On a Unied Representation of Some Interval Analytic. Dedicated to the professors of mathematics. L. Berg, W. Engel, G. Pazderski, and H.- W. Stolle.

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1 Rostock. Math. Kolloq. 49, 75{88 (995) Subject lassication (AMS) 65F5, 65G, 65H Gunter Mayer On a Unied Representation of Some Interval Analytic Algorithms Dedicated to the professors of mathematics L. erg, W. Engel, G. Pazderski, and H.- W. Stolle. ASTRAT. In this article we show that several interval analytic algorithms for verifying solutions x of various mathematical problems can be viewed as special cases of some iterative method involving the rst and the second derivative of the underlying function. Starting with an approximation of x a way is given how to construct interval vectors which contain x, and how to improve these interval bounds. KEY WORDS. Systems of nonlinear equations, second order methods, enclosure methods, verication methods, interval methods for nonlinear systems, algebraic eigenproblem, singular value decomposition, quadratic systems, invariant supspaces Introduction The solutions of many mathematical problems can be expressed as zeros of some function f : D R n R n. Among these problems are the algebraic eigenproblem, the generalized eigenproblem, the singular value problem and the generalized singular value problem, see []{[4], [8]. For example, let A; 2 R nn ; 2 Rnfg and i 2 f; : : : ; ng be given. Then with x := (v T ; ) T, v 2 R n, 2 R the zeros x = ((v ) T ; ) T of the function f(x) := Av? v v i? are obviously eigenpairs of the generalized eigenproblem Av = v with the eigenvector v = (v i ) being normalized by v i = 6=. We will show how certain algorithms for verifying and enclosing solutions of the above problems can be derived from one verication method for general systems of nonlinear equations. To this end we will consider the interval ()

2 76 Gunter Mayer function [g]([x];ex) := t(ex) + ft (ex) + [H]([x];ex)g([x]? ex) (2) for which the function t : D R n R n is assumed to be twice continuously dierentiable on a given open set D. The matrix t (ex) is the Jacobian of t at some xed vector ex 2 D. The function [H] = [H]([x];ex) is dened for all interval vectors [x] D and has n n interval matrices as values. It is supposed to be continuous and inclusion monotone with respect to [x]. For t, [H] and all [x] D we require and t(x) 2 [g]([x];ex) for all x 2 [x] (3) k j[h]([x];ex)j k k j[x]? exj k: (4) The constant is positive and xed for all [x] D; it may depend on ex. Throughout the paper k k denotes the maximum norm of a real vector and the row sum norm of a real matrix, respectively; j j is the absolute value which we shall dene for interval quantities in our next section. There, we also show how t and [H] normally are related to the given function f, the zeros of which we are interested in. We shall derive criteria (Theorem ) for guaranteeing the existence of a xed point x of t which then turns out to be a zero of f. In verifying x we will construct an interval vector [x] which contains x and which in a natural way provides lower and upper bounds for it. Some of our criteria will yield the subset property [g]([x] ;ex) [x] which, together with rouwer's xed point theorem, forms the basis for many verication algorithms. We also will improve the enclosure [x] of x by considering the iteration [x] k+ := [g]([x] k ;ex) \ [x] k ; k = ; ; : : : ; where the intersection can be dropped if the above{mentioned subset property holds. Under slight additional assumptions on [H] we will show that x is the unique xed point of t within [x], that all the iterates [x] k contain it as element and that they contract to it for k. For the standard choice of t and [H] it will turn out that the function [g] reduces to that in Platzoder [4] and Alefeld [5]. Therefore, parts of our criteria are generalizations of results of these authors. 2 Results In order to formulate our results we rst list some notations needed later on. y IR, IR n, IR mn, respectively, we denote the set of real compact intervals, the set of vectors with n

3 On a Unied Representation of Some Interval Analytic Algorithms 77 interval components, and the set of m n matrices with entries from IR. For degenerate intervals [a; a] we simply write a identifying in this way R with f[a; a] j a 2 Rg IR. We proceed similarly for degenerate interval vectors and degenerate interval matrices. Examples are the null matrix O, the identity matrix I, the i{th column e (i) of I and the vector e := (; ; : : : ; ) T. In order to indicate I 2 R nn we sometimes write I n instead of I. As usual, we equip R n and R mn, respectively, with the natural semi{ordering `' which is dened to hold entrywise. We use the notation [A] = ([a] ij ) 2 IR mn simultaneously without further reference, and we assume the same for the elements of R n ; R mn ; IR and IR n. For [a] = [a; a] 2 IR we dene the absolute value j[a]j by j[a]j := maxfjaj j a 2 [a]g = maxfjaj; jajg and the diameter d([a]) by d([a]) := a? a, and we denote the convex hull of [a]; [b] 2 IR by [a][[b]. For interval vectors and interval matrices these terms are applied entrywise. If f(x) is an expression for some function f, we write f([x]) for the interval arithmetic evaluation of this expression (cf. [5] assuming that f([x]) exists. further details on interval analysis we refer to [5] or [3]. We start by presenting the main result of our paper. Among others, this result lists criteria for proving the existence of a xed point x of t from (2) and for guaranteeing the subset property for some interval vector [x]. [g]([x] ; ex) = t(ex) + ft (ex) + [H]([x] ;ex)g([x]? ex) [x] (5) Theorem With D; [g]; [H]; t; ex as in (2) { (4) and with > from (4) choose r 2 R with r > such that [x] := ex + [?r; r]e D, and dene ; by := k t(ex)? ex k; := k t (ex) k: For Let < ; := (? ) 2? 4 and let r? := (?? p )=(2); r + := (? + p )=(2): a) If r r? then t has at least one xed point x 2 [x]. The iteration [x] k+ := [g]([x] k ;ex) \ [x] k ; k = ; ; : : : converges to some interval vector [x] with x 2 [x] [x] k [x] k? : : : [x] ; k 2 N : b) If r 2 [r? ; r + ] then t has at least one xed point x 2 [x]. In addition, [g]([x] ; ex) [x] holds and the iteration [x] k+ := [g]([x] k ;ex); k = ; ; : : :

4 78 Gunter Mayer converges to some interval vector [x] with x 2 [x] [x] k [x] k? : : : [x] ; k 2 N : c) In addition to (4), let [H] fulll k d ([H]([x];ex)) k k d([x]) k (6) for all interval vectors [x] D and for some positive number which is independent of [x] but which may depend on ex. Dene ^; ^r? ; ^r + as ; r? ; r +, with being replaced by ^ := maxf; g. If ^ and if r 2 [^r? ; (^r? + ^r + )=2) then the function t has exactly one xed point x 2 [x] ; [g]([x] ;ex) [x] holds, and the iteration [x] k+ := [g]([x] k ; ex); k = ; ; : : : converges to x with x 2 [x] k [x] k? : : : [x] ; k 2 N : Proof: First we remark that the assumptions < and guarantee r? r +. b) Let [x] := ex + [?r; r]e. Then (5) is equivalent to [g]([x] ; ex)? ex [x]? ex: (7) Property (7) certainly holds if jt(ex)? exj +? jt (ex)j + j[h]([x] ;ex)j re re; and this, in turn, is true if + ( + r)r r; (8) where we used (4). Now (8) can be rewritten as r 2 + (? )r + (9) with equality for r = r? by (3) and r = r +. Hence (9) is fullled for each r 2 [r? ; r + ], and t(x) 2 [g]([x] ;ex) [x] holds for any x 2 [x]. Therefore, rouwer's xed point theorem guarantees that t has at least one xed point x 2 [x]. Since [H] was assumed to be inclusion monotone the iterates [x] k decrease monotonically with respect to the semi{ordering `'; hence they are convergent to some limit [x], and holds for k = ; ; : : : by induction. x = t(x ) 2 [g]([x] k ;ex) = [x] k+

5 On a Unied Representation of Some Interval Analytic Algorithms 79 a) hoose r r such that r 2 [r? ; r + ]. Then b) applies with [^x] := ex + [?r ; r ]e [x] yielding a xed point x 2 [^x] [x] of t. y the intersection in the denition of [x] k+ the iterates [x] k decrease monotonically with respect to `', thus the proof of a) terminates analogously to that of b). c) Since ^ we have ^ and r? = 2? + p ^r? ^r + r + = 2?? p : Therefore, r is contained in [r? ; r + ], and c) is proved by b) with the exception of the uniqueness of x and of the degeneracy of [x] = [x ; x ]. In order to show d([x] ) = apply d() to the equality [x] = [g]([x] ;ex). Then by the subdistributivity of the interval arithmetic and by elementary rules for the diameter (cf. [5] for example) one obtains d([x] ) d (t (ex)([x]? ex) + [H]([x] ;ex) ([x]? ex)) = jt (ex)j d([x] ) + d ([H]([x] ;ex) ([x]? ex)) jt (ex)j d([x] ) + d([h]([x] ;ex)) j[x]? exj + j[h]([x] ;ex)j d([x] ) jt (ex)j d([x] ) + d([h]([x] ;ex)) j[x]? exj + j[h]([x] ;ex)j d([x] ) Let d := kd([x] )k and apply k k to this inequality in order to get d d + rd + rd d + 2r^d : If d >, we obtain + 2r^ which yields the contradiction r? 2^ = ^r? + ^r + : 2 Therefore, d =, and x 2 [x] implies [x] = [x ; x ]. In particular, this proves uniqueness. Note that if ex is a suciently good approximation of a xed point x of t then will be small and the assumption will certainly be fullled provided that <. In practical applications, one often chooses [H]([x];ex) := 2 t ([x][ex)([x]? ex) 2 IR nn () where t (x) : ( R n R n R n (y; z) 7 t (x)(y; z) ()

6 8 Gunter Mayer is the second derivative of t = (t i ) at x 2 D. In () we assume, that t(x) (y) is dened by t (x)(y) := and in () we dene t(x) (y; z) by y T t (x). y T t n(x) A 2 R nn for x 2 D and y 2 R n ; t (x)(y; z) :=? y T t (x)z; : : : ; yt t n (x)z 2 R n ; with 2 t t i (x) i (x) := = x l x k 2 t i (x) x 2 2 t i (x) x x 2. 2 t i (x) x x n 2 t i (x) x 2 x : : : 2 t i (x) x 2 2. : : : 2 t i (x) : : : x 2 x n. 2 t i (x) x n x 2 t i (x) x n x 2. 2 t i (x) x 2 n A = (grad t i (x)) 2 R nn being the Hessian associated with t i (x). Note that k counts the rows while l counts the columns. The reason behind the choice of [H] according to () is the Taylor expansion of t at ex 2 [x] which we write in the form t(x) = t(ex) + ft (ex) + R(x;ex)g(x? ex) with the remainder term R(x; ex) (x?ex), where R(x;ex) 2 R nn. We remind that according to [7], p. 284, and by applying the extended mean value theorem, the entries r ij (x;ex) of R(x; ex) can be expressed as Z r ij (x;ex) = (x? ex) T 2 t i (ex + (x? ex)) (? ) d x j x k = Z 2 (? ) d (x? ex) T t i ( (ijk) ) x j x k = 2 (x? ex)t 2 t i x j x k ( (ijk) ) k=;::: ;n with (ijk) 2 R n between x and ex for i; j; k = ; : : : n. Hence k=;::: ;n k=;::: ;n R(x; ex) 2 2? ([x]? ex) T t i ([x][ex) = [H]([x];ex)

7 On a Unied Representation of Some Interval Analytic Algorithms 8 and t(x) 2 [g]([x];ex) for all x 2 [x] and [H] from (). In addition, [H]([x]; ex) is continuous and inclusion monotone since the function c([x]) := [x][ex has these properties and since [H]([x];ex) can be interpreted as the interval arithmetic evaluation of the expression 2 details.? (x? ex) T t (c(x)) ; cf.[5] or [3] for Assume now that a function f : D R n R n is given and that [x] D. We are interested in the zeros x 2 [x] of f. To this end use the transformation t(x) := x? f(x); 2 R nn nonsingular and independent of x: (2) Then the zeros of f are the xed points of t and vice versa. With t (x) = I? f (x) and with [H] from () we get [g]([x];ex) = ex? f(ex) + I? f (ex)? 2 (f) ([x][ex)([x]? ex) ([x]? ex): (3) For ex 2 [x] this is just the function k 3 in [5], p. 239, and k 7 in [4], p. 3. Therefore, it is not astonishing that Theorem a) and b) reduces to similar results as in [4], x 4, and in [5], x 9. For a comparison take into account the factor in (2). ut note that we will also 2 use [H] and [g] in Example 2 in a dierent meaning as in () and (3). This is caused by the possibility of representing the remainder term R(x; ex)(x? ex) in dierent ways, as is also shown in the following example. Example Let f(x) := x2? 2xy + y 2 f (x)(y; z) = for all x; y; z 2 R 2, whence Let 2?2 yt?2 2 f (x)(y) = A(y) := z 2 R 2, := I, t(x) := x? f(x). Then A = 2y (z? z 2 ) + 2y 2 (?z + z 2 ) 2?2 yt?2 2 2?4 yt 2 A 2 R 22 : (4) A 2 R 22 : (5)

8 82 Gunter Mayer 2y Then apparently f 2 (x)(y; y) =? 4y y 2 + 2y2 2 A(y) and therefore f (x)(y; z) 6= A(y)z, in general. = A(y)y holds although f (x)(y) 6= So we can represent t(x) by means of (4) as well as by means of (5). hoosing [H]([x];ex) := 2 (f) ([x][ex)([x]? ex) and [H]([x];ex) := A([x]? ex), respectively, yields two dierent admissible interval functions 2 [g], where the second one diers from (3). If ex is a suciently good approximation of a zero of f, if f (ex)? exists, and if f (ex)? then,, and r?, r + = for the quantities in Theorem. In particular, the assumptions < and of this theorem are fullled. We want to apply now Theorem with (2) and (2) to various problems of numerical analysis. Example 2 (The generalized eigenproblem with a simple real eigenvalue) onsider the generalized algebraic eigenproblem Av = v as in x. Let ev 2 R n be an approximation of an eigenvector which belongs to an algebraic simple eigenvalue. Let e be an approximation of this eigenvalue and use f from (), 2 R (n+)(n+) nonsingular, ex := (ev T ;e )T 2 R n+, and [v] 2 IR n. In [5] the interval function [g]([x];ex) := ex? f(ex) + ( I n+? A? e?[v]) (e (i ) ) T ) ([x]? ex) (6) was applied in order to verify eigenpairs of the generalized eigenproblem. With t(x) = x? f(x) as in (3) one gets t (ex) = I n+? A? e?ev (e (i ) ) T In [2] it was mentioned that for degenerate interval vectors [x] x the expression [g](x; ex) from (6) is the complete Taylor expansion of t(x) at ex even if ex 62 [x]. Therefore, t(x) 2 [g]([x];ex) holds trivially for all x 2 [x]. Nevertheless [g]([x];ex) is not in the form (3), i. e., [H]([x];ex) is not given by (). In fact, in order to obtain the representation (2) we have to dene [H]([x];ex) := O ([v]? ev) O : 2 IR (n+)(n+) : (7) One can recover this function if one expresses the last, i. e. third, Taylor summand 2 t (x) (x? ex; x? ex) appropriately and if one evaluates this expression interval arithmeti-

9 On a Unied Representation of Some Interval Analytic Algorithms 83 cally. We want to show that [H] fullls (4) and (6). From (7) we get O jj j[v]? evj j[h]([x];ex)j jj O and With := k jj jj d([h]([x];ex)) = jj k this implies k j[h]([x];ex)j k max i max i k= j= O jjd([v]) O jcj ik k= = k j[x]? exj k j= : jbj kj j[v]? evj j jcj ik jbj kj k j[x]? exj k and, analogously, Therefore, Theorem applies with ^ :=. k d([h]([x];ex)) k k d([x]) k: For practical applications there is also a modication of [g] which one gets by choosing in the form = A (8) with 2 R (n?)n, 2 2 R n?, T 2 2 R n?, 22 2 R. Theorem then reduces to a result in [4]. For details see [2]. Example 3 (The algebraic eigenproblem with a simple real eigenvalue) Here we start again with f as in (), where this time we choose := I. The results in Example 2 remain true, of course, and are therefore omitted. They can be found, in [9]{[2] where they have been derived in a dierent way. For the particular choice of in (8) they are already contained in []. We assumed that the eigenvalue to be enclosed is an algebraic simple one. This is due to the fact that only in this case f (x )? exists where x = (v T ; ) T is a corresponding

10 84 Gunter Mayer eigenpair; cf. Theorem 2 in [2] for details. Thus, for a suciently good approximation ex the inverse of f (ex) exists, and f (ex)? can be chosen as in the remark preceding the Example 2. Example 4 (Two{dimensional invariant subspaces) In order to enclose double or nearly double eigenvalues, Alefeld and Spreuer verify in [6] a basis of a two{dimensional subspace of R n which is invariant with respect to the linear mapping given by A 2 R nn. To this end they start with the function f(x) := Au? m u? m 2 v u i? " u i2? Av? m 2 u? m 22 v v i? v i2? A 2 R 2n+4 where x = (u T ; m ; m 2 ; v T ; m 2 ; m 22 ) T 2 R 2n+4, i 6= i 2 2 f; : : : ; ng and "? 6=. It is obvious that the vectors u ; v, which are part of a zero x = ((u ) T ; m ; m 2 ; (v ) T ; m 2 ; m 22) T of f, form a basis of such an invariant subspace. Note that u ; v are unique within a xed subspace because of the four normalization conditions which are hidden in f. Again we set t(x) := x? f(x) with a nonsingular matrix 2 R (2n+4)(2n+4), and we choose ex = (eu T ; em ; em 2 ;ev T ; em 2 ; em 22 ) T as an approximation of x. With and := [T ] := A? em I n?eu?ev?em 2 I n (e (i ) ) T (e (i2) ) T?em 2 I n A? em 22 I n?eu?ev (e (i ) ) T (e (i 2) ) T O [u]? eu [v]? ev O O O [u]? eu [v]? ev A A 2 R (2n+4)(2n+4) ; 2 IR (2n+4)(2n+4) [g]([x];ex) := ex? f(ex) + fi 2n+4? (? [T ])g([x]? ex)

11 On a Unied Representation of Some Interval Analytic Algorithms 85 we again have t (ex) = I 2n+4? and t(x) 2 [g]([x];ex) for all x; ex 2 R n. Note that [g](x;ex) is again the Taylor expansion of t(x) at ex; cf. [2], for example. Therefore, Theorem applies with and with [H]([x];ex) := [T ] ^ := := := 2jj(e T ; ; ; e T ; ; ) T 2k k; e := (; : : : ; ) T 2 R n : The results coincide with those in [6] and [2]. Example 5 (The singular value problem) Each rectangular real matrix A 2 R mn can be represented as A = V U T () AU = V () A T V = U T with orthogonal matrices U 2 R nn, V 2 R mm and with a rectangular diagonal matrix 2 R mn, where () ij := ( for i 6= j i for i = j ; 2 : : : r > r+ = : : : = minfm;ng = : The product V U T is called a singular value decomposition of A, the positive values i are called (non{trivial) singular values of A. Note that is unique while U and V are not. The i{th singular value and the i{th columns u i ; v i of U and V { so{called singular vectors { can be expressed as the zeros of the function f(x) := Au? v A T v? u u T u? where x := (u T ; v T ; ) T. If x = ((u ) T ; (v ) T ; ) T is a zero of f with 6= then A ; (v ) T v = (v ) T Au =? A T v T u = (u ) T u = : Let ex = (eu T ;ev T ;e) T. In [2] a slight modication of the interval function [g]([x];ex) := ex? f(ex) + 8 >< >: I m+n+?? O O [v]? ev O O [u]? eu ([u]? eu) T 9 A >= A>; ([x]? ex)

12 86 Gunter Mayer was used with := A?eI m?ev?ei n A T?eu 2eu T in order to verify singular values and corresponding singular vectors u; v of A. It is an easy task to prove that [g](x;ex) is again the complete Taylor expansion of t(x) := x? f(x) at x = ex with t (ex) = I?. As in Example 2 one easily checks that (4) and (6) hold for with [H]([x];ex) := A O O [v]? ev O O [u]? eu ([u]? eu) T := := k jj (; : : : ; ; n) T k: A : Therefore, Theorem applies with ^ =. f. also [2]. Example 6 (Quadratic systems) In this example we consider systems of equations of the form with b; x 2 R n ; A 2 R nn and with T (x; y) := t(x) := b + Ax + T (x; x) = x t ijk x k y j j= k= i=;::: ;n Note that t() = b; t () = A and t (x)(y; z) = 2T (y; z). Let [H]([x]; ) := T ([x]) := k= [g]([x]; ) := b + (A + T ([x]))[x]: t ijk [x] k 2 IR nn ; : In particular, ex = in this example. One easily sees that (4) and (6) hold with := := k k= jt ijk j k = max in ( n X j= k= jt ijk j ) (9) Therefore, Theorem applies with := kbk, := kak, with ; as in (9) and with ^ =. Its results coincide with those in [3] and [2].

13 On a Unied Representation of Some Interval Analytic Algorithms 87 References [] Alefeld, G. : erechenbare Fehlerschranken fur ein Eigenpaar unter Einschlu von Rundungsfehlern bei Verwendung des genauen Skalarprodukts. Z. Angew. Math. Mech. 67, 45{52 (987) [2] Alefeld, G. : Rigorous error bounds for singular values of a matrix using the precise scalar product. In: Kaucher, E., Kulisch, U., Ullrich, h. (eds.): omputerarithmetic. pp. 9{3. Stuttgart 987 [3] Alefeld, G. : Errorbounds for quadratic systems of nonlinear equations using the precise scalar product. In: Kulisch, U., Stetter, H. J. (eds.): Scientic omputation with Automatic Result Verication. omputing, Suppl. 6, 59{68 (988) [4] Alefeld, G. : erechenbare Fehlerschranken fur ein Eigenpaar beim verallgemeinerten Eigenwertproblem. Z. Angew. Math. Mech. 68, 8{84 (988) [5] Alefeld, G. and Herzberger, J. : Introduction to Interval omputations. New York 983 [6] Alefeld, G. and Spreuer, H. : Iterative improvement of componentwise errorbounds for invariant subspaces belonging to a double or nearly double eigenvalue. omputing 36, 32{334 (986) [7] Heuser, H. : Lehrbuch der Analysis. Teil 2. Stuttgart 98 [8] Homann, R. : Konstruktion von Fehlerschranken bei der verallgemeinerten Singularwertzerlegung und ihre iterative Verbesserung. Thesis, Universitat Karlsruhe 993 [9] Mayer, G. : Enclosures for eigenvalues and eigenvectors. In: Atanassova, L., Herzberger, J. (eds.): omputer Arithmetic and Enclosure Methods. pp. 49{68. Amsterdam 992 [] Mayer, G. : Taylor-Verfahren fur das algebraische Eigenwertproblem. Z. Angew. Math. Mech. 73, T857{T86 (993) [] Mayer, G. : A unied approach to enclosure methods for eigenpairs. Z. Angew. Math. Mech. 74, 5{28 (994) [2] Mayer, G. : Result verication for eigenvectors and eigenvalues. In: Herzberger, J. (ed.): Topics in Validated omputations. Studies in omputational Mathematics 5, pp. 29{276. Amsterdam 994

14 88 Gunter Mayer [3] Neumaier, A. : Interval Methods for Systems of Equations. ambridge 99 [4] Platzoder, L. : Einige eitrage uber die Existenz von Losungen nichtlinearer Gleichungssysteme und Verfahren zu ihrer erechnung. Thesis, erlin 98 [5] Rump, S. M. : Guaranteed inclusions for the complex generalized eigenproblem. omputing 42, 225{238 (989) received: September, 995 Author: Prof. Dr. G. Mayer Universitat Rostock Fachbereich Mathematik Universitatsplatz 85 Rostock Germany guenter.mayermathematik.uni-rostock.de