 2006b); see Table 4. Again, we considered symmetric interval systems. On the average, our approach is slightly better, and the refinement is more significantly better. 7. Concluding remarks Numerical experiments revealed that a generalization of the Bauer Skeel method is a competitive alternative to the Hansen Bliek Rohn method. It is best to use a combination of both to obtain a tight enclosure. As observed in the numerical experiments, the resulting (direct) algorithm is a competitive alternative to existing direct or iterative algorithms. Moreover, efficient refinements of both methods were proposed in order to compute tighter enclosures. As pointed out by one referee, the performance of this centered form approach is limited (cf. Neumaier and Pownuk, 2007). A non-centered form approach may lead to further improvements. Acknowledgment This paper was partially supported by the Czech Science Foundation Grant P403/2/947.

70 Enclosures for the solution set of parametric interval linear systems 57 Table 2. Toeplitz systems with random data. n R relative sum of radii average execution time (in sec.) BS refined BS HBR refined HBR BS refined BS HBR refined HBR References Alefeld, G., Kreinovich, V. and Mayer, G. (997). On the shape of the symmetric, persymmetric, and skew-symmetric solution set, SIAM Journal on Matrix Analysis and Applications 8(3): Alefeld, G., Kreinovich, V. and Mayer, G. (2003). On the solution sets of particular classes of linear interval systems, Journal of Computational and Applied Mathematics 52( 2): 5. Alefeld, G. and Mayer, G. (993). The Cholesky method for interval data, Linear Algebra and Its Applications 94: Alefeld, G. and Mayer, G. (2008). New criteria for the feasibility of the Cholesky method with interval data, SIAM Journal on Matrix Analysis and Applications 30(4): Beeck, H. (975). Zur Problematik der Hüllenbestimmung von Intervallgleichungssystem en, in K. Nickel (Ed.), Interval Mathematics: Proceedings of the International Symposium on Interval Mathematics, Lecture Notes in Computer Science, Vol. 29, Springer, Berlin, pp Busłowicz, M. (200). Robust stability of positive continuoustime linear systems with delays, International Journal of Applied Mathematics and Computer Science 20(4): , DOI: /v Fiedler, M., Nedoma, J., Ramík, J., Rohn, J. and Zimmermann, K. (2006). Linear Optimization Problems with Inexact Data, Springer, New York, NY. Garloff, J. (200). Pivot tightening for the interval Cholesky method, Proceedings in Applied Mathematics and Mechanics 0(): Hladík, M. (2008). Description of symmetric and skewsymmetric solution set, SIAM Journal on Matrix Analysis and Applications 30(2): Horn, R.A. and Johnson, C.R. (985). Matrix Analysis, Cambridge University Press, Cambridge. Jansson, C. (99). Interval linear systems with symmetric matrices, skew-symmetric matrices and dependencies in the right hand side, Computing 46(3): Kolev, L.V. (2004). A method for outer interval solution of linear parametric systems, Reliable Computing 0(3): Kolev, L.V. (2006). Improvement of a direct method for outer solution of linear parametric systems, Reliable Computing 2(3):

71 572 M. Hladík Table 3. Comparison with the interval Cholesky method. n R relative sum of radii average exec. time (in sec.) HBR BS Cholesky HBR BS Cholesky Merlet, J.-P. (2009). Interval analysis for certified numerical solution of problems in robotics, International Journal of Applied Mathematics and Computer Science 9(3): , DOI: /v Meyer, C.D. (2000). Matrix Analysis and Applied Linear Algebra, SIAM, Philadelphia, PA. Neumaier, A. (990). Interval Methods for Systems of Equations, Cambridge University Press, Cambridge. Neumaier, A. (999). A simple derivation of the Hansen Bliek Rohn Ning Kearfott enclosure for linear interval equations, Reliable Computing 5(2): Neumaier, A. and Pownuk, A. (2007). Linear systems with large uncertainties, with applications to truss structures, Reliable Computing 3(2): Ning, S. and Kearfott, R.B. (997). A comparison of some methods for solving linear interval equations, SIAM Journal on Numerical Analysis 34(4): Padberg, M. (999). Linear Optimization and Extensions, 2nd Edn., Springer, Berlin. Popova, E. (2002). Quality of the solution sets of parameterdependent interval linear systems, Zeitschrift für Angewandte Mathematik und Mechanik 82(0): Popova, E.D. (200). On the solution of parametrised linear systems, in W. Krämer and J.W. von Gudenberg (Eds.), Scientific Computing, Validated Numerics, Interval Methods, Kluwer, London, pp Popova, E.D. (2004a). Parametric interval linear solver, Numerical Algorithms 37( 4): Popova, E.D. (2004b). Strong regularity of parametric interval matrices, ini. Dimovski (Ed.), Mathematics and Education in Mathematics, Proceedings of the 33rd Spring Conference of the Union of Bulgarian Mathematicians, Borovets, Bulgaria, BAS, Sofia, pp Popova, E.D. (2006a). Computer-assisted proofs in solving linear parametric problems, 2th GAMM/IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics, SCAN 2006, Duisburg, Germany, p.35. Popova, E.D. (2006b). Webcomputing service framework, International Journal Information Theories & Applications 3(3): Popova, E.D. (2009). Explicit characterization of a class of parametric solution sets, Comptes Rendus de L Academie Bulgare des Sciences 62(0): Popova, E.D. and Krämer, W. (2007). Inner and outer bounds for the solution set of parametric linear systems, Journal of Computational and Applied Mathematics 99(2): Popova, E.D. and Krämer, W. (2008). Visualizing parametric solution sets, BIT Numerical Mathematics 48(): Rex, G. and Rohn, J. (998). Sufficient conditions for regularity and singularity of interval matrices, SIAM Journal on Matrix Analysis and Applications 20(2): Rohn, J. (989). Systems of linear interval equations, Linear Algebra and Its Applications 26(C): Rohn, J. (993). Cheap and tight bounds: The recent result by E. Hansen can be made more efficient, Interval Computations (4): 3 2. Rohn, J. (2004). A method for handling dependent data in interval linear systems, Technical Report

72 Enclosures for the solution set of parametric interval linear systems 573 Table 4. Comparison with the parametric solver by Popova. n R relative sum of radii HBR-BS refined HBR-BS parametric solver , Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, rohn/publist/ rp9.ps. Rohn, J. (200). An improvement of the Bauer Skeel bounds, Technical Report V-065, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, rohn/publist/ bauerskeel.pdf. Rump, S.M. (983). Solving algebraic problems with high accuracy, in U. Kulisch and W. Miranker (Eds.), ANewApproach to Scientific Computation, Academic Press, New York, NY, pp Rump, S.M. (994). Verification methods for dense and sparse systems of equations, in J. Herzberger (Ed.), Topics in Validated Computations, Studies in Computational Mathematics, Elsevier, Amsterdam, pp Rump, S.M. (2006). INTLAB Interval Laboratory, the Matlab toolbox for verified computations, Version intlab/. Rump, S.M. (200). Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica 9: Schrijver, A. (998). Theory of Linear and Integer Programming, Reprint Edn., Wiley, Chichester. Skalna, I. (2006). A method for outer interval solution of systems of linear equations depending linearly on interval parameters, Reliable Computing 2(2): Skalna, I. (2008). On checking the monotonicity of parametric interval solution of linear structural systems, in R. Wyrzykowski, J. Dangarra, K. Karczewski and J. Wasniewski (Eds.), Parallel Processing and Applied Mathematics, Lecture Notes in Computer Science, Vol. 4967, Springer-Verlag, Berlin/Heidelberg, pp Stewart, G. W. (998). Matrix Algorithms, Vol. : Basic Decompositions, SIAM, Philadelphia, PA. Milan Hladík received his Ph.D. degree in operations research from Charles University in Prague in In 2008, he worked in the Coprin team at INRIA, Sophia Antipolis, France, as a postdoc research fellow. Since 2009, he has been an assistant professor at the Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University in Prague. His research interests are interval analysis, parametric programming and uncertain optimization. Appendix Consider a system of interval linear equations Ax = b, which is a special case of (), and the solution set ˆΣ := {x R n Ax = b, A A, b b}. The Bauer Skeel bounds (Rohn, 200; Stewart, 998) and the Hansen Bliek Rohn bounds (Fiedler et al., 2006; Rohn, 993, Theorem 2.39) on ˆΣ are given below. Theorem 6. (Bauer Skeel) Let A c nonsingular and ρ( (A c ) A Δ ) <. Write ˆx := (A c ) b c, ˆM := (A c ) A Δ and ˆM := (I ˆM). Then for each x ˆΣ we have x ˆx ˆM (A c ) (A Δ ˆx + b Δ ), x ˆx + ˆM (A c ) (A Δ ˆx + b Δ ). Theorem 7. (Hansen Bliek Rohn) Under the same assumption and notation as in the previous theorem, we have { x i max ˆx 0 i +(ˆx i ˆx i )ˆm ii, 0 (ˆx 2ˆm ii i +(ˆx ) } i ˆx i )ˆm ii,

73 574 M. Hladík and x i min { ˆx 0 i +(ˆx i + ˆx i )ˆm ii, ( ˆx 0 2ˆm ii i +(ˆx i + ) } ˆx i )ˆm ii, where ˆx 0 := ˆM ( ˆx + (A c ) b Δ ). Received: July 20 Revised: 24 November 20

74 Soft Comput (203) 7: DOI 0.007/s FOCUS Outer enclosures to the parametric AE solution set Evgenija D. Popova Milan Hladík Published online: 9 February 203 Ó European Union 203 Abstract We consider systems of linear equations, where the elements of the matrix and of the right-hand side vector are linear functions of interval parameters. We study parametric AE solution sets, which are defined by universally and existentially quantified parameters, and the former precede the latter. Based on a recently obtained explicit description of such solution sets, we present three approaches for obtaining outer estimations of parametric AE solution sets. The first approach intersects inclusions of parametric united solution sets for all combinations of the end-points of the universally quantified parameters. Polynomially computable outer bounds for parametric AE solution sets are obtained by parametric AE generalization of a single-step Bauer Skeel method. In the special case of parametric tolerable solution sets, we derive an enclosure based on linear programming approach; this enclosure is optimal under some assumption. The application of these approaches to parametric tolerable and controllable solution sets is discussed. Numerical examples accompanied by graphic representations illustrate the solution sets and properties of the methods. Communicated by V. Kreinovich. E. D. Popova (&) Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 8, 3 Sofia, Bulgaria epopova@bio.bas.bg M. Hladík Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské nám. 25, 800 Prague, Czech Republic milan.hladik@matfyz.cz Keywords Linear systems Dependent data AE solution set Tolerable solution set Controllable solution set Introduction Consider a system of linear algebraic equations AðpÞx ¼ bðpþ which has a linear uncertainty structure AðpÞ ¼A 0 þ XK k¼ A k p k ; bðpþ ¼b 0 þ XK k¼ b k p k ; ðþ ð2þ where A k 2 R nn ; b k 2 R n ; k ¼ 0; ;...; K; and p ¼ ðp ;...; p K Þ: The parameters p k ; k 2K¼f;...; Kg are considered as uncertain and varying within given intervals p k ¼½p k ; p k Š: Such systems are common in many engineering analysis or design problems (see Elishakoff and Ohsaki (200) and the references therein), control engineering (Matcovschi and Pastravanu 2007; Sokolova and Kuzmina 2008; Busłowicz 200), robust Monte Carlo simulations (Lagoa and Barmish 2002), etc. Usually, the set of solutions to () (2) which is sought for is the so-called parametric united solution set R p uni ¼ R uniðaðpþ; bðpþ; pþ :¼ fx 2 R n jð9p2pþðaðpþx ¼ bðpþþg: The united parametric solution set generalizes the united non-parametric solution set to an interval linear system Ax ¼ b; which is defined R uni ¼ R uni ða; bþ :¼ fx 2 R n jð9a2aþð9b 2 bþðax ¼ bþg: 23

75 404 E. D. Popova, M. Hladík However, the solutions of many practical problems involving uncertain (interval) data have quantified formulation involving the universal quantifier (V) besides the existential quantifier (A). We consider quantified solution sets where all universally quantified parameters precede all existentially quantified ones. Such solution sets are called AE solution sets, after Shary (2002). Examples of several mathematical and engineering problems formulated in terms of quantified solution sets can be found, for example, in Shary (2002); Pivkina and Kreinovich (2006); Wang (2008). AE solution sets are of particular interest also for interval-valued fuzzy relational equations, see Wang et al. (2003) where the concepts of so-called tolerable and controllable solution sets of interval-valued fuzzy relational equations are introduced, their structure and relations are discussed. The literature on control engineering contains many papers that explore problems related to linear dynamical systems with uncertainties bounded by interval matrices, see, e.g., Sokolova and Kuzmina (2008) and the references in Matcovschi and Pastravanu (2007); Busłowicz (200). The tolerable solution set is utilized in Sokolova and Kuzmina (2008) for parameter identification problems and in controllability analysis. As in the other problem domains, the uncertain data involve more parameter dependencies than in an interval matrix with independent elements. So, the more realistic approaches consider linear dynamical systems with linear dependencies between state parameters as in Matcovschi and Pastravanu (2007), or structural perturbations of state matrices as in Busłowicz (200). Although the non-parametric AE solution sets are studied, e.g., in Shary (995); Shary (997); Shary (2002); Goldsztejn (2005); Goldsztejn and Chabert (2006); Pivkina and Kreinovich (2006), there are a few results on the more general case of linear parameter dependency. A special case of parametric tolerable solution sets is dealt with in Sharaya and Shary (20). A characterization of the general parametric solution set is given in Popova and Krämer (20), and a Fourier Motzkin type elimination of parameters is applied in Popova (202) to derive explicit descriptions of the parametric AE solution sets. In this paper we are interested in obtaining outer bounds for the parametric AE solution sets. To our knowledge, this is the first systematic approach to outer estimations of parametric AE solution sets in their general form. In Sect. 3 we prove that (inner or outer) estimations of parametric AE solution sets can be obtained by using only some corresponding estimations of parametric united solution sets. In Sect. 4 we generalize a Bauer Skeel method (see Rohn (200) and the references therein), applied so far for bounding (parametric) united solution sets. The method is derived in a form which leads to the same conclusion, proven in Sect. 3, and requires intersecting the bounds of parametric united solution sets for all combinations of the end-points of the universally quantified parameters. Another, single-step single-application parametric Bauer Skeel AE method is derived in Sect. 5 and both approaches are compared on several numerical examples. The derivation of both forms of Bauer Skeel parametric AE method is self-contained and no knowledge of the original method is required. The special cases of parametric tolerable and controllable solution sets are discussed. In the tolerable case, an enclosure based on linear programming approach is derived in Sect. 6 Numerical examples accompanied by graphic representations illustrate the solution sets and properties of the methods. 2 Notations Denote by R n ; R nm the set of real vectors with n components and the set of real n 9 m matrices, respectively. A real compact interval is a ¼½a; aš :¼ fa 2 R j a a ag: As a generalization of real compact intervals, an interval matrix a with independent components is defined as a family A ¼½A; AŠ :¼ fa 2 R nm j A A Ag; where A; A 2 R nm ; A A; are given matrices. Similarly we define interval vectors. By IR n ; IR nm we denote the sets of interval n-vectors and interval n 9 m matrices, respectively. For a 2 IR; define mid-point a c :¼ðaþaÞ=2 and radius a D :¼ða aþ=2: These functions are applied to interval vectors and matrices componentwise. Without loss of generality and in order to have a unique representation (2), we assume that p D k [ 0 for all k 2K: The spectral radius of a matrix M is denoted by q(m). The identity matrix of dimension n is denoted by I. For a given index set I ¼ fi ;...; i k g denote p I :¼ðp i ;...; p ik Þ: Next, Card(S) denotes the cardinality of a set S. The following definitions are recalled from Popova and Krämer (20). Definition A parameter p k,b k B K, isofst class if it occurs in only one equation of the system (). Definition 2 A parameter p k ; k 2K; is of 2nd class if it is involved in more than one equation of the system (). Let E and A be two disjoint sets such that E[A¼K: The parametric AE solution set is defined as R p AE ¼ R AEðAðpÞ; bðpþ; pþ :¼ fx 2 R n jð8p A 2 p A Þð9p E 2 p E ÞðAðpÞx ¼ bðpþþg: Beside the parametric united solution set, there are several other special cases of AE solutions: 23

76 Outer enclosures to the parametric AE solution set 405 A parametric tolerable solution set is such that universal quantifiers concern only the constraint matrix and existential quantifiers only the right-hand side. That is, A k = 0 for every k 2Eand b k = 0 for every k 2A: The parametric tolerable solution set is R p tol ¼ R AEðAðp A Þ; bðp E Þ; pþ :¼ fx 2 R n jð8p A 2 p A Þ; ð9p E 2 p E Þ ðaðp A Þx ¼ bðp E ÞÞg: In contrast to the tolerable solutions, a parametric controllable solution set is such that existential quantifiers concern only the constraint matrix and universal quantifiers only the right-hand side. Thus, A k = 0 for every k 2A and b k = 0 for every k 2E: The parametric controllable solution set is denoted shortly by R p cont : R p cont ¼ R AEðAðp E Þ; bðp A Þ; pþ :¼ fx 2 R n jð8p A 2 p A Þ; ð9p E 2 p E Þ ðaðp E Þx ¼ bðp A ÞÞg: For a given parametric system and index sets A; E; there is a unique non-parametric system, resp. non-parametric AE solution set RðAðp A ; p E Þ; bðp A ; p E ÞÞ; where Aðp A ; p E Þ :¼ A 0 þ X A k p k þ X A k p k ; k2a k2e bðp A ; p E Þ :¼ b 0 þ X b k p k þ X b k p k : k2a k2e On the other hand, every non-parametric system, resp. nonparametric AE solution set, can be considered as a parametric system, resp. parametric AE solution set, involving n 2? n quantified parameters. Thus, every non-parametric AE solution set presents a special case of parametric AE solution set involving n 2? n quantified parameters. For a nonempty and bounded set SR n ; define its interval hull by hs :¼ T fy 2 IR n jsyg: For two intervals u; v 2 IR; u v; the percentage by which v overestimates u is defined by Oðu; vþ :¼ 00ð u D =v D Þ: In Popova and Krämer (20), it was shown that every x 2 R p AE satisfies the following inequality jaðp c Þx bðp c Þj X ja k x b k jp D k k2e X ja k x b k jp D k : ð3þ k2a Moreover, for parametric systems involving only st class existentially quantified parameters, this system of nonlinear inequalities describes exactly the set R p AE : 3 End-point bounds for R p AE It follows from the explicit representation of the parametric AE solution sets (Popova 202) that the interval hull of R p AE is attained at particular end-points of the intervals for the st class existentially quantified parameters and the universally quantified parameters. Here we exploit this property to develop a methodology for obtaining outer bounds of the parametric AE solution set using only solvers for bounding parametric united solution sets. For a given index set I ¼fi ;...; i k g; define B I :¼fðp c i þ d i p D i ;...; p c i k þ d ik p D i k Þjd ;...; d k 2fgg: Theorem R p AE ¼ \ We have ~p A 2B A RðAð~p A ; p E Þ; bð~p A ; p E Þ; p E Þ: ð4þ Proof It follows from the set-theoretic representation of R p AE (see (Popova and Krämer 20, heorem 3.)) that R p AE ¼ \ RðAð~p A ; p E Þ; bð~p A ; p E Þ; p E Þ: ~p A 2p A Then, the assertion of the theorem follows from the relation ð8p 2½pŠ : b f ðpþb 2 Þ, b min f ðpþ ^ max f ðpþb 2 p2½pš p2½pš ð5þ and because the polynomials involved in the explicit description of RðAðp A ; p E Þ; bðp A ; p E Þ; p E Þ are linear with respect to all V-parameters. h The next theorem gives a sufficient condition for a nonempty parametric AE solution set to be bounded. Theorem 2 Let R p AE be non-empty and for some ~p A 2B A the matrix Að~p A ; p E Þ be regular for all p E 2 p E : Then R p AE is bounded. Proof R p AE is not empty iff the intersection in (4) is not empty. If for some ~p A 2B A the matrix Að~p A ; p E Þ is regular for all p E 2 p E ; then RðAð~p A ; p E Þ; bð~p A ; p E Þ; p E Þ is bounded and its intersection (which is not empty) with bounded or unbounded solution sets for the remaining p A 2B A will be bounded. h By Theorem, one can obtain (inner or outer) estimations of a bounded parametric AE solution set by intersecting at most CardðB A Þ corresponding estimations of the united parametric solution sets RðAð~p A ; p E Þ; bð~p A ; p E ÞÞ; ~p A 2B A : In particular, we have 23

77 406 E. D. Popova, M. Hladík Corollary For a bounded parametric AE solution set R p AE 6¼; and a set B0 A ; such that B0 A B A and RðAð~p A ; p E Þ; bð~p A ; p E Þ; p E Þ is bounded for all ~p A 2B 0 A ; we have hr p AE \ hrðað~p A ; p E Þ; bð~p A ; p E Þ; p E Þ: ~p A 2B 0 A If the parametric system involves some st class A- parameters p k ; k 2E; we can further sharpen the estimation of the parametric AE solution set. Denote by E ; E E; the set of indices of all A-parameters which occur in only one equation of the system. Since inf/sup of RðAð~p A ; p E Þ; bð~p A ; p E Þ; p E Þ is attained at particular endpoints of p E ; we have R p AE ¼ \ [ R A;E;E ; ~p E 2B E and ~p A 2B 0 A hr p AE \ where ~p A 2B 0 A 0 [ h R A;E;E A; ð6þ ~p E 2B E R A;E;E :¼ RðAð~p A ; ~p E ; p EnE Þ; bð~p A ; ~p E ; p EnE Þ; p EnE Þ: By a methodology based on solving derivative systems with respect to every parameter (Popova 2006) one can prove that the interval hull of a united parametric solution set can be attained at particular end-points of the parameters, which are not only of st class. The parameters, for which we can prove this property, can be joined to the set E in relation (6). 4 Bauer Skeel method for parametric AE solution sets Bauer Skeel bounds were used to enclose bounded and connected non-parametric united solution sets (Stewart 998; Rohn 200) and later bounded and connected parametric united solution sets (Skalna 2006; Hladík 202). In this section, we extend the Bauer Skeel method to the case of non-empty bounded and connected parametric AE solution sets. Since the following is a generalization of the Bauer Skeel theorem, we do not state the original one explicitly. Theorem 3 For a fixed ~p A 2B A in the form of ~p k ¼ p c k þ ~ d k p D k ; ~ d k 2fg; k 2A; suppose that Að~p A ; p c E Þ be regular and define C :¼ Aðp c Þþ X k2a ~ dk A k p D k! ¼ A ð~p A ; p c E Þ; x :¼ C bðp c Þþ X! ~ dk b k p D k ¼ Cbð~p A ; p c E Þ; k2a M :¼ X jca k jp D k : k2e If q(m) \, then every x 2 R p AE satisfies jx x jði MÞ X jcða k x b k Þjp D k : k2e Proof We precondition () byc, so(2) reads jcaðp c Þx Cbðp c Þj X jcða k x b k Þjp D k k2e X jcða k x b k Þjp D k ; k2a that is jcaðp c Þx Cbðp c Þj þ X jcða k x b k Þjp D k k2a X jcða k x b k Þjp D k : k2e Since u? v C u ± v, we have CAðp c Þx Cbðp c Þþ X ~ dk CðA k x b k Þp D k k2a X jcða k x b k Þjp D k : k2e Rearranging we get C Aðp c Þþ X! dk A k p k2a~ D k x Cbðp c Þþ X! ~ dk b k p D k k2a or, X jcða k x b k Þjp D k ; k2e jx x j X jcða k x b k Þjp D k : k2e Now, we approximate the right-hand side from above jx x j X jcða k x b k Þjp D k k2e X jca k ðx x Þjp D k þ X jcða k x b k Þjp D k k2e k2e X jca k jjx x jp D k þ X jcða k x b k Þjp D k k2e k2e ¼ Mjx x jþ X k2e jcða k x b k Þjp D k : ð7þ ð8þ 23

78 Outer enclosures to the parametric AE solution set 407 Hence ði MÞjx x j X jcða k x b k Þjp D k : k2e Since M C 0 and q(m) \, we have (I - M) - C 0 and jx x jði MÞ X k2e jcða k x b k Þjp D k : The application of Theorem 3 to the special case of parametric united solution set has the following form, which is identical with the Bauer Skeel method generalized to parametric united solution sets in Skalna 2006; Hladík 202. Corollary 2 Let A(p c ) be regular and define C :¼ A ðp c Þ; x :¼ Cbðp c Þ; M :¼ X jca k jp D k : k2e If q(m) \, then every x 2 R p uni satisfies jx x jði MÞ X jcða k x b k Þjp D k : k2e In the special case of parametric tolerable solution set we have Corollary 3 and define For a fixed ~p A 2B A let Að~p A Þ be regular C :¼ Aðp c Þþ X k2a ~ dk A k p D k! ¼ A ð~p A Þ; x :¼ Cbðp c Þ¼Cbðp c E Þ: Then every x 2 R p tol satisfies jx x j X jcb k jp D k : k2e The special case of parametric controllable solution set is discussed thoroughly in Sect. 5. Corollary 4 The intersection of the solution enclosures obtained by Theorem 3 (respectively Corollary 3) for all ~p A 2B A is equal to the intersection of the solution enclosures obtained by Corollary 2 for all ~p A 2B A : Proof The proof follows immediately from the equivalent representation of C and x * presented in the formulation of Theorem 3. h Thus, the derivation of the parametric AE version of Bauer Skeel method confirms Corollary. h Corollary with using Bauer Skeel enclosures for the particular united solution sets gives the same result as the intersection of all enclosures by Theorem 3. However, Corollary with some other methods for enclosing parametric united solution sets may give better bounds. Example Let us consider the example from Popova and Krämer (20) p p þ p x ¼ 3 ; p 2 þ 2p 4 3p 2 þ where p ; p 2 2½0; Š and p 3 ; p 4 2½ ; Š: For the sake of simplicity, R 8p4 9p 23 denotes the parametric AE solution set where the universal quantifier is applied to p 4 and the existential one elsewhere. Similar notation is used for other combinations of quantifiers. In case of R 8p 9p 234 ; see Fig., h R 9p234 ðaðp ÞÞ \ R 9p234 ðaðp ÞÞ ¼ hr 9p234 ðaðp ÞÞ: That is why, enclosing sharply hr 9p234 ðaðp ÞÞ ¼ ð½ 2; 3Š; ½ ; ŠÞ > ; we enclose hr 8p 9p 234 in a best way, although the set R 9p234 ðaðp ÞÞ is unbounded. The parametric Bauer Skeel method for R 9p234 ðaðp Þ; p 234 Þ gives the enclosure ([-/ 3, 3], [-, ]) T and the 25% overestimation of x is because the method cannot account well the rowdependencies in p 2. Therefore, applying (6), we compute h [ R 9p234 ðaðp ÞÞ ¼ ð½ 2; 3Š; ½ ; ŠÞ > : p 2 2f0;g For R 8p2 9p 34 we cannot obtain an enclosure since the assumption q(m) \ is not fulfilled. In case of R 8p4 9p 23 ; see Fig. 2, R 9p23 ðaðp 4 ÞÞ \ R 9p23 ðaðp 4 ÞÞ hr 9p23 ðaðp 4 ÞÞ: Since R 9p23 ðaðp 4 ÞÞ is unbounded, we cannot find hr 8p4 9p 23 and approximate the latter outwardly by hr 9p23 ðaðp 4 ÞÞ: Applying the Bauer Skeel method for parametric united solution sets, we obtain hr 9p23 ðað~p 4 Þ; p 23 Þ¼ ð½ 4:96; 4:4546Š; ½ 2:7203; 2:8742ŠÞ > : The overestimation is due to the row-dependencies in p and p 2. Therefore, applying (6), we compute h [ R 9p3 ðað~p ; ~p 2 ; p 3 ; p 4 ÞÞ ¼ ð½ 2; 3Š; ½ ; ŠÞ > ; ~p ;~p 2 2f0;g which is the interval hull of R 9p23 ðaðp 4 ÞÞ: 23

79 408 E. D. Popova, M. Hladík x 2 2 x 2 2 x 2 x x Fig. Solution sets R 8p9p 234 online) of the linear system from Example for p 2f; 0g (blue, green) and their intersection (yellow) (color figure x 2 2 x x x x x Fig. 2 Solution sets R 8p49p 23 of the linear system from Example for p 4 2f ; g (green, blue) and their intersection (red) (color figure online) Remark The formulation of Bauer Skeel method is in real arithmetic, therefore its implementation in floatingpoint arithmetic will not provide a guaranteed enclosure, especially for intervals with very small radii or ill-conditioned problems. All computations below based on Bauer Skeel method were done in rational arithmetic to avoid uncontrolled round-off errors. Instead of Bauer Skeel method for bounding a parametric united solution set one can use the parametric fixed-point iteration, see Popova and Krämer (2007), which provides guaranteed enclosures of comparable quality under the same requirement for strong regularity of the parametric matrix. In fact, most of the general-purpose methods for bounding a parametric united solution set require strong regularity of the parametric matrix. 5 Another form of the Bauer Skeel method Below, we derive another form of the parametric Bauer Skeel method under stronger assumptions. Theorem 4 Let A(p c ) be regular and define C :¼ A ðp c Þ; x :¼ Cbðp c Þ; M :¼ X jca k jp D k : k2k If q (M) \, then every x 2 R p AE satisfies X jx x jði MÞ jcða k x b k Þjp D k k2e X! jcða k x b k Þjp D k : k2a Proof Consider the preconditioned parametric system C AðpÞx ¼ C bðpþ: The characterization (3) for the preconditioned system reads jx x j¼jcaðp c Þx Cbðp c Þj X jcða k x b k Þjp D k k2e X jcða k x b k Þjp D k : k2a For the right-hand side of the above inequality, due to juj jvjju þ vjjujþjvj we have X jcða k x A k x þ A k x b k Þjp D k k2e X jcða k x A k x þ A k x b k Þjp D k k2a jx x j X jca k jp D k þ X jcða k x b k Þjp D k k2e k2e þjx x j X jca k jp D k X jcða k x b k Þjp D k ; k2a k2a 23

80 Outer enclosures to the parametric AE solution set 409 which implies I X! jca k jp D k jx x j k2k X jcða k x b k Þjp D k X jcða k x b k Þjp D k : k2e k2a Since M C 0 and q(m) \, we have (I - M) - C 0 and thus, the statement of the theorem. h In the special case of a united parametric solution set, Theorem 4 has the same form as Corollary. In the special case of a parametric tolerable solution set, Theorem 4 is the following. Corollary 5 Let Aðp c Þ¼Aðp c AÞ be regular and C :¼ A ðp c A Þ; x :¼ Cbðp c Þ¼Cbðp c E Þ; M :¼ X jca k jp D k : k2a If q(m) \, then every x 2 R p tol satisfies X jx x jði MÞ jcb k jp D k X! jca k x jp D k : k2e k2a In the special case of parametric controllable solution sets, Theorem 4 is the following. Corollary 6 C :¼ A ðp c E Þ; x :¼ Cbðp c Þ¼Cbðp c A Þ; Let Aðp c Þ¼Aðp c EÞ be regular and M :¼ X jca k jp D k : k2e If q(m) \, then every x 2 R p cont satisfies X jx x jði MÞ jca k x jp D k X! jcb k jp D k : k2e k2a The application of Corollary 4 requires strong regularity of the parametric matrix Að~p A ; p E Þ on the domain p E for some ~p A 2B A : Theorem 4 has a stronger requirement: strong regularity of Að~p A ; p E Þ on p E for all ~p A 2 p A ; resp. for all ~p A 2B A : Therefore Corollary, resp. Corollary 4, have a larger scope of applicability (and a bigger computational complexity) than Theorem 4. Let us compare the two approaches for bounding parametric tolerable and controllable solution sets. Example 2 Obtain outer enclosures of the parametric tolerable solution set for AðpÞ ¼ p p 2 þ q 2 ; bðqþ ¼ 2p 2 p þ q q 2 and p 2½0; Š; p 2 2½ 3 ; Š; q ; q 2 2½ ; 2Š: The exact interval hull of the parametric tolerable solution set is ð½ 2 5 ; 4 5 Š; ½ 2 3 ; 4 3 ŠÞ> : Applying Corollary 5 we obtain the enclosure ð½ 36:904; 37:555Š; ½ 24:80; 25:38ŠÞ > which overestimates the hull by more than 95%. The application of Corollary 4 yields the interval hull. The conservative enclosure of the tolerable solution set produced by Corollary 5 is natural. Since every parametric tolerable solution set is a convex polyhedron (Popova 202), its interval hull is attained at particular end-points of the parameters, which is the approach exploited by Corollary 4. Indeed, shrinking the interval for p 2 to ½ ; Š the overestimation produced by Theorem 4 is reduced to 45%, resp. 35%. On the contrary, when we enlarge the interval for p 2 the parametric matrix is no more strongly regular. While the application of Theorem 4 is not suitable for bounding parametric tolerable solution sets, this theorem gives a better enclosure for a parametric controllable solution set than the enclosure obtained by Corollary 4 (the intersection of the solution enclosures obtained by Theorem 3 for all ~p A 2B A ). Proposition Under the same assumptions, the enclosure of the parametric controllable solution set computed by Corollary 6 is a subset of the enclosure computed by Corollary 4. Proof For a fixed end-point of a fixed solution component, the intersection of the solution enclosures obtained by Theorem 3 for all ~p A 2B A is attained at a particular ~p A 2B A : Let us consider an upper bound attained at a particular ~p A 2B A : With the notations from Corollary 6, that particular right end-point of the Bauer Skeel enclosure by Theorem 3 is x þ C X ~d j b j p D j þði MÞ X CA k j2a k2e x þ C X ~d j b j p D p D j k : j2a We estimate the right end-point from below as x X jcb j jp D j þði MÞ X jca k x jp D k j2a k2e ði MÞ X X jca k jp D k jcb j jp D j k2e j2a ¼ x þði MÞ X jca k x jp D k k2e I þði MÞ M X jcb j jp D j : j2a 23

81 40 E. D. Popova, M. Hladík Using (I - M) - = I? (I - M) - M, we obtain X x þði MÞ jca k x jp D k X! jcb j jp D j ; k2e j2a which is the right end-point of the enclosure by Corollary 6. Similarly we prove a corresponding relation between the left end-points of the enclosures. h Example 3 Consider a parametric linear system where AðpÞ ¼ p p 2 ; bðqþ ¼ 2q p 2 p 2q and p 2½0; 2 Š; p 2 2½; 3 2 Š; q 2½; 3 2Š: The exact interval hull of the parametric controllable solution set is ð½2; 2=5Š; ½ 2; 6=5ŠÞ > ; see Fig. 3. Applying Corollary 4 we obtain the enclosure R p cont ð½:86; 2:902Š; ½ 2:286; 0:263ŠÞ> ; overestimating the components of the interval hull by more than 76 %, resp. 60 %. However, by Theorem 4 (Corollary 6), we obtain the enclosure R p cont ð½:7802; 2:8352Š; ½ 2:298; 0:857ŠÞ> ; and the overestimation is 62 %, resp. 4 %. Bauer Skeel method, in any of its forms, requires strong regularity of the parametric matrix. Strong regularity (in the present formulation q(m) \ or(i - M) - C 0) must x 2 x x Fig. 4 The parametric controllable solution set for the linear system from Example 4 be checked when implementing the method. Since it is a sufficient condition for a parametric matrix to be regular, Bauer Skeel method may fail for some regular matrices which are not strongly regular, see the next example. Example 4 Consider the parametric system from Example 3 with other domains for the parameters: p 2½ 2 ; 3 2 Š; p 2 2½0; Š and q 2½; 2Š: The parametric matrix is regular but not strongly regular. Therefore, by Theorem 4 (resp. Corollary 6), we cannot find outer bounds for the parametric controllable solution set which is connected and bounded, see Fig. 4, and has interval hull p ð½8=3; 2ð þ ffiffiffi 2 ÞŠ; ½0; 4ŠÞ > : Example 5 We look for the controllable solution set of the parametric system from Example 3, enlarging the domain for q to q 2½; 5 2Š: Although the parametric matrix is strongly regular on the domain for p, p 2, the inequality X jcða k x b k Þjp D k X jcða k x b k Þjp D k k2e k2a does not hold, which means that R cont ¼;: Thus, by Theorem 4 we not only compute enclosures of the controllable solution set, but also can sometimes detect emptiness. x -3 Fig. 3 The controllable solution set for the linear system from Example 3 represented as intersection of the united solution sets for q = (light gray) and q = 3/2 (dark gray) together with its interval hull and its enclosures obtained by Corollary 5 and Corollary 6 6 LP enclosure for the parametric tolerable solution set Besides the united solution set, tolerable solutions are the most studied AE solutions to interval linear systems. In the 23

82 Outer enclosures to the parametric AE solution set 4 non-parametric case, there are plenty of results, see Shary (995); Beaumont and Philippe (200); Shary (2002); Pivkina and Kreinovich (2006); Rohn (2006); Wang (2008), among others. The only generalization to a special class of parametric tolerable solution sets is found in Sharaya and Shary (20). Corollary 4 provides an enclosure to the tolerable solution set R p tol which is much sharper than the enclosure provided by Theorem 4. By a careful inspection of the characterization (2) we can derive a polyhedral approximation of R p tol : Propositon 2 For every x 2 R p tol there are y k 2 R n ; k 2A; such that Aðp c Þx þ X p D k yk X jb k jp D k þ bðpc Þ; k2a k2e ð9aþ Aðp c Þx þ X p D k yk X jb k jp D k bðpc Þ; k2a k2e ð9bþ A k x y k ; A k x y k ; 8k 2A: ð9cþ Moreover, for parametric systems involving only st class existentially quantified parameters, the x solutions to (6) form R p tol : Proof By (2), each x 2 R p tol satisfies jaðp c Þx bðp c Þj þ X ja k xjp D k X jb k jp D k ; k2a k2e or, Aðp c Þx þ X ja k xjp D k X jb k jp D k þ bðpc Þ; k2a k2e Aðp c Þx þ X ja k xjp D k X jb k jp D k bðpc Þ: k2a k2e Substituting y k : = A k x we get (6). The system (6) consists of linear inequalities with respect to x and y k s, so we can employ linear programming techniques to obtain lower and upper bounds for the components of x. Proposition 2 also shows that the parametric tolerable solution set is a convex polyhedron for parametric linear systems involving only st class parameters. This is in accordance with the results from Sharaya and Shary (20); Popova (202). Linear programming (LP) techniques are well studied for bounding non-parametric AE solution sets, see Beaumont and Philippe (200). Proposition 2 generalizes the LP approach for parametric tolerable solution sets and provides exact bounds when the involved A-parameters are only of st class. Recall that a parametric matrix A(p) is h row-independent if for every k ¼ ;...; K and every i ¼ ;...; n the following set has cardinality at most one: fj 2f;...; ng jða k Þ ij 6¼ 0g: Due to the equality relation in (Popova 202, eq. (5.3)), inner and outer inclusions of a tolerable solution set, where the matrix involves only row-independent parameters and the right-hand side vector involves only st class parameters, can be computed by methods for the non-parametric case. Therefore, Proposition 2 is particularly useful for linear systems involving row-dependent parameters in the matrix and right-hand side vector with independent components. By using a standard linear programming technique to calculate lower and upper bounds on x solutions of (6), we have to solve 2n linear programs, each of them with nð þ CardðAÞÞ variables and 2nð þ CardðAÞÞ constraints. For a non-parametric tolerable system ax ¼ b; this number is too conservative. The system (6) may be furhter reduced (Fiedler et al 2006; Rohn 986) and the interval hull of the tolerable solution set is determined by solving 2n linear programs, each of them with only 2n variables and 4n constraints. If we call Corollary to compute an enclosure and linear programming to calculate the subordinate interval hulls, then we have to solve 2n 2 CardðAÞ linear programs, each with n variables and 2n constraints. Example 6 Motivated by Example 5.2 in Popova (202), let A ðþ ðpþ ¼ p p 2 ; A ð2þ r r þ 2 ðrþ ¼ ; p 3 p þ 2r r þ A ð3þ s s þ q 2 ðsþ ¼ ; bðqþ ¼ ; 2s 2 s 2 þ q q 2 where p ; r; s ; s 2 2½0; Š; p 2 2½ 2 ; 3 2 Š; p 3 2½ 2; 0Š and q ; q 2 2½ ; 2Š: Relaxing the parametric dependencies in the interval systems A () (p)x = b(q), A (2) (r)x = b(q), and A (3) (s)x = b(q) we get a standard interval system ax ¼ b drawing ½0; Š ½ 2 ; 3 2 Š ½ 2; 0Š ½; 2Š x ¼ ½ ; 2Š ½ 3; 3Š Consider first the interval systems A () (p)x = b(q) and A ðþ ðpþx ¼ b: Applying Corollary 4 we obtain : R tol ða ðþ ðpþ; bðqþ; p; qþ ð½ 2 5 ; 4 5 Š; ½ 2 3 ; 4 3 ŠÞ> ; R tol ða ðþ ðpþ; b; pþ ð½ :67; :7Š; ½ 0:667; :334ŠÞ > : The two parametric AE solution sets and the corresponding enclosing boxes are presented on Fig. 5. Theorem 4 cannot be applied since the parametric matrix is not strongly 23

83 42 E. D. Popova, M. Hladík - regular. A linear programming approach based on Proposition 2 gives R tol ða ðþ ðpþ; bðqþ; p; qþ ð½ 0:4; 0:8Š; ½ ; :286ŠÞ > ; R tol ða ðþ ðpþ; b; pþ ð½ :3; :7Š; ½ ; :4ŠÞ > ; R tol ða; bþ ð½ :67; :625Š; ½ 0:667; :334ŠÞ > ; The LP enclosures to R tol ða ðþ ðpþ; bðqþ; p; qþ and R tol ða ðþ ðpþ; b; pþ are not optimal since the system involves a 2nd class existentially quantified parameter p. Since the matrix A () (p) involves only row-independent parameters, R tol ða ðþ ðpþ; b; pþ ¼R tol ða; bþ ð½ :67; :625Š; ½ 0:667; :334ŠÞ > ; which is the interval hull. Now, we consider the systems A (2) (r)x = b(q) and A ð2þ ðrþx ¼ b: For these systems Corollary 4 gives the exact interval hulls R tol ða ð2þ ðrþ; b; rþ ð½ :3; :7Š; ½ ; :4ŠÞ > ; R tol ða ð2þ ðrþ; bðqþ; r; qþ ð½ 0:4; 0:8Š; ½ ; :4ŠÞ > : Proposition 2 gives the same enclosures. For the parametric interval system A ð3þ ðsþx ¼ b; Corollary 4 yields the exact hull R tol ða ð3þ ðsþ; b; sþ ð½ :3; :7Š; ½ ; :4ŠÞ > : Since all parameters are of st class only, Proposition 2 gives the same result. 7 Conclusion x Fig. 5 The tolerable solution sets for the linear systems A () (p)x = b(q) (dark gray) and A ðþ ðpþx ¼ b (light gray) from Example 6 together with the enclosing boxes obtained by Corollary 4 This paper presents a first attempt to propose and investigate methods providing outer bounds for parametric AE solution sets. The methods are general ones they are.5 x applicable to linear systems involving arbitrary linear dependencies between interval parameters; the parametric AE solution sets may be defined so that A- and E-parameters are mixed in both sides of the equations. Being the most general, these methods are applicable to the special cases of non-parametric AE solution sets, in particular nonparametric tolerable or controllable solution sets. From a methodological point of view, the methods we consider are based on a simple (though not always complete) Oettli-Prager-type description (3) of the parametric AE solution sets. This allows us to obtain bounds for the parametric AE solution sets either by bounding only parametric united solution sets or by using only real arithmetic and the properties of classical interval arithmetic. This makes the main methodological and computational difference between the methodology employed in this paper and the methodology that is used so far for estimating nonparametric AE solution sets (Shary 995; Shary 997, 2002; Goldsztejn 2005; Goldsztejn and Chabert 2006), based on the arithmetic of proper and improper intervals (called Kaucher interval arithmetic). The methods we present here provide outer bounds for non-empty, connected and bounded parametric AE solution sets. The first approach intersects inclusions of parametric united solution sets for all combinations of the end-points of A-parameters. This approach has exponential computational complexity, however provides very sharp estimations of the AE solution sets, especially for tolerable solution sets and for general parametric AE solution sets when combined with sharp bounds for the linear E-parameters. The second method we discuss is a parametric AE generalization of the single-step Bauer Skeel method used so far for bounding parametric united solution sets. In the special cases of non-parametric (tolerable, controllable) AE solution sets, this new method expands the range of available methods for outer enclosures. However, while most of the known methods for enclosing nonparametric AE solution sets are based on Kaucher interval arithmetic, the present method is based on the classical interval arithmetic. Also, it is a direct method and therefore a fast one. Finally, for parametric tolerable solution sets, we proposed a linear programming based method, which utilizes a polyhedral approximation of the set. When each existentially quantified parameter is involved in only one equation of the system, this method yields the interval hull of the parametric AE solution set. We demonstrated that the approach intersecting enclosures of parametric united solution sets for all combinations of the end-points of A-parameters is applicable to a larger class of parametric AE solution sets compared to the parametric Bauer Skeel AE-method. Despite its computational complexity, the first approach may be more suitable for bounding tolerable solution set of large-scale parametric 23

84 Outer enclosures to the parametric AE solution set 43 systems if one exploits distributed computations and modern methods for solving large-scale point systems which do not invert the matrix. On the other hand, the parametric Bauer Skeel AE method provides better bounds for the parametric controllable solution sets. This method implies a simple necessary (sometimes and sufficient) condition for any parametric AE solution set to be non-empty. The present formulation of the parametric Bauer Skeel AE method is in real arithmetic, therefore its implementation in floating-point arithmetic will not provide a guaranteed enclosure, unless combined with suitably chosen directed rounding. A self-verified method, which corresponds to the present parametric Bauer Skeel AE method, and provides guaranteed outer bounds for nonempty connected and bounded parametric AE solution sets will be presented in a separate paper. The parametric Bauer Skeel AE method and the intersection of enclosures obtained by a self-verified method can be used for bounding only connected and bounded solution sets. However, the interval Gauss-Seidel method, where the interval division is extended to allow division by interval containing zero (Goldsztejn and Chabert 2006), can be used to enclose bounded disconnected solution sets. So, a parametric generalization of the Gauss-Seidel method, may be helpful sometimes. The parametric Bauer Skeel AE method and most of the general-purpose parametric self-verified methods do not provide sharp enclosures of the parametric united solution set when the system involves row-dependent parameters. A parametric generalization of the right preconditioning process, considered in Goldsztejn (2005) for non-parametric AE systems, may be also helpful. Searching best estimations of parametric AE solution sets one has to take into account the inclusion relations between such solution sets (Popova 202), and the properties of the methods. Acknowledgments This work was inspired by the discussions held during the Dagstuhl seminar 37 in Dagstuhl, Germany, September 20. E. Popova was partially supported by the Bulgarian National Science Fund under grant No. DO /2008. M. Hladík was partially supported by the Czech Science Foundation Grant P403/2/ 947. The authors thank the anonymous reviewers for the numerous remarks improving readability of the paper. References Beaumont O, Philippe B (200) Linear interval tolerance problem and linear programming techniques. Reliab Comput 7(6): Busłowicz M (200) Robust stability of positive conti-nuous-time linear systems with delays. Int J Appl Math Comput Sci 20(4): Elishakoff I, Ohsaki M (200) Optimization and anti-optimization of structures under uncertainty. Imperial College Press, London, p 424 Fiedler M, Nedoma J, Ramík J, Rohn J, Zimmermann K (2006) Linear optimization problems with inexact data. Springer, New York Goldsztejn A (2005) A right-preconditioning process for the formalalgebraic approach to inner and outer estimation of AE-solution sets. Reliab Comput (6): Goldsztejn A, Chabert G (2006) On the approximation of linear AEsolution sets. In: Post-proceedings of 2th GAMM IMACS International Symposion on scientific computing, computer arithmetic and validated numerics, IEEE Computer Society Press, Duisburg Hladík M (202) Enclosures for the solution set of parametric interval linear systems. Int J Appl Math Comp Sci 22(3): Lagoa C, Barmish B (2002) Distributionally robust monte carlo simulation: a tutorial survey. In: Proceedings of the 5th IFAC World Congress, IFAC, pp Matcovschi M, Pastravanu O (2007) Box-const-rained stabilization for parametric uncertain systems. In: Petre, E et al (eds) Proceedings of SINTES 3, Internat. Symposium on system theory, automation, robotics, computers, informatics, electronics and instrumentation, Craiova, pp Pivkina I, Kreinovich V (2006) Finding least expensive tolerance solutions and least expensive tolerance revisions: algorithms and computational complexity. Departmental technical reports (CS) 207, University of Texas at El Paso, utep.edu/cs_techrep/207 Popova ED (2006) Computer-assisted proofs in solving linear parametric problems. In: Post-proceedings of 2th GAMM IMACS International Symposion on scientific computing, computer arithmetic and validated numerics, IEEE Computer Society Press, Duisburg Popova ED (202) Explicit description of AE solution sets for parametric linear systems. SIAM J Matrix Anal Appl 33(4): Popova ED, Krämer W (2007) Inner and outer bounds for the solution set of parametric linear systems. J Comput Appl Math 99(2): Popova ED, Krämer W (20) Characterization of AE solution sets to a class of parametric linear systems. 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85 44 E. D. Popova, M. Hladík Sokolova S, Kuzmina E (2008) Dynamic properties of interval systems. In: SPIIRAS Proceedings, Nauka, 7, pp (in Russian) Stewart GW (998) Matrix algorithms. Basic decompositions, vol., SIAM, Philadelphia Wang Y (2008) Interpretable interval constraint solvers in semantic tolerance analysis. Comput-Aided Des Appl 5(5): Wang S et al (2003) Solution sets of interval-valued fuzzy relational equations. Fuzzy Optimization and Decision Making 2:

86 SIAM J. MATRIX ANAL. APPL. Vol. 3, No. 4, pp c 200 Society for Industrial and Applied Mathematics BOUNDS ON REAL EIGENVALUES AND SINGULAR VALUES OF INTERVAL MATRICES MILAN HLADÍK, DAVID DANEY, AND ELIAS TSIGARIDAS Abstract. We study bounds on real eigenvalues of interval matrices, and our aim is to develop fast computable formulae that produce as-sharp-as-possible bounds. We consider two cases: general and symmetric interval matrices. We focus on the latter case, since on the one hand such interval matrices have many applications in mechanics and engineering, and on the other hand many results from classical matrix analysis could be applied to them. We also provide bounds for the singular values of (generally nonsquare) interval matrices. Finally, we illustrate and compare the various approaches by a series of examples. Key words. interval matrix, interval analysis, real eigenvalue, eigenvalue bounds, singular value AMS subject classifications. 65G40, 65F5, 5A8 DOI. 0.37/ Introduction. Many real-life problems suffer from diverse uncertainties, for example, due to data measurement errors. Considering intervals instead of fixed real numbers is one possible way to tackle such uncertainties. In this paper, we study real eigenvalues of matrices, the entries of which vary simultaneously and independently inside some given intervals. The set of all possible eigenvalues forms a finite union of several compact real intervals (see Proposition 2.), and our aim is to compute as-sharp-as-possible bounds for these intervals. The problem of computing lower and upper bounds for the eigenvalue set is well studied; see, e.g., [3, 0, 7, 27, 28, 29, 30, 32]. In recent years some effort was made in developing and extending diverse inclusion sets for eigenvalues [8, 22] such as Gerschgorin discs or Cassini ovals. Even though such inclusion sets are more or less easy to compute and can be extended to interval matrices, the intervals that they produce are big overestimations of the actual ones. The interval eigenvalue problem has a lot of applications in the field of mechanics and engineering. Let us mention for instance automobile suspension systems [27], mass structures [26], vibrating systems [], principal component analysis [2], and robotics [5]. In many cases, the properties of a system are given by the eigenvalues (or singular values) of a Jacobian matrix. A modern approach is to consider that the parameters of this matrix vary in a set of continuous states; hence it is useful to consider this matrix as an interval matrix. The propagation of an interval representation of the parameters in the matrix allows us to bound the properties of the system over all its states. This is useful for designing a system, as well as to certify its performance. Our goal is to revise and improve the existing formulae for bounding eigenvalues of interval matrices. We focus on algorithms that are useful from a practical point Received by the editors March 26, 2009; accepted for publication (in revised form) by A. Frommer March 29, 200; published electronically June, Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské nám. 25, 800, Prague, Czech Republic (milan.hladik@matfyz.cz), and INRIA Sophia- Antipolis Méditerranée, 2004 route des Lucioles, BP 93, Sophia-Antipolis Cedex, France (Milan.Hladik@sophia.inria.fr). INRIA Sophia-Antipolis Méditerranée, 2004 route des Lucioles, BP 93, Sophia-Antipolis Cedex, France (David.Daney@sophia.inria.fr, Elias.Tsigaridas@sophia.inria.fr). 26 Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

87 BOUNDS ON EIGENVALUES OF INTERVAL MATRICES 27 of view, meaning that sometimes we sacrifice the accuracy of the results for speed. Nevertheless, the bounds that we derive are sharp enough for almost all practical purposes and are excellent candidates for initial estimates for various iterative algorithms [7]. We assume that the reader is familiar with the basics of interval arithmetic; otherwise we refer the reader to [2, 4, 24]. An interval matrix is defined as A := [A, A] ={A R m n ; A A A}, where A, A R m n, A A, are given matrices. By A c := (A + A), 2 A Δ := (A A), 2 we denote the midpoint and the radius of A, respectively. matrix is defined as A symmetric interval A S := {A A A = A T }. By an inner approximation of a set S we mean any subset of S, andbyanouter approximation of S we mean a set containing S as a subset. Our aim is to develop formulae for calculating an outer approximation of the eigenvalue set of a (general or symmetric) interval matrix. Moreover, the following notation will used throughout the paper: v = max{ v, v} magnitude (absolute value) of an interval v A magnitude (absolute value) of an interval matrix A, i.e., A ij = A ij diag(v) a diagonal matrix with entries v,...,v n Ax A p =max p x 0 x p matrix p-norm κ p (A) = A p A p condition number (in p-norm) σ max (A) maximal singular value of a matrix A ρ(a) spectral radius of a matrix A λ Re (A) real part of an eigenvalue of a matrix A λ Im (A) imaginary part of an eigenvalue of a matrix A The paper consists of two parts: the first is devoted to general interval matrices, and the second to symmetric interval matrices. Symmetry causes dependency between interval quantities, but on the other hand stronger theorems are applicable. Moreover, bounds of singular values of interval matrices could be obtained as corollaries. 2. Generalintervalmatrix.Let A be a square interval matrix, and let Λ:={λ R; Ax = λx, x 0,A A} be the set of all real eigenvalues of matrices in A. Proposition 2.. The set Λ is a finite union of compact real intervals. Proof. Suppose Λ ; otherwise we are done. Λ is bounded since λ max{ A 2 ; A A} for all λ Λ. To show the closedness consider a sequence λ i Λ, i =,..., converging to λ C. For every i there is a matrix A i A and a vector x i with x i 2 =suchthata i x i = λ i x i. Choose a subsequence {i ν }, ν =,...,such Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

88 28 MILAN HLADÍK, DAVID DANEY, AND ELIAS TSIGARIDAS that A iν converge to A A and x iν converge to x with Euclidean norm. Going to the limit ν we get Ax = λx showing λ Λ. The finiteness follows from Rohn in [29, Theorem 3.4]. It states that every boundary eigenvalue λ Λ is attained for a matrix A A, which is of the form A = A c diag(y) A Δ diag(z), where y, z {±} n. Therefore there are finitely many boundary eigenvalues in Λ and hence also intervals. Computation of the real eigenvalue set is considered a very difficult task. Even checking whether 0 Λ is an NP-hard problem since it is equivalent to checking regularity of the interval matrix A, which is known to be an NP-hard problem [25]. Therefore, we focus on a fast computation of the initial (hopefully sharp enough) outer approximation of Λ. For other approaches that estimate Λ, we refer the reader to [0, 27, 32]. Some methods do not calculate bounds for the real eigenvalues of A; instead they compute bounds for the real parts of the complex eigenvalues. Denote the set of all possible real parts by Λ r := {λ Re R; Ax = λx, x 0,A A}. As Λ Λ r, any outer approximation to Λ r works for Λ as well. Let us recall a method proposed in Rohn [30, Theorem 2] that we will improve in what follows. Theorem 2.2 (see [30]). Let Then Λ r λ 0 := [λ 0, λ 0 ],where S c := ( Ac + A T ) c, SΔ := 2 2 ( AΔ + A T ) Δ. λ 0 = λ min (S c ) ρ(s Δ ), λ 0 = λ max (S c )+ρ(s Δ ), and λ min (S c ), λ max (S c ) denote the minimal and maximal eigenvalue of S c, respectively. In most of the cases, the previous theorem provides a good estimation of the eigenvalue set Λ (cf. [7]). However, its main disadvantage is the fact that it produces nonempty estimations, even in the case where the eigenvalue set is empty. To overcome this drawback we propose an alternative approach that utilizes the Bauer Fike theorem [3, 8, 33]. Theorem 2.3 (see Bauer and Fike, 960). Let A, B R n n and suppose that A is diagonalizable, that is, V AV =diag(μ,...,μ n ) for some V C n n and μ,...,μ n C. For every (complex) eigenvalue λ of A + B, there exists an index i {,...,n} such that λ μ i κ p (V ) B p. For almost all practical cases the 2-norm seems to be the most suitable choice. In what follows we will use the previous theorem with p =2. Proposition 2.4. Let A c be diagonalizable, i.e., V A c V is diagonal for some V C n n.thenλ r ( n i= λ i), where for each i =,...,n, (2.) (2.2) λ i = λ Re i (κ2 (A c ) (V ) σ max(a Δ ) ) 2 λ Im i (A c ) 2, (κ2 (V ) σ max(a Δ ) ) 2 λ Im i (A c ) 2, λ i = λ Re i (A c )+ Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

89 BOUNDS ON EIGENVALUES OF INTERVAL MATRICES 29 provided that ( κ 2 (V ) σ max (A Δ ) ) 2 λ Im i (A c ) 2 ;otherwiseλ i = for i =,...,n. Proof. EveryA A can be written as A = A c + A,where A A Δ (where the inequality applies elementwise). The Bauer Fike theorem with 2-norm implies that for each complex eigenvalue λ(a) there is some complex eigenvalue λ i (A c ) such that λ(a) λ i (A c ) κ 2 (V ) A 2 = κ 2 (V ) σ max (A ). As A A Δ,wehaveσ max (A ) σ max (A Δ ). Hence λ(a) λ i (A c ) κ 2 (V ) σ max (A Δ ). Thus all complex eigenvalues of all matrices A A lie in the circles with centers in λ i (A c ) s with corresponding radii κ 2 (V ) σ max (A Δ ). The formulae (2.) (2.2) represent an intersection of these circles with the real axis. Notice that both a pair of complex conjugate eigenvalues λ i (A c )andλ j (A c ) yields the same interval λ i = λ j, so it suffices to consider only one of them. Proposition 2.4 is a very useful tool for estimating Λ in the case where the large complex eigenvalues of A c also have large imaginary parts. It is neither provably better nor provably worse than Rohn s theorem; see Example 2.8. Therefore it is advisable, in practice, to use both of them. Proposition 2.4 can be applied only if A c is diagonalizable. For the case where A c is defective we can build upon a generalization of the Bauer Fike theorem due to Chu [6, 7]. We present its special form. Theorem 2.5 (see [6]). Let A, B R n n and let V AV = J be the Jordan canonical form of A. Denotebyp the maximal dimension of the Jordan s blocks in J. Then for every (complex) eigenvalue λ of A + B, thereisi {,...,n} such that { λ λ i (A) max Θ 2, Θ p 2 where p(p +) Θ 2 = κ 2 (V ) B 2. 2 Proceeding in a manner similar to that in the proof of Proposition 2.4 we obtain the following general result for interval matrices. Proposition 2.6. Let V A c V = J be the Jordan canonical form of A c,andlet p be the maximal dimension of the Jordan s blocks in J. Denote p(p +) { } Θ 2 = κ 2 (V ) σ max (A Δ ), Θ=max Θ 2, Θ p 2. 2 Then Λ ( n i= λ i), where for each i =,...,n, λ i = λ Re i (A c ) Θ 2 λ Im i (A c ) 2, λ i = λ Re i (A c )+ Θ 2 λ Im i (A c ) 2, provided that Θ 2 λ Im i (A c ) 2 ;otherwiseλ i =. This result is applicable for any interval matrix A. In our experience, Rohn s bounds are usually more narrow when the input intervals of A are wide. On the other hand, this formula is better as long as the input intervals are narrow; cf. Example 2.9. }, Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

90 220 MILAN HLADÍK, DAVID DANEY, AND ELIAS TSIGARIDAS We present one more improvement for computing bounds of Λ, which is based on a theorem by Horn and Johnson [9]. Theorem 2.7. Let A R n n.then λ min ( A + A T 2 ) ( ) A + A λ Re T (A) λ max for every (complex) eigenvalue λ(a) of the matrix A. The theorem says that any upper or lower bound of the eigenvalue set of the symmetric interval matrix 2 (A + AT ) S is also a bound of Λ r. Symmetric interval matrices are studied in detail in section 3 and the results obtained there can be used here to bound Λ via Theorem 2.7. Note that in this way, Rohn s bounds from Theorem 3. yield the same bounds as those from Theorem 2.2. Note also that if the interval matrix A is pointed (i.e., A = A), then Theorems 2.2 and 2.7 yield the same range. In what follows we present two examples that utilize the bounds of the previous propositions. We should mention that the purpose of all the examples in the present paper is to illustrate the proposed bounds; hence no verified computations were carried out, as should always be the case for real-life applications. Example 2.8. Let [ 5, 4] [ 9, 8] [4, 5] [4.6, 5] [.2, ] [7, 8] [7, 8] [, 2] [4, 5] [0, ] A = [7, 7.2] [ 3.5, 2.7] [.9, 2.] [ 3, 2] [6, 6.4] [8, 9] [2, 3] [8, 9] [5, 6] [6, 7]. [3, 4] [8, 9] [9, 0] [ 8, 7] [0, ] Rohn s theorem provides the outer approximation Λ [ 22.04, ]. Now we utilize Proposition 2.4. The eigenvalues of A c are , 4.067, i, i, and , while κ 2 (V ) σ max (A Δ )= Hence λ =[ , ], λ 2 =[ , 4.526], λ 3 = λ 4 =, λ 5 =[2.327, 29.30]. The resulting outer approximation of Λ is a union of two intervals, i.e., [ , 4.526] [2.327, 29.30]. Proposition 2.6 yields the same result since the eigenvalues of A c are mutually different. If we take into account the results of all the methods, and we consider the intersection of the corresponding intervals, we obtain a sharper result, i.e., [ 22.04, 4.526] [2.327, 29.30]. To estimate the quality of the aforementioned results, it is worth noticing that the exact description of the real eigenvalue set of A could be obtained using the algorithm in [7], Λ=[ 7.56, ] [ ,.4582] [6.7804, ]. 2 Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

91 BOUNDS ON EIGENVALUES OF INTERVAL MATRICES 22 Example 2.9. Let A =[A c A Δ ; A c + A Δ ], where A c = , and all entries of A Δ equal ε. The eigenvalues of A c are ± 2 i (both are double). Let ε =0.0. Rohn s theorem leads to the outer approximation [.9445, ]. Proposition 2.4 is not applicable as A c is defective. Using Proposition 2.6 we calculate p =2andΘ=.062 and conclude that Λ =, i.e., no matrix A A has any real eigenvalue. For ε =, Rohn s outer approximation is [ , ], but Proposition 2.6 results in [ 05.02, 07.02]. 3. Symmetric interval matrix. Let A R n n be a real symmetric matrix. It has n real eigenvalues, which are in decreasing order (including multiplicities): λ (A) λ 2 (A) λ n (A). Let A S be a symmetric interval matrix and denote by λ i (A S ):= { λ i (A) A A S} the set of the ith eigenvalues. Each of these sets is a compact real interval; this is due to the continuity of the eigenvalue function and the compactness and convexity of A S [6]. It can happen that the sets λ i (A S )andλ j (A S ), where i j, overlap. Our aim is to compute as-sharp-as-possible bounds of the eigenvalue sets. The upper bound λ u i (AS ), i {,...,n}, is any real number satisfying λ u i (AS ) λ i (A S ). Lower bounds λ l i (AS )forλ i (A S ), i {,...,n}, can be computed as upper bounds of A S, so we omit their treatment. The symmetric case is very important for real-life applications as symmetric matrices appear very often in engineering problems. Under the concept of interval computations, symmetry induces dependencies between the matrix elements, which are hard to deal with in general. The straightforward approach would be to forget the dependencies and apply the methods from the previous section to obtain bounds on eigenvalues. Unfortunately, these bounds are far from sharp, since the loss of dependency implies a big overestimation on the computed intervals. We should mention that there are very few theoretical results concerning symmetric interval matrices. Let us mention only that computing all the exact boundary points of the eigenvalue set is not known. Such a result could be of extreme practical importance since it can be used for testing the accuracy of existing approximation algorithms. In this line of research, let us mention the work of Deif [0] and Hertz [5, 6]. The former provides an exact description of the eigenvalue set, but it works only under some not easily verified assumptions on sign pattern invariance of eigenvectors; the latter (see also [3]) proposes a formula for computing the exact extremal values λ (A S ), λ (A S ), λ n (A S ), and λ n (A S ), which consists of 2 n iterations. Theoretical results could also be found in the work of Qiu and Wang [28]. However, some results turned out to be incorrect [34]. Since the exact problem of computing the eigenvalue set(s) is a difficult one, several approximation algorithms were developed in recent years. An evolution strategy method by Yuan, He, and Leng [34] yields an inner approximation of the eigenvalue Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

92 222 MILAN HLADÍK, DAVID DANEY, AND ELIAS TSIGARIDAS set. By means of matrix perturbation theory, Qiu, Chen, and Elishakoff [26] proposed an algorithm for approximate bounds, and Leng and He [2] for outer approximation. Outer approximation was also presented by Beaumont [4]; he used a polyhedral approximation of eigenpairs and an iterative improvement. Kolev [20] developed an outer approximation algorithm for the general case with nonlinear dependencies. 3.. Basic bounds. The following theorem (without proof) appeared in [3]; to ensure that this paper is self-contained, we present its proof. Theorem 3.. It holds that λ i (A S ) [λ i (A c ) ρ(a Δ ),λ i (A c )+ρ(a Δ )]. Proof. By Weyl s theorem [3, 8, 23, 33], for any symmetric matrices B,C R n n it holds that λ i (B)+λ n (C) λ i (B + C) λ i (B)+λ (C) i =,...,n. Particularly, for every A A in the form of A = A c + A, A [ A Δ,A Δ ], we have λ i (A) =λ i (A c + A ) λ i (A c )+λ (A ) λ i (A c )+ρ(a ) i =,...,n. As A A Δ,wegetρ(A ) ρ(a Δ ), whence λ i (A) λ i (A c )+ρ(a Δ ). Working similarly, we can prove that λ i (A) λ i (A c ) ρ(a Δ ). The bounds obtained by the previous theorem are usually quite sharp. However, the main drawback is that all the produced intervals λ i (A S ), i n, havethe same width. The following proposition provides an upper bound for the largest eigenvalue of A S, i.e., an upper bound for the right endpoint of λ (A S ). Even though the formula is very simple and the bound is not very sharp, there are cases where it yields a better bound than the one obtained by Rohn s theorem. In particular it provides better bounds for nonnegative interval matrices and for interval matrices such as the ones we consider in subsection 3.3 with the form [ A Δ,A Δ ]. Proposition 3.2. It holds that λ (A S ) λ ( A ). Proof. Using the well-known Courant Fischer theorem [3, 8, 23, 33], we have for every A A λ (A) = max x T Ax max x T Ax x T x= x T x= max x T x= x T A x max x T x= x T A x = max x T A x = λ ( A ). x T x= In the same way we can compute a lower bound for the eigenvalue set of A: λ n (A S ) λ ( A ). However, this inequality is not so useful in practice. Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

93 BOUNDS ON EIGENVALUES OF INTERVAL MATRICES Interlacing approach, direct version. The approach that we propose in this section is based on Cauchy s interlacing property for eigenvalues of a symmetric matrix [3, 8, 23, 33]. Theorem 3.3 (interlacing property; see Cauchy, 829). Let A R n be a symmetric matrix, and let A i be a matrix obtained from A by removing the ith row and column. Then λ (A) λ (A i ) λ 2 (A) λ 2 (A i ) λ n (A i ) λ n (A). We develop two methods based on the interlacing property, the direct and the indirect one. These methods are useful as long as the intervals λ i (A S ), i =,...,n, do overlap, or as long as there is a narrow gap between them. Overlapping happens, for example, when there are multiple eigenvalues in A S. If none of the previous cases occur, then the bounds are not so sharp; see Example 3.6. The first method uses the interlacing property directly. Bounds on the eigenvalues of the principal minor A S i are also bounds on the eigenvalues of matrices in A S (except for λ (A S )andλ n (A S )). The basic idea is to compute the bounds recursively. However, such a recursive algorithm would be of exponential complexity. Therefore, we propose a simple local search approach that requires only a linear number of iterations and the results of which are quite satisfactory. It consists of selecting the most promising principal minor A i and recursively using only this. To obtain even better results in practice, we apply this procedure in the reverse order as well. (That is, we begin with some diagonal element a ii of A S, which is a matrix one-by-one, and iteratively increase its dimension until we obtain A S.) The algorithmic scheme is presented in Algorithm. We often need to compute an upper bound λ u (B S ) for the maximal eigenvalue of any matrix in B S (steps 3 and 2). For this purpose we can call Theorem 3. or Proposition 3.2, or, to obtain the best results, we choose the minimum of the two. Notice that the algorithm computes only upper bounds for λ i (A S ), i =,...,n. Lower bounds for λ i (A S ), i =,...,n,can be obtained by calling the algorithm using A S as input matrix. Algorithm. (interlacing approach for upper bounds, direct version) : B S := A S ; 2: for k =,...,n do 3: compute λ u (BS ); 4: λ u k (AS ):=λ u (BS ); 5: select the most promising index i {,...,n k +}; 6: remove the ith row and the ith column from B S ; 7: end for 8: put I = ; 9: for k =,...,n do 0: select the most promising index i {,...,n}\i, and put I := I {i}; : let B S be a submatrix of A S restricted to the rows and columns indexed by I; 2: compute λ u (BS ); 3: λ u n k+ (AS ):=min { λ u n k+ (AS ),λ u (BS ) } ; 4: end for 5: return λ u k (AS ), k =,...,n. An important ingredient of the algorithm is the selection of index i in steps 5 and 0. We describe the selection for step 5; for step 0 we work similarly. In essence, Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

94 224 MILAN HLADÍK, DAVID DANEY, AND ELIAS TSIGARIDAS there are two basic choices: (3.) i := arg min j=,...,n k+ λu (BS j ), and (3.2) i := arg min j=,...,n k+ r,s j B r,s 2. In both cases we select an index i in order to minimize λ (B S i ). The first formula requires more computation than the second but yields the optimal index in more cases than the second. The latter formula is based on the wellknown result [8, 33] that the square of the Frobenius norm of a normal matrix (i.e., the sum of squares of its entries) equals the sum of squares of its eigenvalues. Therefore, the most promising index is the one that maximizes the sum of squares of the absolute values (magnitudes) of the removed components. The selection rule (3.) causes a quadratic time complexity of Algorithm with respect to the number of calculations of spectral radii or eigenvalues. Using the selection rule (3.2) results in only a linear number of such calculations Interlacing approach, indirect version. The second method also uses the interlacing property and is based on the following idea. Every matrix A A S canbewrittenasa = A c + A δ with A δ [ A Δ,A Δ ] S. We compute the eigenvalues of the real matrix A c and bounds on eigenvalues of matrices in [ A Δ,A Δ ] S,andwe merge them to obtain bounds on eigenvalues of matrices in A S. For the merging step we use a theorem for perturbed eigenvalues. The algorithm is presented in Algorithm 2. It returns only upper bounds λ u i (AS ), i =,...,nfor λ i (A S ), i =,...,n, since lower bounds are likewise computable. The bounds required in step 2 are computed using Algorithm. The following theorem due to Weyl [8, 33] gives very nice formulae for the eigenvalues of a matrix sum, which we use in step 4 of Algorithm 2. Theorem 3.4 (Weyl, 92). Let A, B R n n be symmetric matrices. Then λ r+s (A + B) λ r (A)+λ s (B) r, s {,...,n}, r + s n +, λ r+s n (A + B) λ r (A)+λ s (B) r, s {,...,n}, r + s n +. Algorithm 2. (interlacing approach for upper bounds, indirect version) : Compute eigenvalues λ (A c ) λ n (A c ); 2: compute bounds λ u ( [ AΔ,A Δ ] S),...,λ u n( [ AΔ,A Δ ] S) ; 3: for k =,...,n do 4: λ u k (AS ):=min i=,...,k { λi (A c )+λ u k i+( [ AΔ,A Δ ] S)} ; 5: end for 6: return λ u k (AS ), k =,...,n Diagonal maximization. In this subsection we show that the largest eigenvalues are achieved when the diagonal entries of A A S are the maximum entries. Therefore, we can fix them and consider only a subset of A A S. Similar results can be obtained for the smallest eigenvalues. Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

95 BOUNDS ON EIGENVALUES OF INTERVAL MATRICES 225 Lemma 3.5. Let i {,...,n}. Then there is some matrix A A S with diagonal entries A j,j = A j,j such that λ i (A) =λ i (A S ). Proof. LetA A S be such that λ i (A )=λ i (A S ). Such a matrix always exists, since λ i (A S ) is defined as the maximum of a continuous function on a compact set. We define A A S as follows: A ij := A ij if i j, anda ij := A ij if i = j. By the Courant Fischer theorem [3, 8, 23, 33], we have λ i (A )= max min x T A x V R n ;dimv =i x V ; x T x= max min x T Ax V R n ;dimv =i x V ; x T x= = λ i (A). Hence λ i (A) =λ i (A) = λ i (A S ). This lemma implies that for computing upper bounds λ u i (AS )ofλ i (A S ), i =,...,n, it suffices to consider only the symmetric interval matrix A S r AS defined as A S r := {A AS A j,j = A j,j j =,...,n}. To this interval matrix we can apply all the algorithms developed in the previous subsections. The resulting bounds are sometimes sharper and sometimes not so sharp; see Examples So the best possible results are obtained by using all the methods together Singular values. Let A R m n and denote q := min{m, n}. Byσ (A) σ n (A) we denote the singular values of A. It is well known [3, 8, 23] that the singular values of A are identical with the q largest eigenvalues of the Jordan Wielandt matrix ( ) 0 A T, A 0 which is symmetric. Consider an interval matrix A R m n. By σ i (A) :={σ i (A) A A}, i =,...,q, we denote the singular value sets of A. The problem of approximating the singular value sets was considered, e.g., in [, 9]. Deif s method [9] produces exact singular value sets, but only under some assumptions that are generally difficult to verify. Ahn and Chen [] presented a method for calculating the largest possible singular value σ (A). It is a slight modification of [5] and its time complexity is exponential (2 m+n iterations). They also proposed a lower bound calculation for the smallest possible singular value σ n (A) by means of interval matrix inversion. To get an outer approximation of the singular value set of A we can exhibit the methods proposed in the previous subsections and apply them on the eigenvalue set of the symmetric interval matrix ( ) 0 A T S (3.3). A 0 Diagonal maximization (see subsection 3.4) has no effect, since the diagonal of the symmetric interval matrix (3.3) consists of zeros only. The other methods work well. Even though they run very fast, they can be accelerated a bit, as some of them can be slightly modified and used directly on A instead of (3.3). Particularly, Proposition 3.2 is easy to modify for singular values (σ (A) σ ( A )), and the interlacing property can be applied directly to A; cf.[3,8,9]. Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

96 226 MILAN HLADÍK, DAVID DANEY, AND ELIAS TSIGARIDAS 3.6. Case of study. The aim of the following examples is to show that no presented method is better than the others. In different situations, different variants are the best. Example 3.6. Consider the example given by Qiu, Chen, and Elishakoff [26] (see also [34]): [2975, 3025] [ 205, 985] 0 0 A S = [ 205, 985] [4965, 5035] [ 3020, 2980] 0 0 [ 3020, 2980] [6955, 7045] [ 4025, 3975] 0 0 [ 4025, 3975] [8945, 9055] Proposition 3.2 yields the upper bound λ u (AS ) = , which is by chance the optimal value. The other outer approximations of the eigenvalues sets λ i (A S ), i =,...,n, are listed below. The corresponding items are as follows: (R) bounds computed by Rohn s theorem (Theorem 3.) (D) bounds computed by Algorithm with index selection rule (3.) (D2) bounds computed by Algorithm with index selection rule (3.2) (I) bounds computed by Algorithm 2 with index selection rule (3.) (I2) bounds computed by Algorithm 2 with index selection rule (3.2) (DD) bounds computed by diagonal maximization by using Algorithm and index selection rule (3.) (DI) bounds computed by diagonal maximization by using Algorithm 2 and index selection rule (3.) (B) bounds obtained by using Theorem 3. and Algorithms and 2, and then choosing the best ones; the index selection rule is (3.) (O) optimal bounds; they are known provided that an inner and outer approximation (calculated or known from references) coincide; some of them are determined according to Hertz [5, 6] Table 3. Results for Example 3.6. [λ l (AS ),λ u (AS )] [λ l 2 (AS ),λ u 2 (AS )] [λ l 3 (AS ),λ u 3 (AS )] [λ l 4 (AS ),λ u 4 (AS )] (R) [ , ] [ , ] [ , ] [ , ] (D) [ , ] [ , ] [ , ] [ , ] (D2) [ , ] [ , ] [ , ] [ , ] (I) [ , ] [ , ] [ , ] [ , ] (I2) [ , ] [ , ] [ , ] [ , ] (DD) [ , ] [ , ] [ , ] [ , ] (DI) [ , ] [ , ] [ , ] [ , ] (B) [ , ] [ , ] [ , ] [ , ] (O) [ , ] [ , ] [ , ] [ , ] S. Table 3. shows that the direct interlacing methods (D), (D2), and (DD) are not effective; gaps between the eigenvalue sets λ i (A S ), i =,...,n,aretoowide. The indirect interlacing methods (I) and (I2) yield the same intervals as the Rohn method (R). The indirect interlacing method using diagonal maximization is several times better (e.g., for λ l 4(A S ), λ u 4 (A S )) and several times worse (e.g., for λ l (A S ), λ u (AS )) than (R). The combination (B) of all the methods produces good outer approximation of the eigenvalue set, particularly for that of λ (A S ). Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

97 BOUNDS ON EIGENVALUES OF INTERVAL MATRICES 227 For this example, Qiu, Chen, and Elishakoff [26] obtained the approximate values λ (A S ) , λ (A S ) , λ 2 (A S ) , λ 2 (A S ) , λ 3 (A S ) , λ 3 (A S ) , λ 4 (A S ) , λ 4 (A S ) However, these values form neither inner nor outer approximations of the eigenvalue set. The method of Leng and He [2] based on matrix perturbation theory results in the following bounds: λ l (AS ) = , λ u (AS ) = , λ l 2 (AS ) = , λ u 2 (AS ) = , λ l 3 (AS ) = , λ u 3 (AS ) = , λ l 4 (AS ) = 85.65, λ u 4 (AS ) = In comparison to (B), they are not so sharp. The evolution strategy method proposed by Yuan, He, and Leng [34] returns an inner approximation of the eigenvalue set, which is equal to the optimal result (see (O) in Table 3.) in this example. Example 3.7. Consider the symmetric interval matrix [0, 2] [ 7, 3] [ 2, 2] A S = [ 7, 3] [4, 8] [ 3, 5] [ 2, 2] [ 3, 5] [, 5] Following the notation used in Example 3.6 we display in Table 3.2 results obtained by the presented methods. Table 3.2 Results for Example 3.7. [λ l (AS ),λ u (AS )] [λ l 2 (AS ),λ u 2 (AS )] [λ l 3 (AS ),λ u 3 (AS )] (R) [ , 6.088] [ ,.9734] [ , 9.454] (D) [4.0000, ] [ 2.566, ] [ , ] (D2) [4.0000, ] [ 2.566, ] [ , ] (I) [ , 6.088] [ , ] [ , ] (I2) [ , 6.088] [ , ] [ , ] (DD) [4.0000, ] [ , ] [ , ] (DI) [ 0.95, ] [ 2.95, ] [ , ] (B) [4.0000, ] [ , ] [ , ] (O) [6.3209, ] [?,?] [ 7.884, ] S. This example illustrates the case when direct interlacing methods (D) (D2) yield better results than the indirect ones (I) (I2). The same is true for the diagonal maximization variants (DD) and (DI). Rohn s method (R) is not very effective here. Optimal bounds are known only for λ u (AS )andλ l 3 (AS ). Example 3.8. Herein, we consider an example by Deif [9] on singular value sets of [2, 3] [, ] A = [0, 2] [0, ]. [0, ] [2, 3] Deif s method yields the following estimation of the singular value sets: σ (A) [2.566, 4.543], σ 2 (A) [.334, 2.854]. Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

98 228 MILAN HLADÍK, DAVID DANEY, AND ELIAS TSIGARIDAS Ahn and Chen [] confirmed that σ (A) =4.543, but the real value of σ 2 (A) must be smaller. Namely, it is less than or equal to one since σ 2 (A) =fora T =( ). Our approach using a combination of all presented methods results in an outer approximation of σ (A) [2.0489, 4.543], σ 2 (A) [0.4239, 3.87]. 4. Conclusion and future work. In this paper we considered outer approximations of the eigenvalue sets of general interval matrices and symmetric interval matrices. For both cases, we presented several improvements. Computing sharp outer approximations of the eigenvalue set of a general interval matrix is a difficult problem. The proposed methods provide quite satisfactory results, as indicated by Examples Examples demonstrate that we are able to bound quite sharply the eigenvalues of symmetric interval matrices and the singular values of interval matrices. Our bounds are quite close to the optimal ones. Nevertheless, as suggested by one of the referees, it is worth exploring the possibility of using a more numerically stable decomposition than the Jordan canonical form in Proposition 2.6. Currently, there is no algorithm that computes the best bounds in all the cases. Since the computational cost of the presented algorithms is rather low, it is advisable to use all of them in practice and select the best one depending on the particular instance. Acknowledgments. The authors thank Andreas Frommer and the anonymous referees for their valuable comments. REFERENCES [] H.-S. Ahn and Y. Q. Chen, Exact maximum singular value calculation of an interval matrix, IEEE Trans. Automat. Control, 52 (2007), pp [2] G. Alefeld and J. Herzberger, Introduction to Interval Computations, Academic Press, New York, 983. [3] G. Alefeld and G. Mayer, Interval analysis: Theory and applications, J. Comput. Appl. Math., 2 (2000), pp [4] O. Beaumont, An Algorithm for Symmetric Interval Eigenvalue Problem, Technical report IRISA-PI-00-34, Institut de recherche en informatique et systèmes aléatoires, Rennes, France, [5] D. Chablat, Ph. Wenger, F. Majou, and J.-P. Merlet, An interval analysis based study for the design and the comparison of three-degrees-of-freedom parallel kinematic machines, Int. J. Robot. Res., 23 (2004), pp [6] K.-w. E. Chu, Generalization of the Bauer Fike theorem, Numer. Math., 49 (986), pp [7] K.-w. E. Chu, Perturbation theory and derivatives of matrix eigensystems, Appl. Math. Lett., (988), pp [8] L.Cvetkovic,V.Kostic,andR.S.Varga, A new Geršgorin-type eigenvalue inclusion set, Electron. Trans. Numer. Anal., 8 (2004), pp [9] A. S. Deif, Singular values of an interval matrix, Linear Algebra Appl., 5 (99), pp [0] A. S. Deif, The interval eigenvalue problem, Z. Angew. Math. Mech., 7 (99), pp [] A. D. Dimarogonas, Interval analysis of vibrating systems, J. Sound Vibration, 83 (995), pp [2] F. Gioia and C. N. Lauro, Principal component analysis on interval data, Comput. Statist., 2 (2006), pp [3] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD, 996. [4] E. Hansen and G. W. Walster, Global Optimization Using Interval Analysis, 2nd ed., Marcel Dekker, New York, Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

99 BOUNDS ON EIGENVALUES OF INTERVAL MATRICES 229 [5] D. Hertz, The extreme eigenvalues and stability of real symmetric interval matrices, IEEE Trans. Automat. Control, 37 (992), pp [6] D. Hertz, Interval analysis: Eigenvalue bounds of interval matrices, in Encyclopedia of Optimization, C. A. Floudas and P. M. Pardalos, eds., Springer, New York, 2009, pp [7] M. Hladík, D. Daney, and E. Tsigaridas, An Algorithm for the Real Interval Eigenvalue Problem, Research report RR-6680, INRIA, Sophia-Antipolis, France, 2008, inria.fr/inria /en/, submitted to J. Comput. Appl. Math. [8] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 985. [9] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 994. [20] L. V. Kolev, Outer interval solution of the eigenvalue problem under general form parametric dependencies, Reliab. Comput., 2 (2006), pp [2] H. Leng and Z. He, Computing eigenvalue bounds of structures with uncertain-but-nonrandom parameters by a method based on perturbation theory, Comm. Numer. Methods Engrg., 23 (2007), pp [22] H.-B. Li, T.-Z. Huang, and H. Li, Inclusion sets for singular values, Linear Algebra Appl., 428 (2008), pp [23] C. D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, Philadelphia, [24] A. Neumaier, Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, UK, 990. [25] S. Poljak and J. Rohn, Checking robust nonsingularity is NP-hard, Math. Control Signals Systems, 6 (993), pp. 9. [26] Z. Qiu, S. Chen, and I. Elishakoff, Bounds of eigenvalues for structures with an interval description of uncertain-but-non-random parameters, Chaos Solitons Fractals, 7 (996), pp [27] Z. Qiu, P. C. Müller, and A. Frommer, An approximate method for the standard interval eigenvalue problem of real non-symmetric interval matrices, Comm. Numer. Methods Engrg., 7 (200), pp [28] Z. Qiu and X. Wang, Solution theorems for the standard eigenvalue problem of structures with uncertain-but-bounded parameters, J. Sound Vibration, 282 (2005), pp [29] J. Rohn, Interval matrices: Singularity and real eigenvalues, SIAM J. Matrix Anal. Appl., 4 (993), pp [30] J. Rohn, Bounds on eigenvalues of interval matrices, ZAMM Z. Angew. Math. Mech., 78 (998), pp. S049 S050. [3] J. Rohn, A Handbook of Results on Interval Linear Problems, handbook (2005). [32] J. Rohn and A. Deif, On the range of eigenvalues of an interval matrix, Computing, 47 (992), pp [33] J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford University Press, New York, 988. [34] Q. Yuan, Z. He, and H. Leng, An evolution strategy method for computing eigenvalue bounds of interval matrices, Appl. Math. Comput., 96 (2008), pp Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

100 Journal of Computational and Applied Mathematics 235 (20) Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: An algorithm for addressing the real interval eigenvalue problem Milan Hladík a,b,, David Daney b, Elias P. Tsigaridas b a Charles University, Faculty of Mathematics and Physics, Department of Applied Mathematics, Malostranské nám. 25, 8 00, Prague, Czech Republic b INRIA Sophia-Antipolis Méditerranée, 2004 route des Lucioles, BP 93, Sophia-Antipolis Cedex, France a r t i c l e i n f o a b s t r a c t Article history: Received 20 October 2008 Received in revised form 28 June 200 MSC: 65G40 65F5 5A8 Keywords: Interval matrix Real eigenvalue Eigenvalue bounds Regularity Interval analysis In this paper we present an algorithm for approximating the range of the real eigenvalues of interval matrices. Such matrices could be used to model real-life problems, where data sets suffer from bounded variations such as uncertainties (e.g. tolerances on parameters, measurement errors), or to study problems for given states. The algorithm that we propose is a subdivision algorithm that exploits sophisticated techniques from interval analysis. The quality of the computed approximation and the running time of the algorithm depend on a given input accuracy. We also present an efficient C++ implementation and illustrate its efficiency on various data sets. In most of the cases we manage to compute efficiently the exact boundary points (limited by floating point representation). 200 Elsevier B.V. All rights reserved.. Introduction Computation of real eigenvalues is a ubiquitous operation in applied mathematics, not only because it is an important mathematical problem, but also due to the fact that such computations lie at the core of almost all engineering problems. However, in these problems, which are real-life problems, precise data are very rare, since the input data are influenced by diverse uncertainties. We study these problems through models that reflect the real-life situations as well as possible. A modern approach is to consider that the parameters to be defined are not exact values, but a set of possible values. The nature of these variations is not physically homogeneous, mainly due to measurement uncertainties, or due to tolerances that come from fabrication and identification, or simply because we want to study the system in a set of continuous states. Contrary to adopting a statistical approach, which, we have to note, is not always possible, it may be more simple or realistic to bound the variations of the parameters by intervals. Interval analysis turns out to be a very powerful technique for studying the variations of a system and for understanding its properties. One of the most important properties of this approach is the fact that it is possible to certify the results of all the states of a system. Such an approach motivates us to look for an algorithm that computes rigorous bounds on eigenvalues of an interval matrix. Interval-based problems have been studied intensively in the past few decades, for example in control in order to analyse the stability of interval matrices []. The interval eigenvalue problem, in particular, also has a variety of applications throughout diverse fields of science. Let us mention automobile suspension systems [2], vibrating systems [3], principal component analysis [4], and robotics [5], for instance. Corresponding author at: Charles University, Faculty of Mathematics and Physics, Department of Applied Mathematics, Malostranské nám. 25, 8 00, Prague, Czech Republic. addresses: milan.hladik@matfyz.cz (M. Hladík), david.daney@sophia.inria.fr (D. Daney), elias.tsigaridas@sophia.inria.fr (E.P. Tsigaridas) /$ see front matter 200 Elsevier B.V. All rights reserved. doi:0.06/j.cam

101 276 M. Hladík et al. / Journal of Computational and Applied Mathematics 235 (20) Motivation As a motivating example, let us mention the following problem from robotics, that usually appears in experimental planning, e.g. [6]. We consider the following simple robotic mechanism. Let X = (x, y) be a point in the plane, which is linked to two points, M = (a, b) and N = (c, d), using two prismatic joints, r and r 2 respectively. In this case, the end-effector C has two degrees of freedom for moving in the plane. The joints r and r 2 are measured in a range [min{r k }, max{r k }], where k =, 2. The range of the joints is obtained due to mechanical constraints and describes the workspace of the mechanism. If we are given r and r 2 and we want to estimate the coordinates of X, then we solve the polynomial system F = F 2 = 0, where F = X S 2 r 2 and F 2 = X T 2 r 2 2, which describes the kinematics problem. For the calibration problem things are quite different [7,8]. In this case we want to compute, or estimate, the coordinates M and N as a function of several measurements of X, that is X = (x, y ), X 2 = (x 2, y 2 ), X 3 = (x 3, y 3 ),.... This is so because M and N are not known exactly, due to manufacturing tolerances. We have four unknowns, a, b, c and d, expressed as a function of the measurements X i, where i n. If n 2, then we can compute a, b, c, and d using the classical approach of the least squares method. However, we have to take into account the noise in the measurements l,i and l 2,i. To get a robust solution, we choose the position of the measurements by also selecting the values of l,i, l 2,i in [min{l k }, max{l k }], where k =, 2. We estimate the several criteria of selection using the eigenvalues of the observability matrix [8], that is the eigenvalues of J T J, where elements of J are partial derivatives of F k with respect to kinematic parameters. Such an approach requires bounds on the eigenvalues of the observability matrix, which is what we propose in this paper. We present a detailed example in Example 5. Further motivation comes from polynomial system real solving. Consider a system of polynomials in R[x,..., x n ] and let I be the ideal that they define. The coordinates of the solutions of the system can be obtained as eigenvalues of the so called multiplication tables, e.g. [9]. That is for each variable x i we can construct (using Gröbner basis or normal form algorithms) a matrix M xi that corresponds to the operator of multiplication by x i in the quotient algebra R[x,..., x n ]/I. The eigenvalues of these matrices are the coordinates of the solutions; thus the real eigenvalues are the coordinates of the real solutions. If the coefficients of the polynomials are not known exactly, then we can consider the multiplications as interval matrices. For an algorithm for solving bivariate polynomial systems that is based on the eigenvalues and eigenvectors of the Bézoutian matrix, the reader may refer to [0]. For the great importance of eigenvalue computations in polynomial systems solving with inexact coefficients we refer the reader to []..2. Notation and preliminaries In what follows we will use the following notation: sgn(r) The sign of a real number r, i.e., sgn(r) = if r 0, and sgn(r) = if r < 0 sgn(z) The sign of a vector z, i.e., sgn(z) = (sgn(z ),..., sgn(z n )) T e A vector of all ones (with convenient dimension) diag(z) The diagonal matrix with entries z,..., z n ρ(a) The spectral radius of a matrix A A,i The ith column of a matrix A S The boundary of a set S S The cardinality of a set S For basic interval arithmetic the reader may refer to e.g. [2 4]. A square interval matrix is defined as A := [A, A] = {A R n n ; A A A}, where A, A R n n and A A are given matrices and the inequalities are considered elementwise. By A c (A + A), 2 A (A A), 2 we denote the midpoint and radius of A, respectively. We use analogous notation for interval vectors. An interval linear system of equations Ax = b, is a short form for a set of systems Ax = b, A A, b b. The set of all real eigenvalues of A is defined as Λ := {λ R; Ax = λx, x 0, A A}, and is compact set. It seems that Λ is always composed of at most n compact real intervals, but this conjecture has not been proven yet and is proposed as an open problem. In general, computing Λ is a difficult problem. Even checking whether 0 Λ is an NP-hard problem, since the problem is equivalent to checking regularity of the interval matrix A, which is known to be NP-hard [5]. An inner approximation of Λ is any subset of Λ, and an outer approximation of Λ is a set containing Λ as a subset.

102 M. Hladík et al. / Journal of Computational and Applied Mathematics 235 (20) Previous work and our contribution The problem of computing (the intervals of) the eigenvalues of interval matrices has been studied since the nineties. The first results were due to Deif [6] and Rohn and Deif [7]. They proposed formulae for calculating exact bounds; the former case bounds real and imaginary parts for complex eigenvalues, while the latter case bounds the real eigenvalues. However, these results apply only under certain assumptions on the sign pattern invariance of the corresponding eigenvectors; such assumptions are not easy to verify (cf. [8]). Other works by Rohn concern theorems for the real eigenvalues [9] and bounds of the eigenvalue set Λ [20]. An approximate method was given in [2]. The related topic of finding verified intervals of eigenvalues for real matrices is studied in [2]. If A has a special structure, then it is possible to develop stronger results, that is to compute tighter intervals for the eigenvalue set. This is particularly true when A is symmetric; we postpone this discussion to a forthcoming communication. Our aim is to consider the general case, and to propose an algorithm for the eigenvalue problem, when the input is a generic interval matrix, without any special property. Several methods are known for computing the eigenvalues of scalar (non-interval) matrices. It is not possible to directly apply them to interval matrices, since this causes enormous overestimation of the computed eigenvalue intervals. For the same reason, algorithms that are based on the characteristic polynomial of A are rarely, if at all, used. Even though intervalvalued polynomials can be handled efficiently [22], this approach cannot yield sharp bounds, due to the overestimation of the intervals that correspond to the coefficients of the characteristic polynomial. A natural way of computing the set of the eigenvalue intervals Λ, is to try to solve directly the interval nonlinear system Ax = λx, x =, A A, λ λ 0, where λ 0 Λ is some initial outer estimation of the eigenvalue set, and is any vector norm. Interval analysis techniques for solving nonlinear systems of equations with interval parameters are very developed nowadays [23,4]. Using filtering, diverse consistency checking, and sophisticated box splitting they achieve excellent results. However, the curse of dimensionality implies that these techniques are applicable only to problems of relative small size. Recall that the curse of dimensionality refers to the exponential increase of the volume, when additional dimensions are added to a problem. For the eigenvalue problem (), this is particularly the case (cf. Section 4). We present an efficient algorithm for approximating the set of intervals of the real eigenvalues of a (generic) interval matrix, Λ, within a given accuracy. Our approach is based on a branch and prune scheme. We use several interval analysis techniques to provide efficient tests for inner and outer approximations of the intervals in Λ. The rest of the paper is structured as follows. In Section 2 we present the main algorithm, the performance of which depends on checking intervals for being outer (containing no eigenvalue) or inner (containing only eigenvalues). These tests are discussed in Sections 2.3 and 2.4, respectively. Using some known theoretical assertions we can achieve in most cases the exact bounds of the eigenvalue set. This is considered in Section 3. In Section 4 we present an efficient implementation of the algorithm and experiments on various data sets. 2. The general algorithm The algorithm that we present is a subdivision algorithm, based on a branch and prune method [23]. The pseudo-code of the algorithm is presented in Algorithm. The input consists of an interval matrix A and a precision rate ε > 0. Notice that ε is not a direct measure of the approximation accuracy. The output of the algorithm consists of two lists of intervals: L inn which comprises intervals lying inside Λ, and L unc which consists of intervals where we cannot decide whether they are contained in Λ or not, with the given required precision ε. The union of these two lists is an outer approximation of Λ. The idea behind our approach is to subdivide a given interval that initially contains Λ until either we can certify that an interval is an inner or an outer one, or its length is smaller than the input precision ε. In the latter case, the interval is placed in the list L unc. The (practical) performance of the algorithm depends on the efficiency of its subroutines and more specifically on the subroutines that implement the inner and outer tests. This is discussed in detail in Sections 2.3 and Branching in detail We may consider the process of the Algorithm as a binary tree in which the root corresponds to the initial interval that contains Λ. At each step of the algorithm the inner and outer tests are applied to the tested interval. If both are unsuccessful and the length of the interval is greater than ε, then we split the interval into two equal intervals and the algorithm is applied to each of them. There are two basic ways to traverse this binary tree, either depth-first or breadth-first. Even though from a theoretical point of view the two ways are equivalent, this is not the case from a practical point of view. The actual running time of an implementation of Algorithm depends closely on the way that we traverse the binary tree. This is of no surprise. Exactly the same behavior is noticed in the problem of real root isolation of integer polynomials [24 26]. ()

103 278 M. Hladík et al. / Journal of Computational and Applied Mathematics 235 (20) Algorithm (Approximation of Λ) : compute initial bounds λ 0, λ 0 such that Λ λ 0 := [λ 0, λ 0 ]; 2: L := {λ 0 }, L inn :=, L unc := ; 3: while L do 4: choose and remove some λ from L; 5: if λ Λ = then 6: {nothing}; 7: else if λ Λ then 8: L inn := L inn {λ}; 9: else if λ < ε then 0: L unc := L unc {λ}; : else 2: λ := [λ, λ c ], λ 2 := [λ c, λ], L := L {λ, λ 2 }; 3: end if 4: end while 5: return L inn and L unc ; A closely related issue is the data structure that we use to implement the various lists of the algorithm and in particular L. Our experience suggests that we should implement L as a stack, so that the last inserted element to be chosen at step 4 is the next candidate interval λ. Hereby, at step 2 we insert λ 2 first, and λ afterwards. Note that, in essence, the stack implementation of L closely relates to the depth-first search algorithm for traversing a binary tree. In this case, nodes correspond to intervals handled. Each node is a leaf if it is recognized as an outer or inner interval, or if it is small enough. Otherwise, it has two descendants: the left one is for the left part of the interval and the right one is for the right part. The main advantage of the depth-first exploration of the tree, and consequently of the choice to use a stack to implement L, in stack implementation, is that it allows us to exhibit some useful properties of the tested intervals. For example, if a tested interval λ is an inner interval, then the next interval in the stack, which is adjacent to it, cannot be an outer interval. Thus, for this interval we can omit steps 5 6 of the algorithm. Similarly, when a tested interval is an outer interval, then the next in the stack cannot be inner. These kinds of properties allow us to avoid many needless computations in a lot of cases, and turn out to be very efficient in practice. Another important consequence of the choice of traversing the tree depth-first is that it allows us to improve the time complexity of the inner tests. This is discussed in Section Initial bounds During the first step of Algorithm we compute an initial outer approximation of the eigenvalue set Λ, i.e. an interval that is guaranteed to contain the eigenvalue set. For this computation we use a method proposed in [20, Theorem 2]: Theorem. Let S c := Ac + A T c, 2 S := A + A T. 2 Then Λ λ 0 := [λ 0, λ 0 ], where λ 0 = λ min (S c ) ρ(s ), λ 0 = λ max (S c ) + ρ(s ), and λ min (S c ), λ max (S c ) denote the minimal and maximal eigenvalue of S c, respectively. The aforementioned bounds are usually very tight, especially for symmetric interval matrices. Moreover, it turns out, as we will discuss in Section 4, that λ 0 is an excellent starting point for our subdivision algorithm. Other bounds can be developed if we use Gerschgorin discs or Cassini ovals. None of these bounds, however, provide in practice approximations as sharp as the ones of Theorem The outer test In this section, we propose several outer tests, which can be used in step 5 of Algorithm. Even though their theoretical (worst-case) complexities are the same, their performances in practice differ substantially. Consider an interval matrix A and a real closed interval λ. We want to decide whether λ Λ =, that is, there is no matrix A A that has a real eigenvalue inside λ. In this case, we say that λ is an outer interval.

104 M. Hladík et al. / Journal of Computational and Applied Mathematics 235 (20) The natural idea is to transform the problem to the problem of checking regularity of interval matrices. An interval matrix M is regular if every M M is nonsingular. Proposition. If the interval matrix A λi is regular, then λ is an outer interval. Proof. Let λ λ and A A. The real number λ is not an eigenvalue of A if and only if the matrix A λi is nonsingular. Thus, if A λi is regular, then for every λ λ and A A we have that A λi is nonsingular (not conversely), and hence λ is not an eigenvalue. In general, Proposition gives a sufficient but not necessary condition for checking the outer property (due to the dependences caused by multiple appearances of λ). Nevertheless, the smaller the radius of λ, the stronger the condition. We now review some of the applicable conditions and methods. Recall that testing regularity of an interval matrix is an NP-hard problem [5]; therefore we exhibit a sufficient condition as well A sufficient regularity condition There are diverse sufficient conditions for an interval matrix to be regular [27]. The very strong one, which turned out to very useful (cf. Section 4), is formulated below. Proposition 2. An interval matrix M is regular if M c is nonsingular and ρ( M c M ) <. Proof. See e.g. [27, Corollary 3.2.] The Jansson and Rohn method Herein we recall the Jansson and Rohn method [28] for testing regularity of an interval matrix M. Its great benefit is that the time complexity is not a priori exponential. Its modification is also is very useful for the inner test (Section 2.4). That is why we describe the method here in more detail. Choose an arbitrary vector b R n and consider the interval system of equations Mx = b. The solution set X = {x R n ; Mx = b, M M} is described by M c x b M x. This solution set is formed by a union of convex polyhedra, since a restriction of X on an orthant is characterized by a linear system of inequalities (M c M diag(z))x b, (M c + M diag(z))x b, diag(z)x 0, (2) where z {±} n is a vector of signs corresponding to the orthant. Regularity of M closely relates to unboundedness of X. Indeed, Jansson and Rohn [28] obtained the following result. Theorem 2. Let C be a component (maximal connected set) of X. Then M is regular if and only if C is bounded. The algorithm starts by selecting an appropriate vector b. The component C is chosen so that it includes the point M c b. We check the unboundedness of C by checking the unboundedness of (2), for each orthant that C intersects. The list L comprises the sign vectors (orthants) to be inspected, and V consists of the already visited orthants. To speed up the process, we notice that there is no need to inspect all the neighboring orthants. It suffices to inspect only that orthants possibly connected to the actual one. Thus we can skip the ones that are certainly disconnected. Jansson and Rohn proposed an improvement in this way; we refer the reader to [28] for more details. The performance of Algorithm 2 depends strongly on the choice of b. It is convenient to select b so that the solution set X intersects a (possibly) small number of orthants. The selection procedure of b, proposed by Jansson and Rohn, consists of a local search The ILS method The ILS (interval linear system) method is a simple but efficient approach for testing regularity of an interval matrix M. It is based on transforming the problem to solving an interval linear system and using an ILS solver. The more effective the ILS solver is, the more effective the ILS method. Proposition 3. The interval matrix M is regular if and only if the interval linear system Mx = 0, x 0 (5) has no solution.

105 2720 M. Hladík et al. / Journal of Computational and Applied Mathematics 235 (20) Algorithm 2 (Jansson and Rohn method checking regularity of M) : if M c is singular then 2: return M is not regular ; 3: end if 4: select b; 5: z := sgn(a c b); 6: L := {z}, V := ; 7: while L do 8: choose and remove some z from L; 9: V := V {z}; 0: solve the linear program max z T x; (M c M diag(z))x b, (M c + M diag(z))x b, diag(z)x 0 ; (3) : if (3) is unbounded then 2: return M is not regular ; 3: else if (3) is feasible then 4: L := L N(z) \ (L V) }, where 5: end if 6: end while 7: return M is regular ; N(z) = {(z,..., z i, z i, z i+,..., z n ) T ; i n, }; (4) Algorithm 3 (ILS method) : for i =,..., n do 2: b := M,i {the ith column of M}; 3: M := M,,..., M,i, M,i+,..., M,n {the matrix M without the ith column}; 4: solve (approximately) the interval linear system M x = b, e x e; (7) 5: if (7) has possibly a solution then 6: return λ needn t be outer ; 7: end if 8: end for 9: return λ is an outer interval ; As x can be normalized, we replace the inequation x 0 by x =, where the maximum norm is defined as x := max{ x i ; i =,..., n}. Moreover, since both x and x satisfy (5), we derive the following equivalent formulation of (5): Mx = 0, x =, x i = for some i {,..., n}, (6) the solvability of which can be tested using Algorithm 3. The efficiency of the ILS method depends greatly on the selection of an appropriate ILS solver. It is not necessary to solve (7) exactly, as this is time-consuming. In fact, checking solvability is known to be NP-hard [29]. It suffices to exploit a (preferably fast) algorithm to produce an outer approximation of (6); that is, an approximation that contains the whole solution set. Experience shows that the algorithm proposed in [30] modified so as to work for overconstrained linear systems is a preferable choice. It is sufficiently fast and produces a good approximation of the solution set of (7) Direct enumeration The ILS method benefits us even when M is not recognized as a regular matrix. In this case, we have an approximation of the solution set of (7), at each iteration of step 4. By embedding them into n-dimensional space and joining them together, we get an outer approximation of the solution set of (6). This is widely usable; see also Section 3. We will present some more details of this procedure. Let v R n be an interval vector. We consider the sign vector set sgn(v), that is the set of vectors z {±} n with components +, vi 0, z i =, v i < 0, v i 0, (8) ±, otherwise (v i < 0 < v i ).

106 M. Hladík et al. / Journal of Computational and Applied Mathematics 235 (20) Clearly, the cardinality of sgn(v) is always a power of 2. Notice that this set does not always consist of the sign vectors of all v v; the difference is caused when v i < 0, v i = 0 holds for some i =,..., n. Let x be an outer approximation of the solution set of (6), and let Z := sgn(x). As long as Z has reasonably small cardinality we can check the regularity of M by inspecting all the corresponding orthants and solving the linear programs of Eq. (3) with b = 0. There is no need to check the other orthants, since x is a solution of (6) if and only if it is a feasible solution of (3) with b = 0, z = sgn(x) and z T x > 0. Algorithm 4 gives a formal description of this procedure. Algorithm 4 (Direct enumeration via Z) : for all z Z do 2: solve the linear program (3) with b = 0; 3: if (3) is unbounded then 4: return M is not regular ; 5: end if 6: end for 7: return M is regular ; Practical implementation Our implementation exhibits and combines all the methods mentioned in this section. We propose the following procedure (Algorithm 5) for the outer test of Algorithm : Algorithm 5 (Outer test) : M := A λi; 2: if M c is singular then 3: return λ is not an outer interval ; 4: end if 5: if ρ( M c M ) < then 6: return λ is an outer interval ; 7: end if 8: call Algorithm 2 (Jansson and Rohn) with the number of iterations limited by a constant K 3 ; 9: if the number of iteration does not exceed K 3 then 0: return its output; : end if 2: call Algorithm 3 (ILS method); 3: if Algorithm 3 recognize λ as an outer interval then 4: return λ is an outer interval ; 5: end if 6: use the obtained approximation x to define Z; 7: if Z < K 4 then 8: call Algorithm 4; 9: return its output; 20: end if 2: return no decision on λ ; Jansson and Rohn method is very fast as long as radii of M are small and λ is not close to the border of Λ. If this is not the case, then it can be time-consuming. We limit the number of iterations of this procedure to K 3, where K 3 := n 3. If after this number of iterations the result is not conclusive, then we call the ILS method. Finally, if ILS does not succeed, then we compute Z, and if its cardinality is less than K 4, then we call Algorithm 4. Otherwise, we cannot reach a conclusion about λ. We empirically choose K 4 := 2 α with α := 2 log(k ) 8. Notice that in step 2 of Algorithm 5 we obtain a little more information. Not only is λ not an outer interval, but also its half-intervals [λ, λ c ], [λ c, λ] cannot be outer. Remark. The interval Newton method [3,4] applied to the nonlinear system Ax = λx, x 2 = did not turn out to be efficient. Using the maximum norm was more promising; however, at each iteration the interval Newton method solves an interval linear system that is a consequence of (6), and therefore cannot yield better results than the ILS method.

107 2722 M. Hladík et al. / Journal of Computational and Applied Mathematics 235 (20) The inner test This section is devoted to the inner test (step 7 in Algorithm ). We are given a real closed interval λ and an interval matrix A. The question is whether every λ λ represents an eigenvalue of some A A. If so, then λ is called an inner interval. Using inner testing in interval-valued problems is not a common procedure. It depends greatly on the problem under consideration, since interval parameters are usually correlated, and such correlations are, in general, hard to deal with. However, utilization of inner testing provides two great benefits: it decreases the running time and allows us to measure the sharpness of the approximation. Our approach is a modification of Jansson and Rohn method. Proposition 4. We have that λ R is an eigenvalue of some A A if the linear programming problem max z T x; (A c λi A diag(z))x b, (A c λi + A diag(z))x b, diag(z)x 0 (9) is unbounded for some z {±} n. Proof. It follows from [28, Theorems 5.3 and 5.4]. Proposition 5. We have that λ is an inner interval if the linear programming problem max z T x z T x 2 ; (A c A diag(z))(x x 2 ) λx + λx 2 b, (A c + A diag(z))(x x 2 ) λx + λx 2 b, diag(z)(x x 2 ) 0, x, x 2 0 (0) is unbounded for some z {±} n. Proof. Let z {±} n and let (0) be unbounded. That is, there exists a sequence of feasible points (x, k x2 k ), k =, 2,..., such that lim k (z T x k zt x 2 k ) =. We show that (9) is unbounded for every λ λ, and thereby λ is an inner interval. Let λ λ be arbitrary. Define a sequence of points x k := (x k x2), k k =, 2,.... Every x k is a feasible solution to (9), since and and (A c λi A diag(z))x k = (A c λi A diag(z))(x k x2 k ) (A c A diag(z))(x x 2 ) λx + λx 2 b, (A c λi + A diag(z))x k = (A c λi + A diag(z))(x k x2 k ) diag(z)x k = diag(z)(x k x2 k ) 0. (A c + A diag(z))(x x 2 ) λx + λx 2 b, Next, lim k z T x k = lim k z T (x k x2 k ) =. Therefore the linear program (9) is unbounded. Proposition 5 gives us a sufficient condition for checking whether λ is an inner interval. The condition becomes stronger and stronger as λ becomes more narrow. The natural question is that of how to search for a sign vector z that guarantees the unboundedness of (0). We can modify the Jansson and Rohn method and inspect all orthants intersecting a given component. In our experience, slightly better results are obtained by the variation described by Algorithm 6. This approach has several advantages. First, it solves the linear program (0), which has twice as many variables as (3), at most (n + ) times. Next, we can accelerate Algorithm by means of the following properties: Algorithm 6 returns that λ is not an inner interval only if λ c is not an eigenvalue. In this case, neither of the half-intervals [λ, λ c ] and [λ c, λ] can be inner. If (0) is unbounded for some sign vector z, then it is sometimes probable that the interval adjacent to λ is also inner and the corresponding linear program is (0), unbounded for the same sign vector z. Therefore, the sign vector z is worth remembering for the subsequent iterations of step 3 in Algorithm 6. This is particularly valuable when the list L is implemented as a stack; see the discussion in Section Complexity In this section we discuss the complexity of Algorithm. Recall that even testing the regularity of a matrix is an NP-hard problem; thus we cannot expect a polynomial algorithm. By LP(m, n) we denote the (bit) complexity of solving a linear program with O(m) inequalities and O(n) unknowns. Our algorithm is a subdivision algorithm. Its complexity is the number of tests it performs, times the complexity of each step. At each step we perform, in the worst case, an outer and an inner test.

108 M. Hladík et al. / Journal of Computational and Applied Mathematics 235 (20) Algorithm 6 (Inner test) : call Algorithm 2 (Jansson and Rohn) with M := A λ c I; 2: if M is regular then 3: return λ is not inner interval; 4: end if 5: let z be a sign vector for which (3) is unbounded; 6: solve (0); 7: if (0) is unbounded then 8: return λ is an inner interval ; 9: else if (0) is infeasible then 0: return λ is possibly not inner ; : end if 2: for all y N(z) do 3: solve (0) with y as a sign vector; 4: if (0) is unbounded then 5: return λ is an inner interval ; 6: end if 7: end for 8: return λ is possibly not inner ; Let us first compute the number of steps of the algorithm. Let max{a ij } := max{max{a ij }, max{a ij }} 2 τ, i.e. we consider a bound on the absolute value on the numbers used to represent the interval matrix. From Section 2.2 we deduce that the real eigenvalue set of A is contained in an interval, centered at zero and with radius bounded by the sum of the spectral radii of S S and S. Evidently the bound 2 τ holds for the elements of these matrices, as well. Since for an n n matrix M the absolute value of its (possible complex) eigenvalues is bounded by n max ij M ij, we deduce that the spectral radius of S S and S is bounded by n2 τ and thus the real eigenvalues of A are in the interval [ n2 τ+, n2 τ+ ]. Let ε = 2 k be the input accuracy. In this case the total number of intervals that we need to test, or in other words the total number of steps that the algorithm performs, is n2 τ+ /2 k = n2 τ+k+. It remains to compute the complexity of each step. At each step we perform an inner and an outer test. For each of these tests we should solve, in the worst case, 2 O(n) linear programs that consist of O(n) variables and inequalities. The exponential number of linear programs is a consequence of the fact that we should enumerate all the vertices of a hypercube in n dimensions (refer to Algorithm 4). Thus the total complexity of the algorithm is O(n 2 k+τ+ 2 n LP(n, n)) The interval hull We can slightly modify Algorithm to approximate the interval hull of Λ, i.e., the smallest interval containing Λ. Let λ L (resp. λ U ) be the lower (resp. upper) boundary of Λ, i.e., λ L := inf{λ; λ Λ} and λ U := sup{λ; λ Λ}. In order to compute λ L, we consider the following modifications of Algorithm. We remove all the steps that refer to the list L inn, and we change step 8 to. 8: return L unc ;. The modified algorithm returns L unc as output. The union of all the intervals in L unc is an approximation of the lower boundary point λ L. If the list L unc is empty, then the eigenvalue set Λ is empty, too. An approximation of the upper boundary point, λ U, can be computed as a negative value of the lower eigenvalue boundary point of the interval matrix ( A). 3. Exact bounds Algorithm yields an outer and an inner approximation of the set of eigenvalues Λ. In this section we show how to use it for computing the exact boundary points of Λ. This exactness is limited by the use of floating point arithmetic. Rigorous results would be obtained by using interval arithmetic, but this is a direct modification of the proposed algorithm and we do not discuss it in detail. As long as interval radii of A are small enough, we are able, in most of the cases, to determine the exact bounds in a reasonable time. Surprisingly, sometimes computing exact bounds is faster than high precision approximation.

109 2724 M. Hladík et al. / Journal of Computational and Applied Mathematics 235 (20) We build on [9, Theorem 3.4]: Theorem 3. Let λ Λ. Then there exist nonzero vectors x, p R n and vectors y, z {±} n such that Ac diag(y)a diag(z) x = λx, A T c diag(z)at diag(y) p = λp, diag(z)x 0, diag(y)p 0. () Theorem 3 asserts that the boundary eigenvalues are produced by special matrices A y,z A of the form of A y,z := A c diag(y)a diag(z). Here, z is the sign vector of the right eigenvector x, and y is the sign vector of the left eigenvector p. Recall that a right eigenvector is a nonzero vector x satisfying Ax = λx, and a left eigenvector is a nonzero vector p satisfying A T p = λp. In our approach, we are given an interval λ and we are trying to find outer approximations of the corresponding left and right eigenvectors, i.e. p and x, respectively. If no component of p and x contains zero, then the sign vectors y := sgn(p) and z := sgn(x) are uniquely determined. In this case we enumerate all the eigenvalues of A y,z. If only one of them belongs to λ, then we have succeeded. If the eigenvectors in p and x are normalized according to (5), then we must inspect not only A y,z, but also A y,z (the others, A y, z and A y, z, are not needed due to symmetry). The formal description is given in Algorithm 7. Algorithm 7 (Exact bound) : M := A λi; 2: call Algorithm 3 (ILS method) with the input matrix M T to obtain an outer approximation p of the corresponding solutions. 3: if p i 0 p i for some i =,..., n then 4: return bound is possibly not unique ; 5: end if 6: y := sgn(p); 7: call Algorithm 3 (ILS method) with the input matrix M to obtain an outer approximation x of the corresponding solutions. 8: if x i 0 x i for some i =,..., n then 9: return bound is possibly not unique ; 0: end if : z := sgn(x); 2: let L be a set of all eigenvalues of A y,z and A y,z ; 3: if L λ = then 4: return no boundary point in λ ; 5: else if L λ = {λ } then 6: return λ is a boundary point candidate ; 7: else 8: return bound is possibly not unique ; 9: end if We now describe how to integrate this procedure into our main algorithm. Suppose that at some iteration of Algorithm we have an interval λ recognized as outer. Suppose next that the following current interval λ 2 is adjacent to λ (i.e., λ = λ 2 ); it is not recognized as outer and it fulfills the precision test (step 9). According to the result of Algorithm 7 we distinguish three possibilities: If L λ 2 = then there cannot be any eigenvalue boundary point in λ 2, and therefore it is an outer interval. If L λ 2 = {λ } then λ is the exact boundary point required, and moreover [λ, λ 2 ] Λ. If L λ 2 > then the exact boundary point is λ := min{λ; λ L λ 2 }. However, we cannot say anything about the remaining interval [λ, λ 2 ]. A similar procedure is applied when λ is inner and λ 2 is adjacent and narrow enough. We can simply extend Algorithm 7 to the case where there are some zeros in the components of p and x. In this case, the sign vectors y and z are not determined uniquely. Thus, we have to take into account the sets of all the possible sign vectors. Let v be an interval vector and sgn (v) be a sign vector set, that is, the set of all sign vectors z {±} n satisfying +, vi > 0, z i =, v i < 0, ±, otherwise (v i 0 v i ).

110 M. Hladík et al. / Journal of Computational and Applied Mathematics 235 (20) The definition of sgn (v) slightly differs from that of sgn(v) in (8). Herein, we must take into account the both signs of z i whenever v i contains zero (even on a boundary). Assume Y := sgn (p), Z := sgn (x). Instead of two matrices, A y,z and A y,z, we must inspect all possible combinations with y Y and z Z. In this way, step 2 of Algorithm 7 will we replaced by. 2 : L := {λ; λ is an eigenvalue of A y,z or of A y,z, y Y, z Z};. The cardinality of Y is a power of 2, and the cardinality of Z as well. Since we have to enumerate eigenvalues of Y Z matrices, step 2 is tractable for only reasonably small sets Y and Z. 4. Numerical experiments In this section we present results of some numerical experiments. They confirm the quality of the algorithm presented. We are able to determine the eigenvalue set exactly or at least very sharply for dimensions up to about 30. The running time depends heavily not only on the dimension, but also on the widths of matrix intervals. We also compared our implementation with another techniques that solve directly the interval nonlinear system (). It turned out that such techniques are comparable only for very small dimensions, i.e. 5. Results of our numerical experiments are displayed in tables that follow and can be interpreted using the following notation: n Problem dimension ε Precision R Maximal radius Exactness Indication of whether exact bounds of Λ were achieved; if not, we display the number of uncertain intervals Time Computing time in hours, minutes and seconds Hull time Computing time of the interval hull of Λ; see Section 2.6 Note that ε refers to the precision used in the step 9 of Algorithm. For the additional computation of exact boundary points we use 0 4 ε precision. Generally, better results were obtained for smaller R, as both the Jansson and Rohn method and the sufficient regularity condition are more efficient for smaller radii of matrix intervals. The results were carried on an Intel Pentium(R) 4, CPU 3.4 GHz, with 2 GB RAM, and the source code was written in C++ using GLPK v.4.23 [3] for solving linear programs, CLAPACK v.3.. for its linear algebraic routines, and PROFIL/BIAS v [32] for interval arithmetics. Notice, however, that routines of GLPK and CLAPACK [33] do not produce verified solutions, and for real-life problems preferably verified software or interval arithmetic should be used. Example (Random Matrices). The entries of the midpoint matrix A c are chosen randomly with uniform distribution in [ 20, 20]. The entries of the radius matrix A are chosen randomly with uniform distribution in [0, R], where R is a positive real number. The results are displayed in Table. Example 2 (Random Symmetric Matrices). The entries of A c and A are chosen randomly in the same manner as in Example ; the only difference is that both of these matrices are composed to be symmetric. See Table 2 for the results. Example 3 (Random A T A Matrices). The entries of A c and A are chosen randomly as in Example, and our algorithm is applied on the matrix generated by the product A T A. In this case, the maximal radius value R is a bit misleading, since it refers to the original matrix A instead of the product used. The results are displayed in Table 3. Example 4 (Random Nonnegative Matrices). The entries of A c and A are chosen randomly as in Example, and the eigenvalue problem is solved for its absolute value A := { A ; A A}. The absolute value of an interval matrix is again an interval matrix and with entries A ij A ij 0, A ij = A ij A ij 0, [0, max( A ij, A ij )] otherwise. See Table 4 for the results.

111 2726 M. Hladík et al. / Journal of Computational and Applied Mathematics 235 (20) Table Random matrices. n ε R Exactness Time Hull time 5 0. Exact 2 s s Exact 7 s 2 s Exact 9 s 4 s 0 0. Exact 6 s s Exact min 2 s min s Exact 37 s 5 s Exact 0 min 29 s 6 s Exact 20 min 54 s 35 s 5 0. Exact 7 min 59 s min 2 s Exact 2 min 6 s 0 s Exact 7 min 27 s 39 s Exact 2 min 6 s 46 s Exact 5 min 46 s 23 s Exact 0 min 39 s min 34 s Exact 4 min 37 s 54 s Exact 48 min 3 s 29 s Exact 2 min 20 s Exact h 42 min 36 s Exact h 52 min 5 s Exact 9 min 25 s min 34 s Table 2 Random symmetric matrices. n ε R Exactness Time Hull time 5 0. Exact 3 s s Exact s s Exact 7 s s min 8 s s s 0 s Exact 3 min 5 s s min 43 s 4 s Exact 2 min 25 s 3 s Exact 6 min 39 s 4 s Exact 27 min 48 s 8 s min 9 s 8 s Exact 7 min 5 s 2 s Exact h 59 min s s Exact 29 s Exact 6 min 5 s Exact min 23 s Exact h 2 min 43 s Exact 34 min 5 s Table 3 Random A T A matrices. n ε R Exactness Time Hull time Exact 5 s s Exact 37 s 3 s Exact 4 min 0 s s Exact min 35 s 7 s 0 0. Exact min 3 s 56 s Exact min s 3 s Exact 40 s 2 s min 38 s 7 s min 58 s 3 s Exact 39 min 27 s 4 min 48 s Exact h 8 min 6 s Figs. 4 present some examples of the eigenvalue set Λ. Intervals of Λ are colored red while the outer intervals are yellow and green; yellow color is for the intervals recognized by the sufficient regularity condition (step 5 of Algorithm 5), and green is for the remainder.

112 M. Hladík et al. / Journal of Computational and Applied Mathematics 235 (20) Table 4 Random nonnegative matrices. n ε R Exactness Time Hull time Exact 3 s s Exact 8 s s Exact 2 min 22 s 6 s Exact 47 s 4 s Exact min 53 s 27 s Exact 57 s 37 s Exact h 8 min 49 s Exact 3 min 55 s 9 s Exact 8 min 36 s min 9 s Exact 5 min 58 s 2 s Exact 9 min 47 s 49 s Exact 37 min 44 s Exact 2 min 4 s Exact 5 min 57 s Exact 2 h 2 min 22 s Fig.. Random matrix, n = 30, R = 0., computing time 48 min 3 s, initial approximation [ , ]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 2. Random symmetric matrix, n = 5, R = 0.5, computing time 3 min 48 s, initial approximation [ 60.64, ]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 3. Random A T A matrix, n = 5, R = 0.02, computing time min 58 s, initial approximation [ , ]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 4. Random nonnegative matrix, n = 5, R = 0.2, computing time 2 min 22 s, initial approximation [ , 44.64]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Example 5 (Interval Matrices in Robotics). The following problem usually appears in experimental planning, e.g. [6]. We consider a PRRRP planar mechanism, where P stands for a prismatic and R for a rotoid joint; for further details we refer the reader to [34]. We consider the mechanism of Fig. 5. Let X = (x, y) be a point in the plane, which is linked to two points, M and N, using two fixed length bars so that it holds that X M = r, X N = r 2. We can move M (respectively N) between the fixed points A = (0, 0) and B = (L, 0) (respectively C = (L + r 3, 0) and D = (2.L + r 3, 0)) using two prismatic joints, so A M = l, C N = l 2. In the example that we consider, Fig. 5, the points A, B, C, D are aligned. If we control the length l and l 2 by using two linear actuators we allow the displacement of the end-effector X to have two degrees of freedom in a planar workspace that is limited by the articular constraints l, l 2 [0, L]. The two equations F (X, l ) M X 2 r 2 = 0 and F 2(X, l 2 ) N X 2 r 2 2 = 0 link the generalized coordinates (x, y) and the articular coordinates l, l 2. The calibration of such a mechanism is not an easy problem due to assembly and manufacturing errors, and because the kinematic parameters, that is the lengths r, r 2 and r 3, are not well known. The aim is to estimate them using several measurements, k =,..., n, of the end-effector X k and the corresponding measurements of the articular coordinates l k,, l k,2. The identification procedure of r, r 2, and r 3 is based on a classical least square approach for the (redundant) system F [F, (X, l, ), F,2 (X, l,2 ),..., F n, (X n, l n, ), F n,2 (X n, l n,2 )] T = 0. That is, we want to compute r, r 2, and r 3 that minimize the quantity F T F. A natural question is that of how to estimate the measurements positions inside the workspace [7] to improve the robustness of the numerical solution of a least square method. For this we can use the observability index [8], which is a square root of the smallest eigenvalue of J T J, where J is the identification Jacobian. It is defined by the derivatives of F k

113 2728 M. Hladík et al. / Journal of Computational and Applied Mathematics 235 (20) A M B C N D Fig. 5. PRRRP planar parallel robot. and F 2k with respect to the kinematic parameters r, r 2, and r 3, that is F, (X, l, ) F, (X, l, ) F, (X, l, ) r r 2 r 3 F,2 (X, l,2 ) F,2 (X, l,2 ) F,2 (X, l,2 ) r r 2 r 3 J =.... F n, (X n, l n, ) F n, (X n, l n, ) F n, (X n, l n, ) r r 2 r 3 F n,2 (X n, l n,2 ) F n,2 (X n, l n,2 ) F n,2 (X n, l n,2 ) r r 2 r 3 The observability index can be equivalently defined as the third-largest eigenvalue of the matrix 0 J J T. 0 (2) We employ this approach since it gives rise to more accurate estimates. Recall that due to measurement errors, it is not possible to obtain the actual values of the kinematic parameters. However, if the set of measurements is chosen so as to maximize this index, the error of the end-effector positions after calibration is minimized. We demonstrate our approach by setting n = 2. Let r = r 2 = 5, r 3 = 5, and L = 0. If l, [0, 5], l,2 [5, 0], l 2, [5, 0] and l 2,2 [0, 5] then [ 30, 30] 0 0 J = 0 [ 30, 30] [ 30, 30], 0 [5.2, 25] [5, 4.8] and the third-largest eigenvalue λ 3 of (2) lies in the interval [0.25, 2.53]. Similarly, if l, [0, 2], l,2 [8, 0], l 2, [8, 0], and l 2,2 [0, 2], this is workspace, ws, in Fig. 6; then λ 3 = [7.56, 2.53], where [ 30, 30] 0 0 J = 0 [ 30, 30] [ 30, 30]. 0 [2, 25] [5, 9] If l, [4, 7], l,2 [9, 0], l 2, [9, 0], and l 2,2 [4, 7], this is workspace 2, ws 2, in Fig. 6; then λ 3 = [2.52, 7.56], where [ 30, 30] 0 0 J = 0 [ 30, 30] [ 30, 30]. 0 [7, 2] [9, 3] As regards the last two examples, it is always better to chose the measurement poses in ws, in red in Fig. 6, than the ones in ws 2, in blue in Fig. 6.

114 M. Hladík et al. / Journal of Computational and Applied Mathematics 235 (20) Fig. 6. Workspaces ws in red and ws 2 in blue. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 5. Conclusion In this paper we considered the problem of computing the real eigenvalues of matrices with interval entries. Sharp approximation of the set of the (real) eigenvalues is an important subroutine in various engineering applications. We proposed an algorithm based on a branch and prune scheme and splitting only along one dimension (real axis) to compute the intervals of the real eigenvalues. The algorithm approximates the real eigenvalues with an accuracy depending on a given positive parameter ε. Numerical experiments demonstrate that the algorithm is applicable in high dimensions. An exact bound can be achieved in real time up to the dimension of 30, but more or less sharp approximations can be produced in any dimension. To the best of our knowledge there is no comparable method for dimension greater that 5. Our algorithm could be also seen as a first step of an algorithm that produces intervals (in the complex plane) that contain all the eigenvalues of a given interval matrix. This is work in progress. References [] W. Karl, J. Greschak, G. Verghese, Comments on A necessary and sufficient condition for the stability of interval matrices, Internat. J. Control 39 (4) (984) [2] Z. Qiu, P.C. Müller, A. Frommer, An approximate method for the standard interval eigenvalue problem of real non-symmetric interval matrices, Comm. Numer. Methods Engrg. 7 (4) (200) [3] A.D. Dimarogonas, Interval analysis of vibrating systems, J. Sound Vibration 83 (4) (995) [4] F. Gioia, C.N. Lauro, Principal component analysis on interval data, Comput. Statist. 2 (2) (2006) [5] D. Chablat, P. Wenger, F. Majou, J. Merlet, An interval analysis based study for the design and the comparison of 3-DOF parallel kinematic machines, Int. J. Robot. Res. 23 (6) (2004) [6] E. Walter, L. Pronzato, Identification of Parametric Models, Springer, Heidelberg, 997. [7] D. Daney, Y. Papegay, B. Madeline, Choosing measurement poses for robot calibration with the local convergence method and Tabu search, Int. J. Robot. Res. 24 (6) (2005) 50. [8] A. Nahvi, J. Hollerbach, The noise amplification index for optimal pose selection in robot calibration, in: IEEE International Conference on Robotics and Automation, Citeseer, 996, pp [9] D. Cox, J. Little, D. O Shea, Ideals, Varieties, and Algorithms, 2nd ed., in: Undergraduate Texts in Mathematics, Springer-Verlag, New York, 997. [0] L. Busé, H. Khalil, B. Mourrain, Resultant-based methods for plane curves intersection problems, in: V. Ganzha, E. Mayr, E. Vorozhtsov (Eds.), Proc. 8th Int. Workshop Computer Algebra in Scientific Computing, in: LNCS, vol. 378, Springer, 2005, pp [] H. Stetter, Numerical Polynomial Algebra, Society for Industrial Mathematics, [2] G. Alefeld, J. Herzberger, Introduction to Interval Computations, Academic Press, London, 983. [3] E. Hansen, G.W. Walster, Global Optimization Using Interval Analysis, 2nd ed., Marcel Dekker, New York, 2004, revised and expanded. [4] A. Neumaier, Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, 990. [5] S. Poljak, J. Rohn, Checking robust nonsingularity is NP-hard, Math. Control Signals Systems 6 () (993) 9. [6] A.S. Deif, The interval eigenvalue problem, Z. Angew. Math. Mech. 7 () (99) [7] J. Rohn, A. Deif, On the range of eigenvalues of an interval matrix, Computing 47 (3 4) (992) [8] A. Deif, J. Rohn, On the invariance of the sign pattern of matrix eigenvectors under perturbation, Linear Algebra Appl. 96 (994) [9] J. Rohn, Interval matrices: singularity and real eigenvalues, SIAM J. Matrix Anal. Appl. 4 () (993) [20] J. Rohn, Bounds on eigenvalues of interval matrices, ZAMM Z. Angew. Math. Mech. 78 (Suppl. 3) (998) S049 S050. [2] G. Alefeld, G. Mayer, Interval analysis: theory and applications, J. Comput. Appl. Math. 2 ( 2) (2000) [22] E.R. Hansen, G.W. Walster, Sharp bounds on interval polynomial roots, Reliab. Comput. 8 (2) (2002) [23] L. Jaulin, M. Kieffer, O. Didrit, É. Walter, Applied Interval Analysis. With Examples in Parameter and State Estimation, Robust Control and Robotics, Springer, London, 200. [24] F. Rouillier, Z. Zimmermann, Efficient isolation of polynomial s real roots, J. Comput. Appl. Math. 62 () (2004) [25] I.Z. Emiris, B. Mourrain, E.P. Tsigaridas, Real algebraic numbers: complexity analysis and experimentation, in: P. Hertling, C. Hoffmann, W. Luther, N. Revol (Eds.), Reliable Implementations of Real Number Algorithms: Theory and Practice, in: LNCS, vol. 5045, Springer-Verlag, 2008, pp Also available in:

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116 Computers and Mathematics with Applications 62 (20) Contents lists available at SciVerse ScienceDirect Computers and Mathematics with Applications journal homepage: Characterizing and approximating eigenvalue sets of symmetric interval matrices Milan Hladík a,b, David Daney b, Elias Tsigaridas c, a Charles University, Faculty of Mathematics and Physics, Department of Applied Mathematics, Malostranské nám. 25, 800, Prague, Czech Republic b INRIA Sophia-Antipolis Méditerranée, 2004 route des Lucioles, BP 93, Sophia-Antipolis Cedex, France c Computer Science Department, Aarhus University, Denmark a r t i c l e i n f o a b s t r a c t Article history: Received 23 March 20 Received in revised form 26 June 20 Accepted 8 August 20 Keywords: Interval matrix Symmetric matrix Interval analysis Eigenvalue Eigenvalue bounds We consider the eigenvalue problem for the case where the input matrix is symmetric and its entries are perturbed, with perturbations belonging to some given intervals. We present a characterization of some of the exact boundary points, which allows us to introduce an inner approximation algorithm, that in many case estimates exact bounds. To our knowledge, this is the first algorithm that is able to guarantee exactness. We illustrate our approach by several examples and numerical experiments. 20 Elsevier Ltd. All rights reserved.. Introduction Computing eigenvalues of a matrix is a basic linear algebraic task used throughout in mathematics, physics and computer science. Real life makes this problem more complicated by imposing uncertainties and measurement errors on the matrix entries. We suppose we are given some compact intervals in which the matrix entries can vary. The set of all possible real eigenvalues forms a compact set, and the question that we deal with in this paper is how to characterize and compute it. The interval eigenvalue problem has its own history. The first results are probably due to Deif [] and Rohn & Deif [2]: bounds for real and imaginary parts for complex eigenvalues were studied by Deif [], while Rohn & Deif [2] considered real eigenvalues. Their theorems are applicable only under an assumption on sign pattern invariancy of eigenvectors, which is not easy to verify (cf. [3]). A boundary point characterization of the eigenvalue set was given by Rohn [4], and it was used by Hladík et al. [5] to develop a branch & prune algorithm producing an arbitrarily tight approximation of the eigenvalue set. Another approximate method was given by Qiu et al. [6]. The related topic of finding verified intervals of eigenvalues for real matrices was studied in, e.g. [7 9]. In this paper we consider the case of the symmetric eigenvalue problem. Symmetric matrices naturally appear in many practical problems, but symmetric interval matrices are hard to deal with. This is so, mainly due to the so-called dependencies, that is, correlations between the matrix components. If we forget these dependencies and solve the problem by reducing it to the previous case, then the results will be greatly overestimated, in general (but not the extremal points, see Theorem 2). From now on we consider only the symmetric case. Due to the dependencies just mentioned, the theoretical background for the eigenvalue problem of symmetric interval matrices is not well established enough and there are few practical methods. The known results are by Deif [] and Hertz [0]. Corresponding author. addresses: milan.hladik@matfyz.cz, milan.hladik@sophia.inria.fr (M. Hladík), david.daney@sophia.inria.fr (D. Daney), elias@cs.au.dk (E. Tsigaridas) /$ see front matter 20 Elsevier Ltd. All rights reserved. doi:0.06/j.camwa

117 M. Hladík et al. / Computers and Mathematics with Applications 62 (20) Deif [] gives an exact description of the eigenvalue set together with restrictive assumptions. Hertz [0] (cf. []) proposed a formula for computing two extremal points of the eigenvalue set the largest and the smallest ones. As the problem itself is very hard, it is not surprising conjectures on the problem [2] turned out to be wrong [3]. In recent years, several approximation algorithms have been developed. By means of matrix perturbation theory, Qiu et al. [4] proposed an algorithm for approximate bounds, and Leng & He [5] for an outer estimation. An outer estimation was also considered by Kolev [6], but for the general case with nonlinear dependencies. Some initial bounds that are easy and quick to compute were discussed by Hladík et al. [7], and an iterative refinement in [8]. An iterative algorithm for outer estimation was given by Beaumont [9]. In this paper we focus more on the inner approximations (subsets) of the eigenvalue sets. There are much fewer papers devoted to inner approximation. Let us mention an evolution strategy method by Yuan et al. [3] or a general method for nonlinear systems [9]. The interval eigenvalue problem has a lot of applications in the field of mechanics and engineering. Let us mention for instance automobile suspension systems [6], mass structures [4], vibrating systems [20], principal component analysis [2], and robotics [22]. Another applications arise from the engineering area concerning singular values and condition numbers. Using the well-known Jordan Wielandt transformation [23, Section 8.6], [24, Section 7.5] we can simply reduce a singular value calculation to a symmetric eigenvalue one. This paper is organized as follows. In Section 2 we introduce the notation that we use throughout the paper. In Section 3 we present our main theoretical result that enables to exactly determine some of the eigenvalue set. It is a basis for the algorithms that we present in Section 4. The algorithms calculate inner approximations of the eigenvalue sets. Even though outer approximation is usually considered in literature, inner approximation is of interest, too. Moreover, due to the main theorem, we can obtain exact eigenvalue bounds in some cases. Finally, in Section 5 we demonstrate our approach by a number of examples and numerical experiments. 2. Basic definitions and theoretical background Let us introduce some notions first. An interval matrix is denoted by boldface and defined as A := [A, A] = {A R m n ; A A A}, where A, A R m n, A A, are given matrices. By A c := 2 (A + A), A := (A A) 2 we denote the midpoint and the radius of A, respectively. By an interval linear system of equations Ax = b we mean a family of systems Ax = b, such that A A, b b. In a similar way we introduce interval linear systems of inequalities and mixed systems of equations and inequalities. A vector x is a solution of Ax = b if it is a solution of Ax = b for some A A and b b. We assume that the reader is familiar with the basics of interval arithmetic; for further details we refer to e.g. [25 27]. Let F be a family of n n matrices. We denote the eigenvalue set of the family F by Λ(F ) := {λ R; A F x 0 : Ax = λx}. A symmetric interval matrix as defined as A S := {A A A = A T }. It is usually a proper subset of A. Considering the eigenvalue set Λ(A), it generally represents an overestimation of Λ(A S ). That is why we focus directly on the eigenvalue set of the symmetric portion, even though we must take into account the dependencies between the elements, in the definition of A S. A real symmetric matrix A R n n has always n real eigenvalues, let us sort them in non-increasing order λ (A) λ 2 (A) λ n (A). We extend this notation for symmetric interval matrices λ i (A S ) := {λ i (A) A A S }. These sets represent n compact intervals λ i (A S ) = [λ i (A S ), λ i (A S )], i =,..., n; cf. [7]. The intervals can be disjoint, can overlap, or some of them, can be identical. However, what cannot happen is that one interval is a proper subset of another interval. The union of these intervals produces Λ(A S ). For instance, consider an interval matrix [2, 3] A S 0 =. () 0 [, 4] Then λ (A S ) = [2, 4], λ 2 (A S ) = [, 3] and Λ(A S ) = [, 4].

118 354 M. Hladík et al. / Computers and Mathematics with Applications 62 (20) Throughout the paper we use the following notation: λ i (A) the ith eigenvalue of a symmetric matrix A (in non-increasing order) σ i (A) the ith singular value of a matrix A (in non-increasing order) v i (A) the eigenvector associated to the ith eigenvalue of a symmetric matrix A ρ(a) the spectral radius of a matrix A S the boundary of a set S conv S the convex hull of a set S diag(y) the diagonal matrix with entries y,..., y n sgn(x) the sign vector of a vector x, i.e., sgn(x) = (sgn(x ),..., sgn(x n )) T x 2 the Euclidean vector norm, i.e., x 2 = x T x x the Chebyshev (maximum) vector norm, i.e., x = max{ x i ; i =,..., n} x y, A B vector and matrix relations are understood component-wise. 3. Main theorem The following theorem is the main theoretical result of the present paper. We remind the reader that the principal m m submatrix of a given n n matrix is any submatrix obtained by eliminating any n m rows and the corresponding n m columns. Theorem. Let λ Λ(A S ). Then there is k {,..., n} and a principal submatrix ˆ A S R k k of A S such that: If λ = λ j (A S ) for some j {,..., n}, then λ {λ i (Â c + diag(z)â diag(z)); z {±} k, i =,..., k}. (2) If λ = λ j (A S ) for some j {,..., n}, then λ {λ i (Â c diag(z)â diag(z)); z {±} k, i =,..., k}. (3) Proof. Let λ Λ(A S ). Then either λ = λ j (A S ) or λ = λ j (A S ), for some j {,..., n}. We assume the former case. The latter can be proved similarly. The eigenvalue λ corresponds to a matrix A A. Without loss of generality we assume that the corresponding eigenvector x, x 2 =, is of the form x = (0 T, y T ) T, where y R k and y i 0, for all i k, and for some k {,..., n}. The symmetric interval matrix A S can be written as A S B S C =, C T D S where B S R (n k) (n k), C R (n k) k, D S R k k. This can be achieved by a suitable permutation P T A S P, where P is a permutation matrix. Notice that P T A S P remains symmetric with the same eigenvalues and eigenvectors, and no overestimation occurs since P T A S P has the same entries as A S but at different positions. From the basic equality Ax = λx it follows that and Cy = 0 for some C C, (4) Dy = λy for some D D S. (5) We focus on the latter relation; it says that λ is an eigenvalue of D. We will show that D S is the required principal submatrix ˆ A S thanks to the proposed permutation, and D could be written as in (2). From (5) we have that λ = y T Dy, and hence the partial derivatives are λ d ij = y i y j 0, i, j =,..., k. This relation strongly influences the structure of D. If y i y j > 0, then d ij = d ij. This is so, because otherwise by increasing d ij we also increase the value of λ, which contradicts our assumption that λ lies on the upper boundary of Λ(A S ). Likewise, y i y j < 0 implies d ij = d ij. This allows us to write D in the following more compact form D = D c + diag(z)d diag(z), (6) where z = sgn(y) {±} k. Therefore, λ belongs to a set as the one presented in the right-hand side of (2), which completes the proof.

119 M. Hladík et al. / Computers and Mathematics with Applications 62 (20) Note that not every λ j (A S ) or λ j (A S ) is a boundary point of Λ(A S ); see (). Theorem is also true for such λ j (A S ) or λ j (A S ) that are non-boundary, but represent no multiple eigenvalue (since the corresponding eigenvector is uniquely determined). However, correctness of Theorem for all λ j (A S ) and λ j (A S ), j =,..., n, is still an open question. Moreover, full characterization of all λ j (A S ) and λ j (A S ), j =,..., n, is lacking too. As we have already mentioned, in general, the eigenvalue set of an interval matrix is larger than the eigenvalue set of its symmetric portion. This is true even if both the midpoint and radius matrices are symmetric (see Example ). The following theorem says that overestimation caused by the additional matrices is somehow limited by the convex hull area. An illustration will be given in Example, where Λ(A S ) = [3.732, ] [ , ] [ 4.072,.0000], Λ(A) = [3.732, ] [ , ] [ 4.072,.0000]. The lower bounds and the upper bounds Λ(A S ) and Λ(A) are always the same, but the other boundary points may differ. Theorem 2. Let A c, A R n n be symmetric matrices. Then conv Λ(A S ) = conv Λ(A). Proof. The inclusion conv Λ(A S ) conv Λ(A) follows from the definition of the convex hull. Let A A be arbitrary, λ one of its real eigenvalues, and x the corresponding eigenvector, where x 2 B := (A + 2 AT ) A S, then the following holds: =. Let Similarly, λ = x T Ax max y T Ay = max y T By = λ (B) conv Λ(A S ). y 2 = y 2 = λ = x T Ax min y T Ay = min y T By = λ n (B) conv Λ(A S ). y 2 = y 2 = Therefore λ conv Λ(A S ), and so conv Λ(A) conv Λ(A S ), which completes the proof. 4. Inner approximation algorithms Theorem naturally yields an algorithm to compute a very sharp inner approximation of Λ(A S ), which could also be exact in some cases. We will present the algorithm in the sequel (Section 4.3). First, we define some notions and propose two simple but useful methods for inner approximations. Any subset of S is called an inner approximation. Similarly, any set that contains S is called an outer approximation. In our case, an inner approximation of the eigenvalue set λ i (A S ), is denoted by µ i (A S ) = [µ i (A S ), µ i (A S )] λ i (A S ), and an outer approximation is denoted by ω i (A S ) = [ω i (A S ), ω i (A S )] λ i (A S ), where i n. From a practical point of view, an outer approximation is usually more useful. However, an inner approximation is also important in some applications. For example, it could be used to measure quality (sharpness) of an outer approximation, or it could be used to prove the (Hurwitz or Schur) instability of certain interval matrices, cf. [28]. We introduce three inner approximation algorithms. The first one, a local improvement, is an efficient algorithm, but needn t be very accurate. On the contrary, vertex enumeration gives more accurate results (two bounds are exact), but it is more costly. Eventually, submatrix vertex enumeration yields the tightest inner approximation but on the account of the time complexity. 4.. Local improvement The first algorithm that we present is based on a local improvement search technique. A similar method, but for interval matrices A with A c and A symmetric, was proposed by Rohn [28]. The basic idea of the algorithm is to start with an eigenvalue, λ i (A c ), and the corresponding eigenvector, v i (A c ), of the midpoint matrix, A c, and then move to an extremal matrix in A S according to the sign pattern of the eigenvector. The procedure is repeated until no improvement is possible. Algorithm outputs the upper boundaries µ i (A S ) of the inner approximation [µ i (A S ), µ i (A S )], where i n. The lower boundaries, µ i (A S ), can be obtained similarly. The validity of the procedure follows from the fact that every considered matrix, A, belongs to A S. The algorithm terminates after at most 2 n + iterations since we can normalize v i (A) such that the first entry is non-negative. However, usually in practice the number of iterations is much smaller, which makes the algorithm attractive for applications. Our numerical experiments (Section 5) indicate that the number of iterations is rarely greater than two, even for matrices of dimension 20. Moreover, the resulting inner approximation is quite sharp, depending on the width of intervals in A S. This is not surprising as whenever the input intervals are narrow enough, the algorithm produces, sometimes

120 356 M. Hladík et al. / Computers and Mathematics with Applications 62 (20) Algorithm (Local improvement for µ i (A S ), i =,..., n) : for i =,..., n do 2: µ i (A S ) = ; 3: A := A c ; 4: while λ i (A) > µ i (A S ) do 5: µ i (A S ) := λ i (A); 6: D := diag(sgn(v i (A))); 7: A := A c + DA D; 8: end while 9: end for 0: return µ i (A S ), i =,..., n. even after the first iteration, exact bounds; see []. This is due to sign invariancy of eigenvectors, which enables to set up an optimal scenario in steps 6 and 7. If the eigenvectors have no invariant signs of their entries, then we still can achieve the optimal bound by the local improvement. We refer the reader to Section 5 for a more detailed presentation of the experiments Vertex enumeration The second method that we present is based on enumeration of some special boundary matrices of A. It consists of inspecting all matrices A z := A c + diag(z)a diag(z), z {±} n, z =, (7) and continuously improving an inner approximation µ i (A S ), whenever λ i (A z ) > µ i (A S ), where i n. The lower bounds, µ i (A S ), could be obtained in a similar way using the matrices A c diag(z)a diag(z), where z {±} n, and z =. The condition z = follows from the fact that diag(z)a diag(z) = diag( z)a diag( z), which gives us the freedom to fix one component of z. The number of steps that the algorithm performs is 2 n. Therefore, this method is suitable only for matrices of moderate dimensions. The main advantages of the vertex enumeration approach are the following. First, it provides us with a sharper inner approximation of the eigenvalue sets than the local improvement; in local improvement we inspect only some of the matrices in (7). Second, two of the computed bounds are exact; by Hertz [0] (cf. []) and Hertz [29] we have that µ (A S ) = λ (A S ) and µ n (A S ) = λ n (A S ). Concerning the other bounds calculated by vertex enumeration, even though it was conjectured that there were exact [2], it turned out that they were not exact, in general [3]. The assertion by Hertz [29, Theorem ] that µ (A S ) = λ (A S ) and µ n (A S ) = λ n (A S ) is wrong, too; see Example 3. Nevertheless, Theorem and its proof indicate a sufficient condition: if no eigenvector corresponding to an eigenvalue of A S has a zero component, then the vertex enumeration yields exactly the eigenvalue sets λ i (A S ), i =,..., n. This is easy to see from the proof of Theorem ; the submatrices in question is only the matrix A itself, and the values (2) (3) correspond to matrices that are processed by vertex enumeration. The efficient implementation of this approach is quite challenging. In order to overcome in practice the exponential complexity of the algorithm, we implemented a branch & bound algorithm, which is in accordance with the suggestions of Rohn [28]. However, the adopted bounds are not that tight, and the actual running times are usually worse than the direct vertex enumeration; it is probably because of weak pruning part of the exhaustive search, so one has to go through almost all the search tree. That is why we do not consider further this variant. The direct vertex enumeration scheme for computing the upper bounds, µ i (A S ), is presented in Algorithm 2. Algorithm 2 (Vertex enumeration for µ i (A S ), i =,..., n) : for i =,..., n do 2: µ i (A S ) = λ i (A c ); 3: end for 4: for all z {±} n, z =, do 5: A := A c + diag(z)a diag(z); 6: for i =,..., n do 7: if λ i (A) > µ i (A S ) then 8: µ i (A S ) := λ i (A); 9: end if 0: end for : end for 2: return µ i (A S ), i =,..., n.

121 M. Hladík et al. / Computers and Mathematics with Applications 62 (20) Submatrix vertex enumeration In this section we present an algorithm that is based on Theorem, and it usually produces very tight inner approximations, even exact ones in some cases. The basic idea underlying the algorithm is to enumerate all the vertices of all the principal submatrices of A S including A S itself. Thus we go through more matrices than vertex enumeration and the method yields more accurate approximation, but with higher time complexity. The number of steps performed with this approach is n 2 n + n2 n n n2 0 = 2 2 (3n ). To overcome the obstacle of the exponential number of iterations, at least in practice, we notice that not all eigenvalues of the principal submatrices of the matrices in A S belong to some of the eigenvalue sets λ i (A S ), where i n. For this we will introduce a condition for checking such an inclusion. Assume that we are given an inner approximation µ i (A S ) and an outer approximation ω i (A S ) of the eigenvalue sets λ i (A S ); that is µ i (A S ) λ i (A S ) ω i (A S ), where i n. As we will see in the sequel, the quality of the output of our methods depends naturally on the sharpness of the outer approximation used. Let D S R k k be a principal submatrix of A S and, without loss of generality, assume that it is situated in the right-bottom corner, i.e., A S B S C =, C T D S where B S R (n k) (n k) and C R (n k) k. This can be obtained by an appropriate permutation P T A S P, where P is a permutation matrix as in the proof of Theorem. Let λ be an eigenvalue of some vertex matrix D D S, which is of the form (6), and let y be the corresponding eigenvector. If the eigenvector is not unique then λ is a multiple eigenvalue and therefore it is a simple eigenvalue of some principal submatrix of D S due to Cauchy s interlacing property for eigenvalues [23, Theorem 8..7] [24, Example 7.5.3]; in this case we restrict our consideration to this submatrix. Let p {,..., n} be fixed. We want to determine whether λ is equal to λ p (A S ) Λ(A S ), or, if this is not possible, to improve the upper bound µ p (A S ); the lower bound can be handled accordingly. In view of (4), Cy = 0 must hold for some C C, whence 0 Cy. So λ is an eigenvalue of some matrix in A S. Now, we are sure that λ Λ(A S ) and it remains to determine whether λ also belongs to λ p (A S ). If λ µ p (A S ), then it is useless to further consider λ, since it would not improve the inner approximation of the pth eigenvalue set. Suppose λ > µ p (A S ). If p = or λ < ω p (A S ), then λ must belong to λ p (A S ), and we can improve the inner bound µ p (A S ) := λ. In this case the algorithm terminates early, and that is the reason we need ω i (A S ), i n, to be as tight as possible. If p > and λ ω p (A S ), we proceed as follows. We pick an arbitrary C C, such that Cy = 0; we refer to, e.g. [30] for details on the selection process. Next, we select an arbitrary B B S and let B C A := C T. D We compute the eigenvalues of A, and if µ p (A S ) < λ p (A), then we set µ p (A S ) := λ p (A), otherwise we do nothing. However, it can happen that λ = λ i (A S ), and we do not identify it, and hence we do not enlarge the inner estimation µ p (A S ). Nevertheless, if we apply the method for all p =,..., n and all principal submatrices of A S, then we touch all the boundary points of Λ(A S ). If λ Λ(A S ), then λ is covered by the resulting inner approximation. In the case when λ is an upper boundary point, we consider the maximal i {,..., n} such that λ = λ i (A S ) and then the ith eigenvalue of the matrix (8) must be equal to λ. Similar tests are valid for a lower boundary point. Now we have all the ingredients at hand for the direct version of the submatrix vertex enumeration approach that is presented in Algorithm 3, which improves the upper bound µ p (A S ) of an inner approximation, where the index p is still fixed. Let us also mention that in step 4 of Algorithm 3, the decomposition of A S according to the index set J means that D S is a restriction of A S to the rows and the columns indexed by J, B S is a restriction of A S to the rows and the columns indexed by {,..., n} \ J, and C is a restriction of A S to the rows indexed by {,..., n} \ J and the columns indexed by J Branch & bound improvement In order to tackle the exponential worst case complexity of Algorithm 3, we propose the following modification. Instead of inspecting all non-empty subsets of {,..., n} in step 3, we exploit a branch & bound method, which may skip some useless subsets. Let a non-empty J {,..., n} be given. The new, possibly improved, eigenvalue λ must lie in the interval (8)

122 358 M. Hladík et al. / Computers and Mathematics with Applications 62 (20) Algorithm 3 (Direct submatrix vertex enumeration for µ p (A S )) : compute outer approximation ω i (A S ), i =,..., n; 2: call Algorithm to get inner approximation µ i (A S ), i =,..., n; 3: for all J {,..., n}, J, do B 4: decompose A S S C = according to J; C T 5: for all z {±} J, z =, do 6: D := D c + diag(z)d diag(z); 7: for i =,..., J do 8: λ := λ i (D); 9: y := v i (D); 0: if λ > µ p (A S ) and λ ω p (A S ) and 0 Cy then D S : if p = or λ < ω p (A S ) then 2: µ p (A S ) := λ; 3: else 4: find C C such that Cy = 0; Bc C 5: A := C T ; D 6: if λ p (A) > µ p (A S ) then 7: µ p (A S ) := λ p (A); 8: end if 9: end if 20: end if 2: end for 22: end for 23: end for 24: return µ p (A S ). λ := [ µ p (A S ), ω p (A S )]. If this is the case, then the interval matrix A S λi must be irregular, i.e., it contains a singular matrix. Moreover, the interval system (A S λi)x = 0, x =, has a solution x, where x i = 0 for all i J. We decompose A S λi according to J, and, without loss of generality, we may assume that J = {n J +,..., n}, then A S B λi = S λi C. C T D S λi The interval system becomes Cy = 0, (D S λi)y = 0, y =, (9) where we considered x = (0 T, y T ) T. This is a very useful necessary condition. If (9) has no solution, then we cannot improve the current inner approximation. We can also prune the whole branch with J as a root; that is, we will inspect no index sets J J. The strength of this condition follows from the fact that the system (9) is overconstrained, it has more equations than variables. Therefore, with high probability that it has no solution, even for larger J. Let us make two comments about the interval system (9). First, this system has a lot of dependencies. They are caused from the multiple occurrences of λ, and by the symmetry of D S. If no solver for interval systems that can handle dependencies is available, then we can solve (9) as an ordinary interval system, forgetting the dependencies. The necessary condition will be weaker, but still valid. This is what we did in our implementation. The second comment addresses the expression y =. We have chosen the maximum norm in order that the interval system be linear. The expression could be rewritten as y (for checking solvability of (9) we can use either normalization y = or y ). Another possibility is to write y, y i = for some i {,..., J }. This indicates that we can split the problem into solving J interval systems Cy = 0, (D S λi)y = 0, y, y i =, where i runs, sequentially, through all the values {,..., J }; cf. the ILS method proposed in [5]. The advantage of this approach is that the overconstrained interval systems have (one) more equation than the original overconstrained system,

123 M. Hladík et al. / Computers and Mathematics with Applications 62 (20) and hence the resulting necessary condition could be stronger. Our numerical results discussed in Section 5 concern this variant. As a solver for interval systems we utilize the convex approximation approach by Beaumont [3]; it is sufficiently fast and produces narrow enough approximations of the solution set How to conclude for exact bounds? Let us summarize properties of the submatrix vertex enumeration method. On the one hand the worst case complexity of the algorithm is rather prohibitive, O(3 n ), but on the other hand, we obtain better inner approximations, and sometimes we get exact bounds of the eigenvalue sets. Theorem and the discussion in the previous section allow us to recognize exact bounds. Namely, for any i {2,..., n}, we have that if λ i (A S ) < λ i (A S ), then µ i (A S ) = λ i (A S ); a similar inequality holds for the lower bound. This is a rather theoretical recipe because we may not know a priori whether the assumption is satisfied. However, we can propose a sufficient condition: if ω i (A S ) < ω i (A S ), then two successive eigenvalue sets do not overlap and the assumption is obviously true. In this case we conclude µ i (A S ) = λ i (A S ); otherwise we cannot conclude. This sufficient condition is another reason why we need a sharp outer approximation. The sharper it is, the more often we are able to conclude that the exact bound is achieved. Exploiting the condition we can also decrease the running time of submatrix vertex enumeration. We call Algorithm 3 only for p {,..., n} such that p = or ω p (A S ) < ω p (A S ). The resulting inner approximation may be a bit less tight, but the number of exact boundary points of Λ(A S ) that we can identify remains the same. Notice that there is enough open space for developing better conditions. For instance, we do not know whether µ i (A S ) < µ i (A S ) (computed by submatrix vertex enumeration) can serve also as a sufficient condition for the purpose of determining exact bounds. 5. Numerical experiments In this section we present some examples and numerical results illustrating properties of the proposed algorithms. We performed the experiments on a PC Intel(R) Core 2, CPU 3 GHz, 2 GB RAM, and the source code was written in C++. We use GLPK v.4.23 [32] for solving linear programs, CLAPACK v.3.. for its linear algebraic routines, and PROFIL/BIAS v [33] for interval arithmetic and basic operations. Notice, however, that routines of GLPK and CLAPACK [34] do not produce verified solutions; for real-life problems this may not be acceptable. Example. Consider the following symmetric interval matrix 2 [, 5] S A S = 2. [, 5] Local improvement (Algorithm ) yields an inner approximation µ (A S ) = [3.732, ], µ 2 (A S ) = [0.0888, ], µ 3 (A S ) = [ 4.072,.0000]. The same result is obtained by the vertex enumeration (Algorithm 2). Therefore, µ (A S ) = λ (A S ) and µ 3 (A S ) = λ 3 (A S ). An outer approximation that is needed by the submatrix vertex enumeration (Algorithm 3) is computed using the methods of Hladík et al. [7,8]. It is ω (A S ) = [3.5230, ], ω 2 (A S ) = [0.0000,.059], ω 3 (A S ) = [ 4.24, 0.209]. Now, the submatrix vertex enumeration algorithm yields the inner approximation µ (AS ) = [3.732, ], µ 2 (AS ) = [0.0000, ], µ 3 (AS ) = [ 4.072,.0000]. Since the outer approximation intervals do not overlap, we can conclude that this approximation is exact, that is, λ i (A S ) = µ i (AS ), i =, 2, 3. This example shows two important aspects of the interval eigenvalue problem. First, it demonstrates that the vertex enumeration does not produce exact bounds in general. Second, the symmetric eigenvalue set can be a proper subset of the

124 360 M. Hladík et al. / Computers and Mathematics with Applications 62 (20) unsymmetric one, i.e., Λ(A S ) Λ(A). This could be easily seen by the matrix It has three real eigenvalues , and.0000, but the second one does not belong to Λ(A S ). Indeed, using the method by Hladík et al. [5] we obtain Λ(A) = [3.732, ] [ , ] [ 4.072,.0000]. Example 2. Consider the example given by Qiu et al. [4] (see also [7,3]): S [2975, 3025] [ 205, 985] 0 0 A S [ 205, 985] [4965, 5035] [ 3020, 2980] 0 = 0 [ 3020, 2980] [6955, 7045] [ 4025, 3975]. 0 0 [ 4025, 3975] [8945, 9055] The local improvement (Algorithm ) yields an inner approximation µ (A S ) = [ , ], µ 2 (A S ) = [ , ], µ 3 (A S ) = [ , ], µ 4 (A S ) = [ , ]. The vertex enumeration (Algorithm 2) produces the same result. Hence we can state that µ (A S ) and µ 4 (A S ) are optimal. To call the last method, submatrix vertex enumeration (Algorithm 3) we need an outer approximation. We use the following by [7] ω (A S ) = [ , ], ω 2 (A S ) = [ , ], ω 3 (A S ) = [ , ], ω 4 (A S ) = [ , ]. Now, submatrix vertex enumeration yields the same inner approximation as the previous methods. However, now we have more information. Since the outer approximation intervals are mutually disjoint, the obtained results are the best possible. Therefore, µ i (A S ) = λ i (A S ), where i =,..., 4. Example 3. Herein, we present two examples for approximating the singular values of an interval matrix. Let A R m n and q := min{m, n}. By the Jordan Wielandt theorem [23, Section 8.6], [24, Section 7.5] the singular values σ (A) σ q (A) of A are identical to the q largest eigenvalues of the symmetric matrix 0 A T A 0. Thus, if we consider the singular value sets σ (A),..., σ q (A) of some interval matrix A R m n, we can identify them as the q largest eigenvalue sets of the symmetric interval matrix 0 A M := T S. A 0 () Consider the following interval matrix from [35] [2, 3] [, ] A = [0, 2] [0, ]. [0, ] [2, 3] Both the local improvement and the vertex enumeration result in the same inner approximation, i.e. µ (M) = [2.566, 4.543], µ 2 (M) = [.220, 2.854]. Thus, σ (A) = Additionally, consider the following outer approximation from [7]. ω (M) = [2.0489, 4.543], ω 2 (M) = [0.4239, 3.87]. Using Algorithm 3, we obtain µ (M) = [2.566, 4.543], µ 2 (M) = [.0000, 2.854]. Now we can claim that σ 2 (A) =, since ω 2 (M) > 0. Unfortunately, we cannot conclude about the exact values of the remaining quantities, since the two outer approximation intervals overlap. We only know that σ (A) [2.0489, 2.566] and σ 2 (A) [2.854, 3.87].

125 M. Hladík et al. / Computers and Mathematics with Applications 62 (20) (2) The second example comes from Ahn & Chen [36]. Let A be the following interval matrix [0.75, 2.25] [ 0.05, 0.005] [.7, 5.] A = [3.55, 0.65] [ 5.,.7] [.95, 0.65]. [.05, 3.5] [0.005, 0.05] [ 0.5, 3.5] Both local improvement and vertex enumeration yield the same result, i.e. µ (M) = [4.66, 3.937], µ 2 (M) = [2.240,.5077], µ 3 (M) = [0.296, 2.97]. Hence, σ (A) = As an outer approximation we use the following intervals calculated by a method from [7] ω (M) = [4.3308, 4.05], ω 2 (M) = [.9305,.6], ω 3 (M) = [0.0000, 5.000]. Running the submatrix vertex enumeration, we get the inner approximation µ (M) = [4.5548, 3.937], µ 2 (M) = [2.240,.5077], µ 3 (M) = [0.296, 2.957]. We cannot conclude that σ 3 (A) = µ 3 (A) = 0.296, because ω 3 (M) has a nonempty intersection with the fourth largest eigenvalue set, which is equal to zero. Also the other singular value sets remain uncertain, but within the computed inner and outer approximations. Notice that µ (M) < µ (M), whence µ (M) < λ (M) = σ (A) disproving the Hertz s theorem from [29] that the lower and upper limits of λ (M) and λ n (M) are computable by the vertex enumeration method. It is true only for λ (M) and λ n (M). Example 4. In this example we present some randomly generated examples of large dimensions. The entries of the midpoint matrix, A c, are taken randomly in [ 20, 20] using the uniform distribution. The entries of the radius matrix A are taken randomly, using the uniform distribution in [0, R], where R is a positive real number. We applied our algorithm on the interval matrix M := A T A, because it has a convenient distribution of eigenvalue set some are overlapping and some are not. Sharpness of results is measured using the quantity et µ (M S ) e T ω (M S ), where e = (,..., ) T. This quantity lies always within the interval [0, ]. The closer to zero it is, the tighter the approximation. In addition, if it is zero, then we achieved exact bounds for every eigenvalue set λ i (M S ), i n. The initial outer approximation, ω i (M S ), i n, was computed using the method due of Hladík et al. [7], and filtered by the method proposed by Hladík et al. in [8]. Finally, it was refined according to the comment in Section For the submatrix vertex enumeration algorithm we implemented the branch & bound improvement, which is described in Sections 4.3. and The results are displayed in Table ; the values are appropriately rounded. We see that local improvement yields almost as tight inner approximation as vertex enumeration, but with much lower effort. Submatrix vertex enumeration is even more costly, but it can sometimes conclude for exact bounds, so the approximation is more accurate, particularly for narrow input intervals. Example 5. In this example we present some numerical results on approximating singular value sets as introduced in Example 3. The input consists of an interval (rectangular) matrix A R m n which is selected randomly as in the previous example. Table 2 presents our experiments. The time in the table corresponds to the computation of the approximation of only the q largest eigenvalue sets of the Jordan Wielandt matrix. The behavior of the three algorithms is similar to that in Example Conclusion and future directions We proposed a new solution theorem for the symmetric interval eigenvalue problem, which describes some of the boundary points of the eigenvalue set. Unfortunately, the complete characterization is still a challenging open problem. We developed an inner approximation algorithm (submatrix vertex enumeration), which in the case where the eigenvalue sets are disjoint, and the intermediate gaps are wide enough, outputs exact results. To our knowledge, even under this assumption, this is the first algorithm that can guarantee exact bounds. Thus, it can be used in correspondence with outer approximation methods to produce exact eigenvalue sets. We carried out comparisons with other inner approximation methods, local improvement and vertex enumeration. The local improvement method is very efficient with sufficiently tight bounds. The vertex enumeration is more time

126 362 M. Hladík et al. / Computers and Mathematics with Applications 62 (20) Table Eigenvalues of random interval symmetric matrices A T A of dimension n n. n R Algorithm (local improvement) Algorithm 2 (vertex enumeration) Algorithm 3 (submatrix vertex enumeration) Sharpness Time (s) Sharpness Time Sharpness Time s s s s s s s s s s s s s s s s s s s min 29 s s min 58 s s s min 2 s min 46 s min 6 s h 4 min 55 s min 4 s h 5 min 4 s min 33 s min 39 s h 53 min 0 s h 32 min 54 s Table 2 Singular values of random interval matrices of dimension m n. m n R Algorithm (local improvement) Algorithm 2 (vertex enumeration) Algorithm 3 (submatrix vertex enumeration) Sharpness Time (s) Sharpness Time Sharpness Time s s s s s s s s s s s s min 3 s min 36 s min 33 s min 58 s min 32 s min 47 s min 3 s min 9 s min 0 s min 2 s min 52 s h 48 min 38 s h 5 min 53 s h 54 min 56 s consuming with slightly more accurate bounds, two of which are exact. Our numerical experiments suggest that the local search algorithm is superior to the other methods as long as the input matrices have higher dimension. However, for small dimensional problems with possibly narrow input intervals, the submatrix vertex enumeration approach gives very accurate bounds in reasonable time. Thus local improvement is suitable for high dimensional problems or for problems where computing time is important. Contrary, submatrix vertex enumeration is a good choice when accuracy is the main objective. Acknowledgments The authors thank the reviewers for their detailed and helpful comments. ET is partially supported by an individual postdoctoral grant from the Danish Agency for Science, Technology and Innovation, and also acknowledges support from the Danish National Research Foundation and the National Science Foundation of China (under the grant ) for the Sino-Danish Center for the Theory of Interactive Computation, within which part of this work was performed. References [] A.S. Deif, The interval eigenvalue problem, Z. Angew. Math. Mech. 7 () (99) [2] J. Rohn, A. Deif, On the range of eigenvalues of an interval matrix, Computing 47 (3 4) (992) [3] A. Deif, J. Rohn, On the invariance of the sign pattern of matrix eigenvectors under perturbation, Linear Algebr. Appl. 96 (994) [4] J. Rohn, Interval matrices: singularity and real eigenvalues, SIAM J. Matrix Anal. Appl. 4 () (993) [5] M. Hladík, D. Daney, E.P. Tsigaridas, An algorithm for addressing the real interval eigenvalue problem, J. Comput. Appl. Math. 235 (8) (20) [6] Z. Qiu, P.C. Müller, A. Frommer, An approximate method for the standard interval eigenvalue problem of real non-symmetric interval matrices, Comm. Numer. Methods Engrg. 7 (4) (200) [7] G. Alefeld, G. Mayer, Interval analysis: theory and applications, J. Comput. Appl. Math. 2 ( 2) (2000)

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128 Applied Mathematics and Computation 27 (20) Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: A filtering method for the interval eigenvalue problem Milan Hladík a,b,, David Daney b, Elias Tsigaridas c a Charles University, Faculty of Mathematics and Physics, Department of Applied Mathematics, Malostranské nám. 25, 800 Prague, Czech Republic b INRIA Sophia-Antipolis Méditerranée, 2004 route des Lucioles, BP 93, Sophia-Antipolis Cedex, France c Department of Computer Science, Aarhus University, IT-parken, Aabogade 34, DK 8200 Aarhus N, Denmark article Keywords: Interval matrix Symmetric matrix Interval analysis Eigenvalue Eigenvalue bounds info abstract We consider the general problem of computing intervals that contain the real eigenvalues of interval matrices. Given an outer approximation (superset) of the real eigenvalue set of an interval matrix, we propose a filtering method that iteratively improves the approximation. Even though our method is based on a sufficient regularity condition, it is very efficient in practice and our experimental results suggest that it improves, in general, significantly the initial outer approximation. The proposed method works for general, as well as for symmetric interval matrices. Ó 200 Elsevier Inc. All rights reserved.. Introduction In order to model real-life problems and perform computations we must deal with inaccuracy and inexactness; these are due to measurement, to simplification assumption on physical models, to variations of the parameters of the system, and finally due to computational errors. Interval analysis is an efficient and reliable tool that allows us to handle the aforementioned problems, even in the worst case where all together are encountered simultaneously. The input quantities are given with some interval estimation and the algorithms output verified intervals as results that (even though they usually have the drawback of overestimation) cover all the possibilities for the input quantities. We are interested in the interval real eigenvalue problem. Given a matrix the elements of which are real intervals, also called interval matrix, the task is to compute real intervals that contain all possible eigenvalues. For formal definitions we refer the reader to the next section. Moreover, there is a need to distinguish general interval matrices from the symmetric ones. Applications arise mostly in the field of mechanics and engineering. We name, for instance, automobile suspension system [], mass structures [2], vibrating systems [3], robotics [4], and even principal component analysis [5] and independent component analysis [6], which could be considered as a statistics oriented applications. Using the well-known Jordan Wielandt transformation [7,8], if we are given a solution of the interval real eigenvalue problem, we can provide an approximation for the singular values and the condition number; both quantities have numerous applications. The first general results for the interval real eigenvalue problem were produced by Deif [9], and Deif and Rohn [0]. However their solutions depend on theorems that have very strong assumptions. Later, Rohn [], introduced a boundary point characterization of the eigenvalue set. Approximation methods were addressed by Qiu et al. [], Leng et al. [2] and by Hladík et al. [3]. The works [2,3] are based on a branch and prune approach and yield results that depend on a given arbitrarily high accuracy. Corresponding author at: Charles University, Faculty of Mathematics and Physics, Department of Applied Mathematics, Malostranské nám. 25, 800 Prague, Czech Republic. addresses: milan.hladik@matfyz.cz, hladik@kam.mff.cuni.cz (M. Hladík), david.daney@sophia.inria.fr (D. Daney), elias.tsigaridas@gmail.com (E. Tsigaridas) /$ - see front matter Ó 200 Elsevier Inc. All rights reserved. doi:0.06/j.amc