On a root-finding approach to the polynomial eigenvalue problem

Transcription

1 On a root-finding approach to the polynomial eigenvalue problem Dipartimento di Matematica, Università di Pisa bini Joint work with Vanni Noferini and Meisam Sharify Limoges, May, 10, 2012

2 The problem Given m m matrices A i, i = 0,..., n compute the eigenvalues λ such that A(λ)v = 0, v 0, where A(λ) = n λ i A i, det A(λ) 0 i=0 Standard matrix approach: linearization (A λb)w = 0, for suitable mn mn matrices A, B Polynomial approach: solving the scalar equation a(λ) = 0, where a(λ) = det(a(λ))

3 Structured matrix polynomials Skew-symmetric coefficients of even size: eigenvalues with even multiplicities since a(x) = q(x) 2 roots appear in pairs (λ, λ) Palindromic polynomials: A i = A n i T-palindromic polynomials: A i = A T n i roots appear in pairs (λ, 1/λ) including eigenvalues at infinity Hamiltonian polynomials: A 2i is Hamiltonian, A 2i+1 is skew-hamiltonian roots appear in pairs (λ, λ) General property of the roots: roots appear in pairs (λ, f (λ)), where f (z) is such that f (f (z)) = z Problem: to design a polynomial eigensolver which exploits the structure, and delivers approximations in pairs (λ, f (λ))

4 The Ehrlich-Aberth iteration Given a scalar polynomial a(x) of degree N, given approximations x (0) 1,..., x (0) N to the roots λ 1,..., λ N, the Ehrlich-Aberth method generates the sequence x (ν+1) i = x (ν) i N (x (ν) i ) 1 N (x (ν) i ) N j=1, j i x (ν) i 1 x (ν) j, i = 1,..., N where N (x) = a(x)/a (x) is the Newton correction This iteration is just Newton s iteration applied to the rational functions (implicit deflation) g i (x) = a(x) n j=1, j i (x x j)

5 Some features of the E-A iteration Local convergence to simple roots is cubic Local convergence to simple roots is more than cubic in the Gauss-Seidel fashion Local convergence to multiple roots is linear Global convergence is not proved: practically, convergence always holds if initial approximations are equally placed along a circle Cost per iteration: O(N 2 ) + N Cost of computing N (x) Number of iterations for arriving at numerical convergence is practically bounded under a suitable choice of starting approximations

6 Problems related to the E-A implementation computing N (x) choosing initial approximations design an effective stop condition design a posteriori error bounds design effective strategies for accelerating convergence in case of clusters or multiple roots These issues were treated in the package MPSolve for approximating polynomial zeros up to any number of (guaranteed) digits [Bini, Fiorentino, Numer. Algo. 2000] We try to do the same for (structured) matrix polynomials

7 Exploiting the structure: Newton s correction For a general m m matrix polynomial A(x) of degree n, set a(x) = det A(x). Then N (x) = a(x) a (x) = 1 trace(a 1 (x)a (x)) Computational cost: O(m 3 + m 2 n) ops

8 Exploiting the structure: Newton s correction Structured case (palindromic and T-palindromic polynomials): Assume that N = mn is even, set z = z(x) = x + 1/x then q(z) = x(z) mn/2 a(x(z)) is a polynomial of degree mn/2, where x(z) = (z z 2 4)/2 or x(z) = (z + z 2 4)/2 is any of the two branches of the inverse of the function z(x).

9 Exploiting the structure: Newton s correction Structured case (palindromic and T-palindromic polynomials): Assume that N = mn is even, set z = z(x) = x + 1/x then q(z) = x(z) mn/2 a(x(z)) is a polynomial of degree mn/2, where x(z) = (z z 2 4)/2 or x(z) = (z + z 2 4)/2 is any of the two branches of the inverse of the function z(x). Moreover, q(z) q (z) = 1 1/x 2 g(x) mn/(2x), g(x) = trace(a(x) 1 A (x)) This way, the computation of the roots of the palyndromic polynomial a(x) of degree mn is reduced to computing the roots of the polynomial q(z) of degree mn/2

10 Exploiting the structure: Newton s correction Structured case (palindromic and T-palindromic polynomials): Assume mn odd In this case 1 is root. Moreover the matrix polynomial Â(x) = (x + 1)A(x) is such that (n + 1)m is even and 1 has multiplicity at least m + 1. Aberth iteration can be applied to Â(x) with m + 1 components of the initial vector equal to 1.

11 Exploiting the structure: Newton s correction Structured case: Hamiltonian polynomials (mn is even) Set z = x 2. For x(z) = z or x(z) = z, q(z) = a(x(z)) is a polynomial of degree mn/2. Moreover, q(z) q (z) = 2x g(x) 1, g(x) = trace(a(x) 1 A (x)) x If mn is odd, then q(z) = a(x(z))/x(z) is a polynomial and q(z) q (z) = 2x 3 g(x) 1, g(x) = trace(a(x) 1 A (x)) Once the roots z 1,..., z mn/2 have been computed, the roots of a(x) are given by the pairs ( z i, z i ), i = 1,..., mn/2

12 Exploiting the structure: Newton s correction General case: roots in pairs (x, f (x)), f (x) = (αx + β)/(γx + α), α 2 + βγ 0 Set z(x) = xf (x) = αx2 +βx γx α and denote x(z) one of the two branches of the inverse of z(x). If there are no eigenvalues with odd multiplicity, then is a polynomial q(z) = a(x(z)) (γx(z) α) mn/2

13 Computational cost The overall cost of E-A iteration applied to an m m matrix polynomial of degree n is O(m 4 n + m 3 n 2 ) ops The overall cost of the technique based on linearization is O(m 3 n 3 ) ops From the complexity point of view the E-A approach is more convenient if n > m The cost of E-A, i.e., the computation of trace(a(x) 1 A (x)), can be reduced to O(m 2 n 2 ) ops, by using linearization. This leads to the overall cost of O(m 3 n 3 ) ops The drawback is that linearization may increase the condition number of eigenvalues.

14 Choosing initial approximations The choice of initial approximations is crucial to arrive at numerical convergence with a small number of iterations This is particularly important for polynomials having very unbalanced roots The package MPSolve for computing polynomial roots to any guaranteed precision, relies on a very effective tool: the Newton polygon construction We synthesize the main ideas of this tool and then extend it to the case of matrix polynomials.

15 Choosing initial approximations The Rouché theorem: If p(x) and q(x) are polynomials such that p(x) > q(x) for x = r then p(x) and p(x) + q(x) have the same number of roots in the open disk of center 0 and radius r Given a(x) = n i=0 a ix i apply Rouché theorem with p(x) = a k x k and q(x) = n i=0, i k x i a i. It follows that if p(x) = a k x k > n i=0,i k a i x i q(x), x = r then a(x) = p(x) + q(x) has k roots of modulus less than r.

16 Choosing initial approximations Properties: The equation has a k r k = n i=0,i k a i r i one positive solution t 0 if k = 0 one positive solution s n if k = n either no positive solutions or two positive solutions s t otherwise Pellet s Theorem, 1881 If h 1,..., h p are the values of k for which the above equation has either one or two positive solutions s hi t hi then the open annulus with radii s hi, t hi contains no roots of any polynomial equimodular with p(x) the closed annulus with radii t hi 1, s hi contains h i h i 1 roots

17 Choosing initial approximations

18 Choosing initial approximations: strategy of MPSolve Computing the positive roots of n polynomials is expensive. An alternative solution relies on the following properties: Any positive solution r of the blue equation, if it exists, is such that u k := max i<k a i a k 1 k i < r < min i>k a i a k 1 k i =: v k Let k i, i = 1,... q be such that u ki < v ki. Then v ki = u ki+1, moreover any interval [s hj, t hj ] does not contain any u ki The indices k i are the abscissas of the Newton polygon: upper convex hull of the set {(i, log a i ), i = 0,..., n} The values u ki are the exponential of the slopes of the edges of the Newton polygon. They are called tropical roots

19 Choosing initial approximations: strategy of MPSolve

20 Choosing initial approximations: strategy of MPSolve

21 Starting approximations: an example p(x) = x 10 + (1000x + 1) 3 Coefficient vector: (1, 3000, , , 0, 0, 0, 0, 0, 0, 1) There are: 3 roots of modulus close to 1/1000, 7 roots of modulus close to The Newton polygon has three vertices besides 0 and n. The values of u k are , 0.001, 0.003, initial approximations are placed in a circle of radius initial approximations are placed inside a disk of radius 3/1000

22 The Sharify improvement What happens inside a blue annulus? If a vertex of the polynomial is sufficiently sharp then we may arrive at a strong localization of the roots Denote r i = u ki the value of the radius corresponding to the vertex k i m i = k i+1 k i its multiplicity δ i = r i /r i+1 the ratio of two consecutive radii Theorem (Sharify) If δ i, δ i 1 < 1/9 then the polynomial a(x) has m i roots in the annulus with radii r i /3 and 3r i.

23 Extension to matrix polynomials Let A, B Hermitian matrices, define A B if A B is positive definite, and A B if A B is positive semidefinite Let F (x), G(x) be matrix polynomials Theorem (Rouché theorem for matrix polynomials) If F (x) F (x) G(x) G(x) for x = r then F (x) and F (x) + G(x) have the same number of roots in the disc of center 0 and radius r. Corollary If (A k A k)r 2k ( n i=0, i k A ix i ) ( n i=0, i k A ix i ), for x = r then det A(x) has kn roots inside the disk of center 0 and radius r. The above condition requires that det A k 0 and is implied by n r k A 1 k A i r i i=0, i k

24 Generalization Theorem 1. If 0 < k < n is such that det A k 0 then the equation r k = A 1 k A i r i (1) i=0, i k has either no real positive solution or two real positive solutions s k t k. 2. In the latter case, the polynomial a(x) = det A(x) has no roots in the open annulus of radii s k, t k, while it has mk roots in the disk of radius s k. 3. If k = 0 and det A 0 0 then (1) has only one real positive root t 0, moreover, the polynomial a(x) has no root in the open disk of radius t If k = n and det A n 0, then (1) has only one real positive solution s n and the polynomial a(x) has no root outside the disk of radius s n.

25 Corollary (Generalized Pellet Thm.) Let h 0 < h 1 <... < h q be the values of k such that det A k 0 and there exist positive real solution(s) s hi t hi of (1). Then 1. t hi 1 s hi, i = 1,..., q; 2. there are m(h i h i 1 ) roots of a(x) in the annulus of radii t hi 1, s hi ; 3. there are no roots of the polynomial a(x) in the open annulus of radii s hi, t hi, where i = 0, 1,..., q and we assume that s 0 = 0, t n =.

26 Theorem If k is such that (1) has two real positive solutions s k t k, then u k s k t k v k where u k := max i<k A 1 k A i 1/(k i), v k := min i>k A 1 k A i 1/(k i) If, for k = 0 (1) has a solution t 0 then t 0 v 0 := min i>0 A 1 0 A i 1/i If for k = n (1) has a solution s n then s n u n := max i<n A 1 n A i 1/(n i) Moreover, v ki u ki+1, i = 1,..., p 1.

27 A special class Consider the class Q m,n of matrix polynomials A(x) = n i=0 x i A i, where A i = α i Q i, Q i Q i = I, α i 0 The condition given in the blue equation (1) turns into α k r k = n i=0,i k α i r i and leads to the machinery valid in the scalar case for the location of the roots of a polynomial by means of the Newton polygon. This is the best result that we can obtain for polynomials in the class Q m,n. In fact, for Q i = I we obtain exactly n copies of the scalar polynomial p(x) = n i=0 α ix i for which the inclusion result is optimal.

28 Theorem Let S = {h i, k S. Then, i = 0,..., p} be such that u k < v k if and only if 1. {k 0,..., k q } {h 0,..., h p }, moreover {h 0,..., h p } [s ki, t ki ] = ; 2. v ki u ki+1 ; 3. if A(x) Q m,n, then v ki = u ki+1 and v ki coincide with the vertices of the Newton polygon of the polynomial w(x) = n i=0 A i x i = n i=0 α ix i.

29 The Sharify bound for matrix polynomials Let A(x) = n i=0 x i A i Q m,n, i.e., A i C m m, A i = α i Q i, Q i Q i = I. Let a(x) = det A(x). Define r i the radii (tropical roots) corresponding to the i-th vertex of the Newton polygon of the polynomial n x i A i 2 i=0 and m i their multiplicities, define δ i = r i /r i 1 Theorem For A(x) Q m,n, if δ i, δ i 1 < 1/12.12 then the polynomial a(x) det A(x) has mm i roots in the annulus with radii r i /4.37 and 4.37r i.

30 Some numerical experiments: Counting # of iterations Orthogonal random coefficients, degree n = 13, scaling factors: α = [1, , , 10 9, 0,... 0, 10 4, 0, 0, 1] Newton polygon Unit circle m simul it aver it simul it aver it

31 Some numerical experiments: Counting # of iterations Non-orthogonal random coefficients, degree n = 13, scaling factors: α = [1, , , 10 9, 0,... 0, 10 4, 0, 0, 1] Newton polygon Unit circle m simul it aver it simul it aver it

32 Some numerical experiments: Counting # of iterations General random coefficients, scaling factors: coefficient i, factor: (rand < 1/4) Number of average iterations with m = 10 and n = 30 Newton s Polygon 7.2 Unit circle 111

33 Some numerical experiments: Approximation error Ehrlich-Aberth: Polyeig: + Problem: Sommerfeld problem

34 Some numerical experiments: Approximation error Ehrlich-Aberth: Polyeig: + Problem: Plasma drift

35 Some numerical experiments: Approximation error Ehrlich-Aberth: Polyeig: + QuadEig (Hammarling-Munro-Tisseur): Problem: hospital

36 Some numerical experiments: Approximation error Ehrlich-Aberth: Polyeig: + QuadEig (Hammarling-Munro-Tisseur): Problem: power plant

37 Some numerical experiments: Approximation error Structured Ehrlich-Aberth: Ehrlich-Aberth: Polyeig (QZ): + Structured URV (Schroeder): Problem: butterfly

38 Some numerical experiments: Approximation error Structured Ehrlich-Aberth: Ehrlich-Aberth: Polyeig (QZ): + Structured URV (Schroeder): Problem: wiresaw1

39 Some numerical experiments: Approximation error Structured Ehrlich-Aberth: Ehrlich-Aberth: Polyeig (QZ): + Problem with symmetry (λ, λ+1 λ 1 )

40 Some references O. Aberth. Iteration methods for finding all zeros of a polynomial simultaneously. Math. Comput, D. A. Bini and G. Fiorentino. Design, analysis, and implementation of a multiprecision polynomial rootfinder. Numer. Algorithms, D.A. Bini, V. Noferini. Solving polynomial eigenvalue problems by means of the Ehrlich-Aberth method. TR L. Gemignani and V. Noferini. The Ehrlich-Aberth method for palindromic matrix polynomials represented in the Dickson basis. Linear Algebra Appl H. Guggenheimer, Initial Approximations in Durand-Kerner s Root Finding Method, BIT, 26 (1986). S. Hammarling, C. Munro and F. Tisseur. An Algorithm for the Com- plete Solution of Quadratic Eigenvalue Problems. MIMS EPrint , avaliable online. N. J. Higham, D. S. Mackey and F. Tisseur. The conditioning of linearizations of matrix polynomials. SIAM J. Matrix Anal. Appl., V. Noferini. The application of the Ehrlich-Aberth method to structured polynomial eigenvalue problems. Proc. Appl. Math. Mech., V. Noferini. Polynomial Eigenproblems: A Root-Finding Approach, Ph.D. Thesis, 2012, Dipartimento di Matematica Università di Pisa, Italy. A. Ostrowski, On a Theorem by J.L. Walsh Concerning the Moduli of Roots of Algebraic Equations, Bull. A.M.S., 47 (1941)

41 Some references A.E. Pellet, Sur un mode de séparation des racines des équations et la formule de Lagrange, Bulletin des Sciences Mathématiques, (2), vol 5 (1881), pp M. Sharify. Scaling Algorithms and Tropical Methods in Numerical Ma- trix Analysis: Application to the Optimal Assignment Problem and to the Accurate Computation of Eigenvalues. Ph.D. Thesis, Ecóle Polytechnique Paris Tech, 2011 C. Schröder. Palindromic and Even Eigenvalue Problems Analysis and Numerical Methods. PhD thesis, T.U. Berlin, P.P. Vaidyanathan, S.K. Mitra, A unified structural interpretation of some well-known stability test procedures for linear systems. Proceedings of the IEEE, vol 75, no. 4, J.L. Walsh, On Pellet s theorem concerning the roots of a polynomial, Annals of Mathematics, vol 26 (1924), pp