Algorithms for Solving the Polynomial Eigenvalue Problem

Transcription

1 Algorithms for Solving the Polynomial Eigenvalue Problem Nick Higham School of Mathematics The University of Manchester Joint work with D. Steven Mackey and Françoise Tisseur. TU Berlin March 13, 2006 Nick Higham Polynomial eigenproblem 1

2 Polynomial Eigenproblem P(λ) = m λ i A i, A i C n n, A m 0. i=0 P assumed regular (det P(λ) 0). Find scalars λ and nonzero vectors x and y satisfying P(λ)x = 0 and y P(λ) = 0. Nick Higham Polynomial eigenproblem 2

3 Hyperbolic and Overdamped Quadratics Q(λ)x = (λ 2 A + λb + C)x = 0. Q is hyperbolic if A is Hermitian pos. def., B and C Hermitian, and (x Bx) 2 > 4(x Ax)(x Cx) for all x 0. Hyperbolic implies real e vals with λ 1 λ n > λ n+1 λ 2n. Q is overdamped if it is hyperbolic with B Hermitian pos. def. and C Hermitian pos. semidef. Overdamped implies λ 1 0. Nick Higham Polynomial eigenproblem 3

4 Methods Interested in methods for solving dense problems. Solvent. Bandwidth reduction. Structure-preserving transformations. Linearization. Nick Higham Polynomial eigenproblem 4

5 Solvent Approach Q(X) = AX 2 + BX + C. S is a solvent if Q(S) = 0. Then Q(λ) = (B + AS + λa)(s λi). Eigenproblem reduced to two n n problems: standard and generalized. Existence of solvent guaranteed if λ 1 λ n > λ n+1 λ 2n and lin indep e vecs exist for {λ 1,...,λ n } and {λ n+1,...,λ 2n }. Conditions satisfied for overdamped polys. Nick Higham Polynomial eigenproblem 5

6 Solvent via Newton, Bernoulli Newton s method with exact line searches (H & Kim, 2001). Solve a gen Sylvester equation on each step. Nick Higham Polynomial eigenproblem 6

7 Solvent via Newton, Bernoulli Newton s method with exact line searches (H & Kim, 2001). Solve a gen Sylvester equation on each step. Bernoulli iteration (H & Kim, 2000): (AX i + B)X i 1 + C = 0, X 1 = A 1 B. Convergence requires λ n > λ n+1 and existence of dominant solvent: Λ(S 1 ) = {λ 1,...,λ n }, minimal solvent: Λ(S 2 ) = {λ n+1,...,λ 2n }. Linear convergence to S 1 with constant λ n / λ n+1. H & Kim showed can be faster thanpolyeig. Nick Higham Polynomial eigenproblem 6

8 Solvent via Cyclic Reduction Cyclic reduction (Guo & Lancaster, 2005) on infinite block tridiagonal system with rows [C B A]. Matrix iteration X i+1 = X i A i B 1 i C i etc n3 flops per iteration for overdamped Q. Quadratic convergence for overdamped Q. Dominant and minimal solvents are obtained from limit X as X 1 C and A 1 X. E vals not guaranteed real! Total cost if k iterations: (25 + 6k)n 3 flops e vals only (55 + 6k)n 3 flops e vals and e vecs Nick Higham Polynomial eigenproblem 7

9 Linearization for Hyperbolic Quadratic Assume we know σ such that Q(σ) < 0. Q(λ) := Q(λ + σ) = λ 2 Q (σ) + λq (σ) + Q(σ) X λy = λ 2 }{{} A +λ B + C }{{}. >0 <0 [ ] [ ] B A C 0 λ. A 0 0 A Transform X λy G λi using Cholesky of A and C. Real e vals assured. Total cost: 18n 3 flops e vals only 33n 3 flops e vals and e vecs Nick Higham Polynomial eigenproblem 8

10 Bandwidth Reduction Q(λ) = λ 2 A + λb + C. Transform GQ(λ)H = λ 2Ã + λ B + C with Ã, B, C of minimal bandwidth in fte # operations. Then apply some other method. Tridiagonal form is not achievable (2n 2 parameters, 3n 2 equations). Is pentadiagonal form achievable? Nick Higham Polynomial eigenproblem 9

11 Structure-Preserving Transformations Idea is to produce a diagonal poly with the same spectrum as P. Garvey, Prells & Friswell (2002, 2003) Chu & Del Buono (2005) pursue the approach: Form a linearization (in DL(P)). Iteratively transform it to diagonal form preserving its structure. Generate the transformations via isospectral flows? Many open questions. Nick Higham Polynomial eigenproblem 10

12 Linearizations L(λ) = λx + Y, X, Y C mn mn is a linearization of P(λ) = m i=0 λi A i if [ ] P(λ) 0 E(λ)L(λ)F(λ) = 0 I (m 1)n for some unimodular E(λ) and F(λ). Example Companion form linearization ( [ ] [ A2 0 A1 A E(λ) λ I I 0 ]) F(λ) = [ ] λ 2 A 2 + λa 1 + A I Nick Higham Polynomial eigenproblem 11

13 Solution Process Standard way of solving P(λ)x = 0: Linearize P(λ) into L(λ) = λx + Y. Solve generalized eigenproblem L(λ)z = 0. Recover eigenvectors of P from those of L. Usual choice of L: companion linearization, for which λ m 1 x z =. λx. x Left e vec: more complicated formula. Nick Higham Polynomial eigenproblem 12

14 Desiderata for a Linearization Good conditioning. Backward stability. Preservation of structure, e.g. symmetry. Numerical preservation of key qualitative properties, including location and symmetries of spectrum. Preserve partial multiplicities of e vals (strong linearization). Nick Higham Polynomial eigenproblem 13

15 Meta-Algorithm Preprocess P. E.g., Fan, Lin & Van Dooren { (2004): γ = c/a λ 2 A + λb + C µ 2 (γ 2 δa) + µ(γδb) + δc δ = 2/(c + bγ) for one or more linearizations L Balance L Apply QZ or HZ to L Obtain relevant e vals. Recover left and right e vecs Iteratively refine e vecs Compute/estimate b errs and condition numbers Detect nonregular problem end Nick Higham Polynomial eigenproblem 14

16 Balancing Balancing GEP: Ward (1981) Lemonnier & Van Dooren (2005) To investigate: Exploit structure of pencils arising via linearization of a matrix poly. Can we balance a QEP? To what extent balancing can make the results worse? Cf. Watkins (2005): A Case where Balancing is Harmful. Nick Higham Polynomial eigenproblem 15

17 Iterative Refinement Underlying theory for fixed and extended precision residuals in Tisseur (2001). Done for definite GEPs in Davies, H & Tisseur (2001). Details for QEPs in Berhanu (2005), incl. complex conj. pairs in real arith. Issues: Convergence to wrong eigenpair or non-convergence. Exploiting structure of pencil from a linearization. Nick Higham Polynomial eigenproblem 16

18 L 1 and L 2 Linearizations Λ := [λ m 1,λ m 2,...,1] T. Mackey, Mackey, Mehl & Mehrmann (2005) define L 1 (P) = { L(λ) : L(λ)(Λ I n ) = v P(λ), v C m }, L 2 (P) = { L(λ) : (Λ T I n )L(λ) = w T P(λ), w C m }. They show that L 1 and L 2 are vector spaces of dim m(m 1)n 2 + m. Almost all pencils in L 1, L 2 are linearizations of P. Quadratic case (m = 2): L = λx + Y L 1 (P) iff [ ] [ v1 A 2 v 1 A 1 v 1 A 0 X11 X = 12 + Y 11 Y 12 v 2 A 2 v 2 A 1 v 2 A 0 X 21 X 22 + Y 21 Y 22 ]. Nick Higham Polynomial eigenproblem 17

19 L 1 and L 2 Linearizations cont. Recall Note L 1 (P) = { L(λ) : L(λ)(Λ I n ) = v P(λ), v C m }. L(λ)(Λ x) = L(λ)(Λ I n )(1 x) = (v P(λ))(1 x) = v P(λ)x. So (x,λ) is an e pair of P iff (Λ x,λ) is an e pair of L. Right eigenvectors of P can be recovered from right eigenvectors of linearizations in L 1. Left eigenvectors of P can be recovered from left eigenvectors of linearizations in L 2. Nick Higham Polynomial eigenproblem 18

20 DL(P) Linearizations Mackey, Mackey, Mehl & Mehrmann (2005) define DL(P) = L 1 (P) L 2 (P). They show that L DL(P) iff w = v in the definitions of L 1 and L 2. DL(P) is a vector space of dimension m. Almost all pencils in DL(P) are linearizations of P. Example: For Q(λ) = λ 2 A + λb + C, DL(Q) is the pencils [ ] [ ] v1 A v L(λ) = λ 2 A v1 B v + 2 A v 1 C, v C 2. v 2 A v 2 B v 1 C v 1 C v 2 C Nick Higham Polynomial eigenproblem 19

21 Conditioning in DL(P) Let ρ = max i A i 2 min( A 0 2, A m 2 ) 1. H, D. S. Mackey & Tisseur (2005) show that κ L (λ; e 1 ) ρ 2 m 7/2 κ P (λ), A 0 nonsing, λ 1, κ L (λ; e m ) ρ 2 m 7/2 κ P (λ), A m nonsing, λ 1. For Q not heavily damped, ρ = O(1). With FLV scaling, ρ = O(1) for elliptic quadratics. Nick Higham Polynomial eigenproblem 20

22 Conditioning of Companion Form H, D. S. Mackey & Tisseur (2005) find that κ C1 /κ P depends on ratios w 2 / y 2 1 of norms of left e vecs of C 1 and P, rational functions of the A i 2 and λ. Conclude that If A i 1, i = 0: m then κ C1 κ P. If w 2 / y 2 1 or if A i 2 1, i = 0: m, then κ C1 inf v κ L (λ; v) is possible. Nick Higham Polynomial eigenproblem 21

23 Companion versus DL(P) Companion is always a linearization; for DL(P) need spectrum of P distinct from roots of v. DL(P) pencil symmetric if P is, companion not. Scaling can help both. Easier to check suff. conds for DL(P) well conditioned. Role of L 1 and L 2 Can preserve other structures of P. Conditioning analysis can be extended using new left e vec recovery formula. Nick Higham Polynomial eigenproblem 22

24 Concluding Remarks Only general-purpose current methods are those based on linearization. Recent work opens up opportunities for developing more sophisticated algs based on linearization. Interesting possibilities for hyperbolic polys. How to design an LAPACK QEP solver? Nick Higham Polynomial eigenproblem 23

25 Bibliography I M. Berhanu. The Polynomial Eigenvalue Problem. PhD thesis, University of Manchester, Manchester, England, P. I. Davies, N. J. Higham, and F. Tisseur. Analysis of the Cholesky method with iterative refinement for solving the symmetric definite generalized eigenproblem. SIAM J. Matrix Anal. Appl., 23(2): , Nick Higham Polynomial eigenproblem 24

26 Bibliography II H.-Y. Fan, W.-W. Lin, and P. Van Dooren. Normwise scaling of second order polynomial matrices. SIAM J. Matrix Anal. Appl., 26(1): , C.-H. Guo and P. Lancaster. Algorithms for hyperbolic quadratic eigenvalue problems. Math. Comp., 74(252): , N. J. Higham and H.-M. Kim. Numerical analysis of a quadratic matrix equation. IMA J. Numer. Anal., 20(4): , Nick Higham Polynomial eigenproblem 25

27 Bibliography III N. J. Higham and H.-M. Kim. Solving a quadratic matrix equation by Newton s method with exact line searches. SIAM J. Matrix Anal. Appl., 23(2): , N. J. Higham, D. S. Mackey, N. Mackey, and F. Tisseur. Symmetric linearizations for matrix polynomials. MIMS EPrint , Manchester Institute for Mathematical Sciences, The University of Manchester, UK, Submitted to SIAM J. Matrix Anal. Appl. Nick Higham Polynomial eigenproblem 26

28 Bibliography IV N. J. Higham, D. S. Mackey, and F. Tisseur. The conditioning of linearizations of matrix polynomials. Numerical Analysis Report No. 465, Manchester Centre for Computational Mathematics, Manchester, England, To appear in SIAM J. Matrix Anal. Appl. D. Lemonnier and P. M. Van Dooren. Balancing regular matrix pencils. SIAM J. Matrix Anal. Appl., To appear. Nick Higham Polynomial eigenproblem 27

29 Bibliography V D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann. Palindromic polynomial eigenvalue problems: Good vibrations from good linearizations. Numerical Analysis Report No. 466, Manchester Centre for Computational Mathematics, Manchester, England, Submitted to SIAM J. Matrix Anal. Appl. D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann. Vector spaces of linearizations for matrix polynomials. Numerical Analysis Report No. 464, Manchester Centre for Computational Mathematics, Manchester, England, To appear in SIAM J. Matrix Anal. Appl. Nick Higham Polynomial eigenproblem 28

30 Bibliography VI F. Tisseur. Newton s method in floating point arithmetic and iterative refinement of generalized eigenvalue problems. SIAM J. Matrix Anal. Appl., 22(4): , R. C. Ward. Balancing the generalized eigenvalue problem. SIAM J. Sci. Statist. Comput., 2(2): , D. S. Watkins. A case where balancing is harmful. Submitted, Nick Higham Polynomial eigenproblem 29