Solving the Polynomial Eigenvalue Problem by Linearization

Transcription

1 Solving the Polynomial Eigenvalue Problem by Linearization Nick Higham School of Mathematics The University of Manchester Joint work with Ren-Cang Li, Steve Mackey and Françoise Tisseur. IWASEP 6, May

2 Outline PEP and Linearization Background B err and Conditioning of Linearizations Algorithm based on Linearization MIMS Nick Higham Polynomial eigenproblem 2 / 39

3 Polynomial Eigenproblem (PEP) 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. Special case quadratic eigenvalue problem (QEP): (λ 2 M + λd + K)x = 0. MIMS Nick Higham Polynomial eigenproblem 3 / 39

4 Applications classical structural mechanics molecular dynamics gyroscopic systems optical waveguide design MIMO systems in control theory constrained least squares problems. MIMS Nick Higham Polynomial eigenproblem 4 / 39

5 Applications classical structural mechanics molecular dynamics gyroscopic systems optical waveguide design MIMO systems in control theory constrained least squares problems. More specifically: Excitation of rail tracks by high speed trains (SIAM News, Nov. 2004). Extreme designs lead to problems with poor conditioning; physics of system leads to structure. MIMS Nick Higham Polynomial eigenproblem 4 / 39

6 Polyeig (Companion Pencil) 4 x

7 Polyeig on Scaled Quadratic 4 x

8 Methods Interested in methods for solving dense problems (possibly a projection of a sparse problem). Solvent: m i=0 A ix i = 0. Bandwidth reduction. Structure-preserving transformations. Linearization. MIMS Nick Higham Polynomial eigenproblem 8 / 39

9 Current Software MATLAB spolyeig. Solvers in commercial engineering packages (Nastran, etc.). Aim to design and implement LAPACK solvers for general and structured QEPs/PEPs (Manchester and TU Berlin). MIMS Nick Higham Polynomial eigenproblem 9 / 39

10 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 MIMS Nick Higham Polynomial eigenproblem 10 / 39

11 Solution Process for PEP 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. MIMS Nick Higham Polynomial eigenproblem 11 / 39

12 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 Polynomial eigenproblem 12 / 39

13 Outline PEP and Linearization Background B err and Conditioning of Linearizations Algorithm based on Linearization MIMS Nick Higham Polynomial eigenproblem 13 / 39

14 Backward Error P(α,β) = m α i β m i A i i=0 (x,α,β) approx eigenpair of P: (λ α/β). { η P (x,α,β) = min ǫ : Extending Tisseur (2000): m α i β m i (A i + A i )x = 0, i=0 A i 2 ǫ A i 2, i = 0: m }. η P (x,α,β) = P(α,β)x 2 ( m i=0 α i β m i A i 2 ) x 2. MIMS Nick Higham Polynomial eigenproblem 14 / 39

15 Key Question How good an approx eigenpair of P will be produced from an approx eigenpair of L? Here good" refers to relative error (see H, D. S. Mackey & Tisseur, 2005), backward error. A small perturbation to C 1 (λ) = [ ] A I [ ] A1 A 0 I 0 may not correspond to a small perturbation to Q(λ) = λ 2 A 2 + λa 1 + A 0. MIMS Nick Higham Polynomial eigenproblem 15 / 39

16 Want to compare η P (x,α,β) = P(α,β)x 2 ( m i=0 α i β m i A i 2 ) x 2, with η L (z,α,β) = L(α,β)z 2 ( α X 2 + β Y 2 ) z 2. (z, α, β): approx e pair of linearization L(λ) = λx + Y of P(λ). (x,α,β): approx e pair of P with x recovered from z. MIMS Nick Higham Polynomial eigenproblem 16 / 39

17 One Sided Factorization Suppose there exists n nm G(α, β) s.t. G(α,β)L(α,β) = g T P(α,β), g C m. With z i := z((i 1)n + 1: in), G(α,β)L(α,β)z = (g T P(α,β))z z 1 = [g 1 P(α,β)... g m P(α,β)]. z m m = P(α,β) g i z i =: P(α,β)x. i=1 P(α,β)x 2 G(α,β) 2 L(α,β)z 2. MIMS Nick Higham Polynomial eigenproblem 17 / 39

18 Bounding η P /η L Suppose G(α,β)L(α,β) = g T P(α,β), g C m. Let z be approx e vec of L with e val (α,β) and take x = g i z i as approx e vec of P. η P (x,α,β) η L (z,α,β) = α X 2 + β Y 2 m i=0 α i β m i A i 2 P(α,β)x 2 L(α,β)z 2 z 2 x 2. MIMS Nick Higham Polynomial eigenproblem 18 / 39

19 Bounding η P /η L Suppose G(α,β)L(α,β) = g T P(α,β), g C m. Let z be approx e vec of L with e val (α,β) and take x = g i z i as approx e vec of P. η P (x,α,β) η L (z,α,β) α X 2 + β Y 2 m i=0 α i β m i A i 2 G(α,β) 2 z 2 x 2. Separates the dependence on L, P and (α,β) from dependence on G and z. η P is finite if η L is. MIMS Nick Higham Polynomial eigenproblem 18 / 39

20 Bounding η P /η L Suppose G(α,β)L(α,β) = g T P(α,β), g C m. Let z be approx e vec of L with e val (α,β) and take x = g i z i as approx e vec of P. η P (x,α,β) η L (z,α,β) α X 2 + β Y 2 m i=0 α i β m i A i 2 G(α,β) 2 z 2 x 2. Separates the dependence on L, P and (α,β) from dependence on G and z. η P is finite if η L is. Sim. for left e vecs: assume L(α,β)H(α,β) = h P(α,β), h C m. MIMS Nick Higham Polynomial eigenproblem 18 / 39

21 Vector Spaces L 1, L 2, and DL(P) 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) = ṽ T P(λ), ṽ C } m, DL(P) = L 1 (P) L 2 (P). MIMS Nick Higham Polynomial eigenproblem 19 / 39

22 Vector Spaces L 1, L 2, and DL(P) 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) = ṽ T P(λ), ṽ C } m, DL(P) = L 1 (P) L 2 (P). Dimensions: L 1, L 2 : m(m 1)n 2 + m, DL: m. Pencils in DL(P) are block symmetric. Almost all L in the above are linearizations. MIMS Nick Higham Polynomial eigenproblem 19 / 39

23 Vector Spaces L 1, L 2, and DL(P) 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) = ṽ T P(λ), ṽ C } m, DL(P) = L 1 (P) L 2 (P). Dimensions: L 1, L 2 : m(m 1)n 2 + m, DL: m. Pencils in DL(P) are block symmetric. Almost all L in the above are linearizations. Recall G(α,β)L(α,β) = g T P(α,β), g C m. MIMS Nick Higham Polynomial eigenproblem 19 / 39

24 Companion Linearizations C i (λ) = λx + Y i with X = diag(a m, I n,...,i n ), Y 1 = A m 1 A m 2... A 0 I n I n 0, Y 2 = A m 1 I n A m In A C 1 L 1 (P) (v = e 1 ) and C 2 L 2 (P) (ṽ = e 1 ). C 1 and C 2 always (strong) linearizations. C 2 (P) = C 1 (P T ) T concentrate on C 1 only.. MIMS Nick Higham Polynomial eigenproblem 20 / 39

25 Companion Linearizations C i (λ) = λx + Y i with X = diag(a m, I n,...,i n ), Y 1 = A m 1 A m 2... A 0 I n I n 0, Y 2 = A m 1 I n A m In A C 1 L 1 (P) (v = e 1 ) and C 2 L 2 (P) (ṽ = e 1 ). C 1 and C 2 always (strong) linearizations. C 2 (P) = C 1 (P T ) T concentrate on C 1 only. Are the factorizations G(α,β)C 1 (α,β) = g T P(α,β), possible for C 1? C 1 (α,β)h(α,β) = h P(α,β). MIMS Nick Higham Polynomial eigenproblem 20 / 39

26 First Companion For m = 2, [ αi βa0 βi βa 1 + αa 2 yields two choices: ] [ ] P(α,β) 0 C 1 (α,β) = 0 P(α, β) G(α,β) = [αi βa 0 ], g = e 1 x = z 1, G(α,β) = [βi βa 1 + αa 2 ], g = e 2 x = z 2. Generalizes to arbitrary degrees m. C 1 L 1 (v = e 1 ) C 1 (α,β)(λ α,β I n ) = e 1 P(α,β). Can take H(α,β) = Λ α,β I n and h = e 1 y = w 1. MIMS Nick Higham Polynomial eigenproblem 21 / 39

27 First Companion: Right Eigenvector Theorem Let z be approx right e vec of C 1 with approx e val (α,β). For z k = z((k 1)n + 1: kn), k = 1: m, ( ) 2 η P (z k,α,β) max 1, maxi A i 2 m5/2 η C1 (z,α,β) min ( ) z 2. A 0 2, A m 2 z k 2 MIMS Nick Higham Polynomial eigenproblem 22 / 39

28 First Companion: Right Eigenvector Theorem Let z be approx right e vec of C 1 with approx e val (α,β). For z k = z((k 1)n + 1: kn), k = 1: m, ( ) 2 η P (z k,α,β) max 1, maxi A i 2 m5/2 η C1 (z,α,β) min ( ) z 2. A 0 2, A m 2 z k 2 η P η C1 if min( A 0 2, A m 2 ) max i A i 2 1. z 2 / z k 2 1. Exact z = Λ α,β x z 2 z k 2 m, k = { 1 if α β, m if α β. MIMS Nick Higham Polynomial eigenproblem 22 / 39

29 First Companion: Left Eigenvector Theorem Let w be approx left e vec of C 1 with approx e val (α,β). Then for w 1 = w(1: n), ( ) η P (w 1,α,β) max 1, maxi A i 2 η C1 (w m3/2,α,β) min( A m 2, A 0 2 ) η P (w 1,α,β) η C1 (w,α,β) if min( A 0 2, A m 2 ) max i A i 2 1 w 2 / w w 2 w 1 2. MIMS Nick Higham Polynomial eigenproblem 23 / 39

30 Left Eigenvector recovery for C 1 Lemma y C n is a left e vec of P with simple e val (α,β) iff w = [α m 1 I] [α m 2 βa m αβ m 2 A 1 + β m 1 A 0 ] [α m 2 βa m α 2 β m 3 A 1 + αβ m 2 A 0 ]. [α m 2 βa 0 ] y, α 0, is a left e vec of C 1 corr. to (α,β). Every left e vec of C 1 with e val (α,β) has this form for some left e vec y of P. MIMS Nick Higham Polynomial eigenproblem 24 / 39

31 Left Eigenvector recovery for C 1 Lemma y C n is a left e vec of P with simple e val (α,β) iff w = [β m 1 I] [αβ m 2 A m + β m 1 A m 1 ]. [α m 1 A m + + αβ m 2 A 2 + β m 1 A 1 ] y, β 0 is a left e vec of C 1 corr. to (α,β). Every left e vec of C 1 with e val (α,β) has this form for some left e vec y of P. MIMS Nick Higham Polynomial eigenproblem 24 / 39

32 Left Eigenvector recovery for C 1 Lemma y C n is a left e vec of P with simple e val (α,β) iff w = [β m 1 I] [αβ m 2 A m + β m 1 A m 1 ]. [α m 1 A m + + αβ m 2 A 2 + β m 1 A 1 ] y, β 0 is a left e vec of C 1 corr. to (α,β). Every left e vec of C 1 with e val (α,β) has this form for some left e vec y of P. If A i 2 1 i then exact left e vec w satisfies w 2 / w 1 2 (m 2 /3) 1/2. MIMS Nick Higham Polynomial eigenproblem 24 / 39

33 UnScaled Companion Form C 1 (λ) = λx + Y i with X = diag(a m, I n,...,i n ), A m 1 A m 2... A 0 I Y 1 = n I n 0 MIMS Nick Higham Polynomial eigenproblem 25 / 39

34 Scaled Companion Form C 1 (λ) = λx + Y i with X = diag(a m,µi n,...,µi n ), A m 1 A m 2... A 0 µi Y 1 = n µi n 0 Let D µ = diag(1,µ,...,µ) I n, where µ = max i A i 2. Then D µ C 1 (λ) L 1 (P) with v = e 1. MIMS Nick Higham Polynomial eigenproblem 26 / 39

35 Scaled Companion: Right and Left E vecs ρ = max i A i 2 min( A 0 2, A m 2 ). Theorem Let z and w be approx right and left e vecs of D µ C 1 corr. to the approx e val (α,β). Then η P (z k,α,β) η DµC 1 (z,α,β) m5/2 ρ z 2 z k 2, k = 1: m, η P (w 1,α,β) η DµC 1 (w,α,β) m3/2 ρ w 2 w 1 2. MIMS Nick Higham Polynomial eigenproblem 27 / 39

36 Scaling Q(λ) Write a = A 2, b = B 2, c = C 2. Fan, Lin & Van Dooren (2004): let λ = µγ, Q(λ) = λ 2 A + λb + C Q(µ) = µ 2 (δγ 2 A) + µ(δγb) + δc, where γ = c/a, δ = 2/(c + bγ). For Q(µ) = µ 2Ã + µ B + C we have max( Ã 2, B 2, C 2 ) 2. MIMS Nick Higham Polynomial eigenproblem 28 / 39

37 Growth Factor Bound η eq (z i,α,β) η C1 (z,α,β) 27/2 ω z 2 z i 2, i = 1, 2, where, with α 2 + β 2 = 1, { } 1 1 ω min 1 + τ, 1 + τ, τ = b. αβ ac MIMS Nick Higham Polynomial eigenproblem 29 / 39

38 Growth Factor Bound η eq (z i,α,β) η C1 (z,α,β) 27/2 ω z 2 z i 2, i = 1, 2, where, with α 2 + β 2 = 1, { } 1 1 ω min 1 + τ, 1 + τ, τ = b. αβ ac F, L & VD identify max(1 + τ, 1 + τ 1 ) as growth factor. Our bounds for ω sharper: τ 1 is harmless. if τ 1, min{ } = O(1) if α β = O(1). ω = O(1) if B 2 < A 2 C 2. Hence η C1 η Q for systems not heavily damped. MIMS Nick Higham Polynomial eigenproblem 29 / 39

39 Comparison with Conditioning Results Analysis of H, D. S. Mackey & Tisseur (2005) bounds the ratio κ C1 (α,β)/κ P (α,β). Conditions for those κ C1 (α,β) κ P (α,β) are essentially the same as those for η C1 η P. Backward error results entirely harmonious with e val conditioning results. MIMS Nick Higham Polynomial eigenproblem 30 / 39

40 Polyeig (Companion Pencil) 4 x

41 Polyeig on Scaled Quadratic 4 x

42 Details Before scaling After scaling A B C ρ = ρ = 1, ω = 1 MIMS Nick Higham Polynomial eigenproblem 33 / 39

43 Outline PEP and Linearization Background B err and Conditioning of Linearizations Algorithm based on Linearization MIMS Nick Higham Polynomial eigenproblem 34 / 39

44 Meta-Algorithm for PEP 1 Preprocess P 2 for one or more (scaled) linearizations L 3 Balance (scale) L 4 Apply QZ to L (maybe HZ if structured) 5 Obtain relevant e vals 6 Recover left and right e vecs 7 Iteratively refine e vecs 8 Compute/estimate b errs and condition numbers 9 Detect nonregular problem 10 end MIMS Nick Higham Polynomial eigenproblem 35 / 39

45 Balancing Balancing GEP: Ward (1981) Lemonnier & Van Dooren (2006) To investigate: To what extent can balancing make the results worse? Cf. Watkins (2006): A Case where Balancing is Harmful. Exploit structure of pencils arising via linearization of a matrix poly. Can we balance a QEP? MIMS Nick Higham Polynomial eigenproblem 36 / 39

46 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. MIMS Nick Higham Polynomial eigenproblem 37 / 39

47 Tentative Outline of Algorithm for QEP 1 Preprocess Q: Fan, Lin & Van Dooren (2004) scaling 2 Let L = D µ C 1 : scaled companion linearization 3 Apply QZ to L 4 Obtain relevant e vals 5 Recover left (w 1 ) and right (max i z i 2 ) e vecs 6 Compute/estimate b errs and condition numbers 7 Detect nonregular problem MIMS Nick Higham Polynomial eigenproblem 38 / 39

48 Tentative Outline of Algorithm for QEP 1 Preprocess Q: Fan, Lin & Van Dooren (2004) scaling 2 Let L = D µ C 1 : scaled companion linearization 3 Apply QZ to L 4 Obtain relevant e vals 5 Recover left (w 1 ) and right (max i z i 2 ) e vecs 6 Compute/estimate b errs and condition numbers 7 Detect nonregular problem Mainpolyeig differences: Doesn t scale. Doesn t return left e vecs. Doesn t detect nonregular Q. MIMS Nick Higham Polynomial eigenproblem 38 / 39

49 Concluding Remarks Analysis of cond. & b err for wide variety of lineariz ns. E vector recovery formulae crucial. Scaling crucial. Favour L = (scaled) companion form for general PEPs. L L 1 (P), L 2 (P) or DL(P) for structured problems. Implications for pseudospectra. Further work needed on algorithmics scaling & balancing (pencils & general m) structured problems (symm, odd even, definite)... MIMS Nick Higham Polynomial eigenproblem 39 / 39

50 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): , H.-Y. Fan, W.-W. Lin, and P. Van Dooren. Normwise scaling of second order polynomial matrices. SIAM J. Matrix Anal. Appl., 26(1): , MIMS Nick Higham Polynomial eigenproblem 35 / 39

51 Bibliography II N. J. Higham, R.-C. Li, and F. Tisseur. Backward error of polynomial eigenproblems solved by linearization. MIMS EPrint 2006.xx, Manchester Institute for Mathematical Sciences, The University of Manchester, UK, In preparation. 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. MIMS Nick Higham Polynomial eigenproblem 36 / 39

52 Bibliography III 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., 28(1): , MIMS Nick Higham Polynomial eigenproblem 37 / 39

53 Bibliography IV 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. D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann. Structured polynomial eigenvalue problems: Good vibrations from good linearizations. MIMS EPrint , Manchester Institute for Mathematical Sciences, The University of Manchester, UK, To appear in SIAM J. Matrix Anal. Appl. MIMS Nick Higham Polynomial eigenproblem 38 / 39

54 Bibliography V 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. Electron. Trans. Numer. Anal., 23:1 4, MIMS Nick Higham Polynomial eigenproblem 39 / 39