Scaling, Sensitivity and Stability in Numerical Solution of the Quadratic Eigenvalue Problem

Transcription

1 Scaling, Sensitivity and Stability in Numerical Solution of the Quadratic Eigenvalue Problem Nick Higham School of Mathematics The University of Manchester Joint work with Seamus Garvey, Steve Mackey and Françoise Tisseur

2 NLEVP Toolbox with T. Betcke, V. Mehrmann, C. Schröder, F. Tisseur Collection of Nonlinear Eigenvalue Problems : F(λ)x = 0, where F : C C m n. Provided as a MATLAB Toolbox. Problems from real-life applications + specially constructed problems. Available from numerical-analysis/nlevp.html MIMS Nick Higham Quadratic Eigenproblem 2 / 31

3 NLEVP Toolbox with T. Betcke, V. Mehrmann, C. Schröder, F. Tisseur Collection of Nonlinear Eigenvalue Problems : F(λ)x = 0, where F : C C m n. Provided as a MATLAB Toolbox. Problems from real-life applications + specially constructed problems. Available from numerical-analysis/nlevp.html Further contributions are welcome. MIMS Nick Higham Quadratic Eigenproblem 2 / 31

4 Outline 1 QEP and Linearization Background 2 Conditioning of Linearizations 3 Scaling 4 Algorithm based on Linearization MIMS Nick Higham Quadratic Eigenproblem 3 / 31

5 Quadratic Eigenproblems Consider Q(λ) = λ 2 M + λd + K, M, D, K C n n. QEP: find scalars λ and nonzero x, y C n satisfying Q(λ)x = 0 and y Q(λ) = 0. λ is an e val, x, y are corresponding right and left e vecs. Q(λ) has 2n eigenvalues, solutions of det(q(λ)) = 0. MIMS Nick Higham Quadratic Eigenproblem 4 / 31

6 Quadratic Eigenproblems Consider Q(λ) = λ 2 M + λd + K, M, D, K C n n. QEP: find scalars λ and nonzero x, y C n satisfying Q(λ)x = 0 and y Q(λ) = 0. λ is an e val, x, y are corresponding right and left e vecs. Q(λ) has 2n eigenvalues, solutions of det(q(λ)) = 0. When λ =, consider homogeneous form of Q: Q(α,β) = α 2 M + αβd + β 2 K. E vals are pairs (α,β) (0, 0) s.t. det Q(α,β) = 0. MIMS Nick Higham Quadratic Eigenproblem 4 / 31

7 Linearizations L(λ) = λx + Y, X, Y C 2n 2n is a linearization of Q(λ) = λ 2 M + λd + K if [ ] Q(λ) 0 E(λ)L(λ)F(λ) = 0 I n for some unimodular E(λ) and F(λ). Example [ ] M 0 For companion pencil C 1 (λ) = λ + 0 I n ( ) holds with [ In λm + D E(λ) = 0 I n ( ) [ ] D K, I n 0 ] [ ] λin I, F(λ) = n. I n 0 MIMS Nick Higham Quadratic Eigenproblem 5 / 31

8 Solution Process for QEP Linearize Q(λ) into L(λ) = λx + Y. Solve generalized eigenproblem L(λ)z = 0. Recover eigenvectors of Q from those of L. Usual choice of linearization: companion linearization, [ ] [ ] M 0 D K C 1 (λ) = λ + 0 I I 0 for which right and left e vecs have the form [ ] λx z =, w = x x, y being right and left e vecs of Q(λ). [ y λk y MIMS Nick Higham Quadratic Eigenproblem 6 / 31 ],

9 Beam Problem /////// L /////// //////////// Transverse displacement u(x, t) governed by ρa 2 u t 2 + c(x) u t + EI 4 u x 4 = 0. Boundary conditions: u(x, t) = u (x, t) = 0 at x = 0, L. u(x, t) = e λt v(x,λ) yields e val problem for the free vibrations : λ 2 ρav(x,λ) + λc(x)v(x,λ) + EI 4 v(x,λ) = 0. x4 MIMS Nick Higham Quadratic Eigenproblem 7 / 31

10 Discretized Beam Problem Finite element method leads to Q(λ) = λ 2 M + λd + K with symmetric M, D, K R n n. Roots of x Q(λ)x = 0, λ = (x Dx) ± (x Dx) 2 4(x Mx)(x Kx). 2(x Mx) M > 0, K > 0, D 0 all ei vals have Re(λ) 0. D is rank 1. Can show n pure imaginary ei vals. MIMS Nick Higham Quadratic Eigenproblem 8 / 31

11 Eigenvalues of Q via First Companion C 1 Q(λ) = λ 2 M + λd + K, C 1 (λ) = λ [ ] [ ] M 0 D K +. 0 I I 0 nele = 100; coeffs = nlevp( damped_beam,nele); K = coeffs{1}; D = coeffs{2}; M = coeffs{3}; I = eye(2*nele); O = zeros(2*nele); eval = eig([d K; -I O],-[M O; O I]); plot(eval,.r ) MIMS Nick Higham Quadratic Eigenproblem 9 / 31

12 [ eig on Companion C 1 (λ)=λ M0 ] [ 0 I + D I ] K 0 4 x MIMS Nick Higham Quadratic Eigenproblem 10 / 31

13 eig on Linearization L 1 (λ)=λ[ M0 ] [ 0 K + DK ] K 0 4 x MIMS Nick Higham Quadratic Eigenproblem 11 / 31

14 [ eig on Linearization L 2 (λ)=λ 0M ] [ M M0 ] D + 0 K 4 x MIMS Nick Higham Quadratic Eigenproblem 12 / 31

15 Spectrum of Beam Problem 4 x MIMS Nick Higham Quadratic Eigenproblem 13 / 31

16 Sensitivity and Stability of Linearizations Condition number measures sensitivity of the solution of a problem to perturbations in the data. Backward error measures how well the problem has been solved. error in solution < condition number backward error. MIMS Nick Higham Quadratic Eigenproblem 14 / 31

17 Sensitivity and Stability of Linearizations Condition number measures sensitivity of the solution of a problem to perturbations in the data. Backward error measures how well the problem has been solved. error in solution < condition number backward error. For a given Q(λ), infinitely many linearizations exist: can have widely varying eigenvalue condition numbers, computed eigenpairs can have widely varying backward errors. MIMS Nick Higham Quadratic Eigenproblem 14 / 31

18 Desiderata for a Linearization Good conditioning. Backward stability. Suitable eigenvector recovery formulae. 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). MIMS Nick Higham Quadratic Eigenproblem 15 / 31

19 Vector Spaces L 1, L 2 Mackey, Mackey, Mehl & Mehrmann (2006) define L 1 (Q) = { L(λ) : L(λ)(Λ I n ) = v Q(λ), v C } 2, L 2 (Q) = { L(λ) : (Λ T I n )L(λ) = ṽ T Q(λ), ṽ C } 2, where Λ := [λ, 1] T. MIMS Nick Higham Quadratic Eigenproblem 16 / 31

20 Vector Spaces L 1, L 2 Mackey, Mackey, Mehl & Mehrmann (2006) define L 1 (Q) = { L(λ) : L(λ)(Λ I n ) = v Q(λ), v C 2 }, L 2 (Q) = { L(λ) : (Λ T I n )L(λ) = ṽ T Q(λ), ṽ C 2 }, where Λ := [λ, 1] T. L(λ) = λx + Y L 1 (Q) with v C 2 iff [ ] [ v1 M v 1 D v 1 K X11 X = 12 + Y 11 Y 12 v 2 M v 2 D v 2 K X 21 X 22 + Y 21 Y 22 ]. MIMS Nick Higham Quadratic Eigenproblem 16 / 31

21 Vector Spaces L 1, L 2 Mackey, Mackey, Mehl & Mehrmann (2006) define L 1 (Q) = { L(λ) : L(λ)(Λ I n ) = v Q(λ), v C } 2, L 2 (Q) = { L(λ) : (Λ T I n )L(λ) = ṽ T Q(λ), ṽ C } 2, where Λ := [λ, 1] T. Dimensions: L 1, L 2 : 2n Almost all pencils in L 1 and L 2 are linearizations. MIMS Nick Higham Quadratic Eigenproblem 16 / 31

22 Eigenvector Recovery for L 1 (Q) L 1 (Q) = { L(λ) : L(λ)(Λ I n ) = v Q(λ), v C 2 } Λ := [λ, 1] T. If L L 1 (Q) with vector v then every right e vec of L with finite e val λ is of the form Λ x for some right e vec x of P, [M 4, 2006] if w is a left e vec of L with e val λ then y = (v I n )w is a left e vec of P with e val λ. [H, Li, Tisseur, 2007]. E vecs of Q easily recovered from e vecs of L L 1. MIMS Nick Higham Quadratic Eigenproblem 17 / 31

23 Outline 1 QEP and Linearization Background 2 Conditioning of Linearizations 3 Scaling 4 Algorithm based on Linearization MIMS Nick Higham Quadratic Eigenproblem 18 / 31

24 Eigenvalue Condition Numbers κ Q (λ) Q(λ)x = 0, y Q(λ) = 0. Q(λ) = λ 2 M + λ D + K For λ simple, nonzero and finite, { λ κ Q (λ) = lim sup : [ (Q + Q)(λ + λ) ] (x + x) = 0, ǫ 0 ǫ λ } M 2 ǫm, D 2 ǫd, K 2 ǫk, κ Q (λ) = ( λ 2 m + λ d + k ) y 2 x 2. (Tisseur, 2000) λ y (2λM + D)x MIMS Nick Higham Quadratic Eigenproblem 19 / 31

25 Eigenvalue Conditioning of Linearizations For L(λ) = λx + Y, L(λ)z = 0, w L(λ) = 0, κ L (λ) = ( λ X 2 + Y 2 ) w 2 z 2. λ w Xz Define growth factor φ L : κ L (λ) = φ L (λ) κ Q (λ). Theorem (H, Mackey, Tisseur, 2006) Let L(λ) = λx + Y B(Q) with vector v. For λ simple, nonzero and finite, where Λ = [λ, 1] T. φ L (λ; v) = λ X 2 + Y 2 λ 2 m + λd + k Λ 2 2 Λ T v, MIMS Nick Higham Quadratic Eigenproblem 20 / 31

26 Sufficient conditions for κ Q κ L ρ = max(m, d, k)/ min(m, k), Linearization Eigenvalue Condition [ ] [ M 0 D K L 1 (λ) = λ + 0 K K 0 C 1 No restriction m d k 1 λ > L 1 1 ρ 1 λ 1 not available" λ > L 2 1 not available" λ 1 ρ 1 [ ] [ M 0 D K C 1 (λ) = λ + 0 I I 0 ], L 2 (λ) = λ [ 0 M M D ], ] [ ] M K MIMS Nick Higham Quadratic Eigenproblem 21 / 31

27 Beam Problem M 2 = , D 2 = 5, K 2 = Thus ρ = beam problem is badly scaled. Approximations to growth factors φ L (λ) = κ L (λ)/κ Q (λ): φ C1 (λ) φ L1 (λ) φ L2 (λ) λ = λ = λ = For λ = 10 6, ǫ 10 16, λ < ǫ λ κ Li (λ) = ǫ λ φ Li (λ)κ Q (λ) = O(1), i = 1, 2. E vals on imaginary axis can be perturbed by distance O(1) into the right half-plane. MIMS Nick Higham Quadratic Eigenproblem 22 / 31

28 Computed Spectrum of L 1, L 2 and C 1 4 x x x MIMS Nick Higham Quadratic Eigenproblem 23 / 31

29 Outline 1 QEP and Linearization Background 2 Conditioning of Linearizations 3 Scaling 4 Algorithm based on Linearization MIMS Nick Higham Quadratic Eigenproblem 24 / 31

30 Scaling Q(λ) = λ 2 M + λd + K Let λ = µγ and convert Q(λ) = λ 2 M + λd + K δq(µγ) = µ 2 (δγ 2 M) + µ(δγd) + δk = µ 2 M + µ D + K =: Q(µ), where γ = K 2 / M 2, δ = 2/( K 2 + D 2 γ). Fan, Lin and Van Dooren (2004). 2/3 max( M 2, D 2, K 2 ) 2. Does not affect sparsity of M, D, K. Has no effect on κ Q and η Q. γ minimizes scaling factor ρ. MIMS Nick Higham Quadratic Eigenproblem 25 / 31

31 Effect of Scaling on Beam Problem Before scaling After scaling M D K ρ = ρ = 1 Our theory guarantees optimal conditioning and stability for the companion linearization, E val bound µ 1 2 τκ 2(M) ( /(τ 2 κ 2 (M)) ) = Can show this implies symm linearization L 2 optimal in terms of both conditioning and stability. MIMS Nick Higham Quadratic Eigenproblem 26 / 31

32 Spectrum of C 1, L 2 before/after Scaling 4 x x x x MIMS Nick Higham Quadratic Eigenproblem 27 / 31

33 Outline 1 QEP and Linearization Background 2 Conditioning of Linearizations 3 Scaling 4 Algorithm based on Linearization MIMS Nick Higham Quadratic Eigenproblem 28 / 31

34 Meta-Algorithm for PEP 1 Balance, scale P (Fan, Lin & Van Dooren, 2004) 2 for one or more (scaled) linearizations L 3 Deflate L 4 Balance, scale L 5 Apply QZ to L (maybe HZ if structured) 6 Obtain relevant e vals 7 Recover left and right e vecs 8 Iteratively refine e vecs 9 Compute/estimate b errs and condition numbers 10 Detect nonregular problem 11 end MIMS Nick Higham Quadratic Eigenproblem 29 / 31

35 Balancing Ward (1981) for pencils. Lemonnier & Van Dooren (2006) for pencils. Betcke (2009) for polynomials. MIMS Nick Higham Quadratic Eigenproblem 30 / 31

36 Concluding Remarks Analysis of conditioning & backward error for wide variety of linearizations. E vector recovery formulae crucial. Scaling crucial. Favour L = companion form for general QEPs. Results useful to develop a general QEP algorithm & code. New version ofpolyeig in preparation. For papers and Eprints, MIMS Nick Higham Quadratic Eigenproblem 31 / 31

37 References I T. Betcke. Optimal scaling of generalized and polynomial eigenvalue problems. SIAM J. Matrix Anal. Appl., 30(4): , T. Betcke, N. J. Higham, V. Mehrmann, C. Schröder, and F. Tisseur. NLEVP: A collection of nonlinear eigenvalue problems. research/numerical-analysis/nlevp.html. MIMS Nick Higham Quadratic Eigenproblem 27 / 31

38 References II T. Betcke, N. J. Higham, V. Mehrmann, C. Schröder, and F. Tisseur. NLEVP: A collection of nonlinear eigenvalue problems. MIMS EPrint , Manchester Institute for Mathematical Sciences, The University of Manchester, UK, Apr pp. N. J. Higham, R.-C. Li, and F. Tisseur. Backward error of polynomial eigenproblems solved by linearization. SIAM J. Matrix Anal. Appl., 29(4): , MIMS Nick Higham Quadratic Eigenproblem 28 / 31

39 References III N. J. Higham, D. S. Mackey, N. Mackey, and F. Tisseur. Symmetric linearizations for matrix polynomials. SIAM J. Matrix Anal. Appl., 29(1): , N. J. Higham, D. S. Mackey, and F. Tisseur. The conditioning of linearizations of matrix polynomials. SIAM J. Matrix Anal. Appl., 28(4): , N. J. Higham, D. S. Mackey, F. Tisseur, and S. D. Garvey. Scaling, sensitivity and stability in the numerical solution of quadratic eigenvalue problems. Internat. J. Numer. Methods Eng., 73(3): , MIMS Nick Higham Quadratic Eigenproblem 29 / 31

40 References IV D. Lemonnier and P. M. Van Dooren. Balancing regular matrix pencils. SIAM J. Matrix Anal. Appl., 28(1): , D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann. Structured polynomial eigenvalue problems: Good vibrations from good linearizations. SIAM J. Matrix Anal. Appl., 28(4): , D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann. Vector spaces of linearizations for matrix polynomials. SIAM J. Matrix Anal. Appl., 28(4): , MIMS Nick Higham Quadratic Eigenproblem 30 / 31

41 References V R. C. Ward. Balancing the generalized eigenvalue problem. SIAM J. Sci. Statist. Comput., 2(2): , MIMS Nick Higham Quadratic Eigenproblem 31 / 31