Multiparameter eigenvalue problem as a structured eigenproblem

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1 Multiparameter eigenvalue problem as a structured eigenproblem Bor Plestenjak Department of Mathematics University of Ljubljana This is joint work with M Hochstenbach Będlewo, /28

2 Overview Introduction Backward error and condition numbers Pseudospectrum Distance to the nearest singular problem Będlewo, /28

3 Introduction Two-parameter eigenvalue problem: A 1 x = λb 1 x + µc 1 x A 2 y = λb 2 y + µc 2 y, (W) where A i, B i, C i are n i n i matrices, λ, µ C, x C n 1, and y C n 2 Eigenvalue: a pair (λ, µ) that satisfies (W) for nonzero x and y Eigenvector: the tensor product x y Będlewo, /28

4 u + νu = 0 on Ω, u δω = 0 Rectangular membrane: Ω = [0, a] [0, b] = two S L equations (ν = λ + µ) x + λx = 0, x(0) = x(a) = 0, y + µy = 0, y(0) = y(b) = 0 Będlewo, /28

5 u + νu = 0 on Ω, u δω = 0 Rectangular membrane: Ω = [0, a] [0, b] = two S L equations (ν = λ + µ) x + λx = 0, x(0) = x(a) = 0, y + µy = 0, y(0) = y(b) = 0 Circular membrane: Ω = {x 2 + y 2 a 2 }, polar coordinates = a triangular situation Φ + λφ = 0, Φ(0) = Φ(2π) = 0, r 1 (rr ) + (ν λr 2 )R = 0, R(0) <, R(a) = 0 Będlewo, /28

6 u + νu = 0 on Ω, u δω = 0 Rectangular membrane: Ω = [0, a] [0, b] = two S L equations (ν = λ + µ) x + λx = 0, x(0) = x(a) = 0, y + µy = 0, y(0) = y(b) = 0 Circular membrane: Ω = {x 2 + y 2 a 2 }, polar coordinates = a triangular situation Φ + λφ = 0, Φ(0) = Φ(2π) = 0, r 1 (rr ) + (ν λr 2 )R = 0, R(0) <, R(a) = 0 Elliptical membrane: Ω = {(x/a) 2 + (y/b) 2 1}, elliptic coordinates (c focus) = modified Mathieu s and Mathieu s DE ( 4λ = c 2 ν) v 1 + (2λ cosh(2y 1) µ)v 1 = 0, v 1 (0) = v 1 (d) = 0, v 2 (2λ cos(2y 1) µ)v 2 = 0, v 2 (0) = v 2 (π/2) = 0 Będlewo, /28

7 Tensor product approach A 1 x = λb 1 x + µc 1 x A 2 y = λb 2 y + µc 2 y (W) On C n 1 C n 2 of the dimension n 1 n 2 we define 0 = B 1 C 2 C 1 B 2 1 = A 1 C 2 C 1 A 2 2 = B 1 A 2 A 1 B 2 Two-parameter problem (W) is equivalent to a coupled GEP where z = x y (W) is nonsingular 0 is invertible and commute 1 z = λ 0 z 2 z = µ 0 z, ( ) Będlewo, /28

8 Right definite problem (W ) A 1 x = λb 1 x + µc 1 x A 2 y = λb 2 y + µc 2 y 0 = B 1 C 2 C 1 B 2 1 = A 1 C 2 C 1 A 2 2 = B 1 A 2 A 1 B 2 1 z = λ 0 z 2 z = µ 0 z ( ) (W) is right definite when A i, B i, C i are Hermitian and 0 is positive definite Atkinson (1972): 0 positive definite x B 1 x y B 2 y x C 1 x y C 2 y = (x y) 0 (x y) > 0 for x, y 0 If (W) is right definite then eigenvalues are real, there exist n 1 n 2 linearly independent eigenvectors, eigenvectors of distinct eigenvalues are 0 -orthogonal, ie (x 1 y 1 ) 0 (x 2 y 2 ) = 0 Będlewo, /28

9 Some available numerical methods Blum, Curtis, Geltner (1978), and Browne, Sleeman (1982): gradient method, Bohte (1980): Newton s method for eigenvalues, Slivnik, Tomšič (1986): solving ( ) with standard numerical methods for RD problem, Ji, Jiang, Lee (1992): Generalized Rayleigh Quotient Iteration Shimasaki (1995): continuation method for a special class of RD problems P (1999): continuation method for RD problem, Tensor Rayleigh Quotient Iteration P (2000): continuation method for weakly elliptic problem Hochstenbach, P (2002): Jacobi-Davidson type method for RD problem Hochstenbach, Košir, P (2003): Two-sided Jacobi-Davidson type method Hochstenbach, P (submitted): Harmonic Jacobi-Davidson type method Będlewo, /28

10 Multiparameter eigenvalue problems V 10 x 1 = λ 1 V 11 x 1 + λ 2 V 12 x λ k V 1k x 1 V 20 x 2 = λ 1 V 21 x 2 + λ 2 V 22 x λ k V 2k x 2 V k0 x k = λ 1 V k1 x k + λ 2 V k2 x k + + λ k V kk x k, where V ij is n i n i matrix A k-tuple λ = (λ 1,, λ k ) is an eigenvalue and x = x 1 x k is the corresponding eigenvector Będlewo, /28

11 Multiparameter eigenvalue problems V 10 x 1 = λ 1 V 11 x 1 + λ 2 V 12 x λ k V 1k x 1 V 20 x 2 = λ 1 V 21 x 2 + λ 2 V 22 x λ k V 2k x 2 V k0 x k = λ 1 V k1 x k + λ 2 V k2 x k + + λ k V kk x k, where V ij is n i n i matrix A k-tuple λ = (λ 1,, λ k ) is an eigenvalue and x = x 1 x k is the corresponding eigenvector In the tensor product space of dimension N = n 1 n k we define 0 = V 11 V 12 V 1k V 21 V 22 V V k1 V k2 2k V kk, i = V 11 V 1,i 1 V 21 V 2,i 1 V k1 V k,i 1 V 10 V 1,i+1 V 20 V 2,i+1 V k0 V k,i+1 V 1k V 2k V kk, i = 1,, k, where V ij is defined by V ij (x 1 x i x k ) = x 1 V ij x i x k and linearity A nonsingular MEP (ie, 0 nonsingular) is equivalent to a joined GEP i x = λ i 0 x, i = 1,, k, for decomposable tensors x = x 1 x k, where the matrices 1 0 i commute Będlewo, /28

12 Multiparameter eigenvalue problems V 10 x 1 = λ 1 V 11 x 1 + λ 2 V 12 x λ k V 1k x 1 V 20 x 2 = λ 1 V 21 x 2 + λ 2 V 22 x λ k V 2k x 2 V k0 x k = λ 1 V k1 x k + λ 2 V k2 x k + + λ k V kk x k, where V ij is n i n i matrix A k-tuple λ = (λ 1,, λ k ) is an eigenvalue and x = x 1 x k is the corresponding eigenvector In the tensor product space of dimension N = n 1 n k we define 0 = V 11 V 12 V 1k V 21 V 22 V V k1 V k2 2k V kk, i = V 11 V 1,i 1 V 21 V 2,i 1 V k1 V k,i 1 V 10 V 1,i+1 V 20 V 2,i+1 V k0 V k,i+1 V 1k V 2k V kk, i = 1,, k, where V ij is defined by V ij (x 1 x i x k ) = x 1 V ij x i x k and linearity A nonsingular MEP (ie, 0 nonsingular) is equivalent to a joined GEP i x = λ i 0 x, i = 1,, k, for decomposable tensors x = x 1 x k, where the matrices 1 0 i commute MEP is right definite when all matrices V ij are Hermitian and 0 is positive definite Będlewo, /28

13 Overview Introduction Backward error and condition numbers Pseudospectrum Distance to the nearest singular problem Będlewo, /28

14 Backward error W i (λ) := V i0 kx j=1 λ j V ij for i = 1,, k Let (ex, e λ) be an approximate eigenpair of W and let ex = ex 1 ex k be normalized, ie, ex i 2 = 1 for i = 1,, k We define the normwise backward error of (ex, e λ) by where η(ex, λ) e := min nɛ : W i (e λ) + Wi (e λ) o exi = 0, V ij ɛ E ij for all i, j W i (λ) = V i0 kx j=1 λ j V ij for i = 1,, k, Będlewo, /28

15 Backward error W i (λ) := V i0 kx j=1 λ j V ij for i = 1,, k Let (ex, e λ) be an approximate eigenpair of W and let ex = ex 1 ex k be normalized, ie, ex i 2 = 1 for i = 1,, k We define the normwise backward error of (ex, e λ) by where η(ex, λ) e := min nɛ : W i (e λ) + Wi (e λ) o exi = 0, V ij ɛ E ij for all i, j Can show: W i (λ) = V i0 and e θi := E i0 + kx j=1 η(ex, e λ) = max i=1,,k kx j=1 λ j V ij for i = 1,, k r i eθ i, where r i := W i (e λ)ex i e λj E ij for i = 1,, k, Będlewo, /28

16 MEP vs GEP and PEP GEP (V Frayssé, V Toumazou (1998), Higham, Higham (1998)): Ax = λbx η(ex, e λ) = min n ɛ : (A + A)ex = e λ(b + B)ex, A ɛ E, B ɛ F o = e λbex Aex E + e λ F PEP (Tisseur (2000)): P (λ)x = (λ m A m + λ m 1 A m A 0 )x = 0 o η(ex, λ) e = min nɛ : (P (e λ) + P ( λ))ex e = 0, Ai ɛ E i, i = 0,, m = P (e λ)ex P mi=0 e λ i E i MEP (Hochstenbach, P (2003)): η(ex, e λ) = min nɛ : W i (λ)x i = (V i0 P k j=1 λ j V ij )x i = 0 for i = 1,, k o W i (e λ) + Wi (e λ) exi = 0, V ij ɛ E ij for all i, j = max i=1,,k W i (e λ)xi E i0 + P k j=1 e λj E ij Będlewo, /28

17 Backward error for a Hermitian MEP If W is Hermitian then we consider perturbations where all V ij are Hermitian η H (ex, e λ) := min{ ɛ : (Wi (e λ) + Wi (e λ))ex i = 0, V ij = V ij, V ij ɛ E ij, i = 1,, k; j = 0,, k} Clearly, η H (ex, e λ) η(ex, e λ) Can show: If W is Hermitian and e λ is real then η H (ex, e λ) = η(ex, e λ) Będlewo, /28

18 Backward error for an eigenvalue approximation η(ex, e λ) = max i=1,,k r i eθ i, r i := W i (e λ)ex i, e θi := E i0 + kx j=1 e λj E ij for i = 1,, k If we are interested only in the approximate eigenvalue e λ, then a more appropriate measure of the backward error may be η(e λ) := min{ η(ex, e λ) : ex normalized } Can show: η(e λ) = max i=1,,k σ min (W i (e λ)) eθ i Będlewo, /28

19 Alternative definition η(ex, λ) e = min nɛ : W i (e λ) + Wi (e λ) o exi = 0, V ij ɛ E ij for all i, j = max i=1,,k r i E i0 + P k j=1 e λj E ij Liu and Wang (2005): Backward error in Frobenius norm 8 > 0 < η F (ex, λ) e := X k >: = kx i=1 i=1 kx j=0 1 V ij 2 F A 1/2 E ij 2 r i 2!1/2 P E i0 2 k + j=1 λj e 2 E ij 2 9 : W i (e λ) + Wi (e λ) >= exi = 0>; Będlewo, /28

20 Alternative definition η(ex, λ) e = min nɛ : W i (e λ) + Wi (e λ) o exi = 0, V ij ɛ E ij for all i, j = max i=1,,k r i E i0 + P k j=1 e λj E ij Liu and Wang (2005): Backward error in Frobenius norm 8 > 0 < η F (ex, λ) e := X k >: = kx i=1 i=1 kx j=0 1 V ij 2 F A 1/2 E ij 2 r i 2!1/2 P E i0 2 k + j=1 λj e 2 E ij 2 9 : W i (e λ) + Wi (e λ) >= exi = 0>; Can show: 1 k η F (ex, e λ) η(ex, e λ) ηf (ex, e λ) Będlewo, /28

21 Eigenvector backward error If we assume that E i0 = E i1 = = E ik for i = 1,, k then η F (ex) = inf eλ η F (ex, e λ) = inf eλ where E := diag( E 10,, E k0 ) E 1 0 B@ V 10 ex 1 V 1k ex 1 V 20 ex 2 V 2k ex 2 V k0 ex k V kk ex k 0 1 e λ1 e λk 1 B@ CA 0 0 V 10 ex 1 V = σ 20 ex 2 min B@ B@ E 1 V 1k ex 1 V 2k ex 2 V k0 ex k V kk ex k 10 CA B@ 11 CA CA, 1 1 e λ1 CA e λk η F (ex) k η(ex) η F (ex) Będlewo, /28

22 Matrix B 0 Let x and y be the corresponding left and right eigenvectors of λ, respectively We define B 0 = 2 64 y 1 V 11x 1 y 1 V 12x 1 y 1 V 1kx 1 y 2 V 21x 2 y 2 V 22x 2 y 2 V 2kx 2 y k V k1x k y k V k2x k y k V kkx k 3 75 Notice that det(b 0 ) = y 0 x Košir (1994): λ is algebraically simple = B 0 is nonsingular Będlewo, /28

23 Eigenvalue condition number Let λ be a nonzero algebraically simple eigenvalue of a nonsingular MEP W with normalized right eigenvector x and left eigenvector y A normwise condition number of λ is λ κ(λ) := lim sup ɛ 0 ɛ λ : W i(λ + λ) + W i (λ + λ) (x i + x i ) = 0, V ij ɛ E ij, i = 1,, k; j = 0,, k Będlewo, /28

24 Eigenvalue condition number Let λ be a nonzero algebraically simple eigenvalue of a nonsingular MEP W with normalized right eigenvector x and left eigenvector y A normwise condition number of λ is λ κ(λ) := lim sup ɛ 0 ɛ λ : W i(λ + λ) + W i (λ + λ) (x i + x i ) = 0, V ij ɛ E ij, i = 1,, k; j = 0,, k Can show: where and B 1 0 θ θ i := E i0 + κ(λ) = λ 1 B 1 0 θ, kx j=1 λ j E ij for i = 1,, k := max{ B 1 0 z 2 : z C k, z i = θ i for i = 1,, k } Będlewo, /28

25 Eigenvalue condition number Let λ be a nonzero algebraically simple eigenvalue of a nonsingular MEP W with normalized right eigenvector x and left eigenvector y A normwise condition number of λ is λ κ(λ) := lim sup ɛ 0 ɛ λ : W i(λ + λ) + W i (λ + λ) (x i + x i ) = 0, V ij ɛ E ij, i = 1,, k; j = 0,, k Can show: where and B 1 0 θ θ i := E i0 + κ(λ) = λ 1 B 1 0 θ, kx j=1 λ j E ij for i = 1,, k := max{ B 1 0 z 2 : z C k, z i = θ i for i = 1,, k } 1 θ B B 1 0 θ B θ 2 Będlewo, /28

26 Hermitian MEP κ H (λ) := lim sup ɛ 0 λ ɛ λ : W i(λ + λ) + W i (λ + λ) (x i + x i ) = 0, V ij ɛ E ij, V ij = V ij, i = 1,, k; j = 0,, k Can show: If W is Hermitian and λ is real then κ H (λ) = κ(λ) Będlewo, /28

27 Hermitian MEP κ H (λ) := lim sup ɛ 0 λ ɛ λ : W i(λ + λ) + W i (λ + λ) (x i + x i ) = 0, V ij ɛ E ij, V ij = V ij, i = 1,, k; j = 0,, k Can show: If W is Hermitian and λ is real then κ H (λ) = κ(λ) = λ 1 B 1 0 θ If MEP is Hermitian, then B 0 is real and B 1 0 θ = max{ B 1 0 z 2 : z R k, z i = θ i for i = 1,, k } Będlewo, /28

28 MEP vs GEP and PEP GEP (V Frayssé, V Toumazou (1998), Higham, Higham (1998)): κ(λ) = lim ɛ 0 sup = E + λ F λ y Bx Ax = λbx λ ɛ λ : (A + A)(x + x) = (λ + λ)(b + B)(x + x), PEP (Tisseur (2000)): P (λ)x = (λ m A m + λ m 1 A m A 0 )x = 0 κ(λ) = lim ɛ 0 sup λ ɛ λ : (P (λ + λ) + P (λ + λ))(x + x) = 0, A i ɛ E i = P mi=0 λ i E i λ y P (λ)x MEP (Hochstenbach, P (2003)): ɛ 0 W i (λ)x i = (V i0 P k j=1 λ j V ij )x i = 0 for i = 1,, k λ κ(λ) = lim sup ɛ λ : W i (λ + λ) + W i (λ + λ) (x i + x i ) = 0, = 1 λ 2 4 y 1 V 11x 1 y1 V 1kx 1 yk V k1x k yk V kkx k θ θ 2 B0 1 2 λ Będlewo, /28

29 Overview Introduction Backward error and condition numbers Pseudospectrum Distance to the nearest singular problem Będlewo, /28

30 Pseudospectrum We define the ɛ-pseudospectrum of W by Λ ɛ (W ) = λ C k : (W i (λ) + W i (λ))x i = 0, x i 0, V ij ɛ E ij, i = 1,, k; j = 0,, k Będlewo, /28

31 Pseudospectrum We define the ɛ-pseudospectrum of W by Λ ɛ (W ) = λ C k : (W i (λ) + W i (λ))x i = 0, x i 0, V ij ɛ E ij, i = 1,, k; j = 0,, k Can show: Λ ɛ (W ) = { λ C k : η(λ) ɛ} = { λ C k : σ min (W i (λ)) ɛθ i for i = 1,, k } = { λ C k : W i (λ) 1 1/(ɛθ i ) for i = 1,, k } = { λ C k : u i, u i = 1 st W i (λ)u i ɛθ i for i = 1,, k }, where θ i = E i0 + P k j=1 λ j E ij for i = 1,, k Będlewo, /28

32 Pseudospectrum We define the ɛ-pseudospectrum of W by Λ ɛ (W ) = λ C k : (W i (λ) + W i (λ))x i = 0, x i 0, V ij ɛ E ij, i = 1,, k; j = 0,, k Can show: Λ ɛ (W ) = { λ C k : η(λ) ɛ} = { λ C k : σ min (W i (λ)) ɛθ i for i = 1,, k } = { λ C k : W i (λ) 1 1/(ɛθ i ) for i = 1,, k } = { λ C k : u i, u i = 1 st W i (λ)u i ɛθ i for i = 1,, k }, where θ i = E i0 + P k j=1 λ j E ij for i = 1,, k Λ ɛ (W ) = Λ ɛ (W 1 ) Λ ɛ (W 2 ) Λ ɛ (W k ), where Λ ɛ (W i ) = λ C k : (W i (λ) + W i (λ))x i = 0, x i 0, V ij ɛ E ij, j = 0,, k Będlewo, /28

33 W 1 (λ) = W 2 (λ) = Example λ λ λ λ λ 2 0 λ λ λ λ 2 0 λ λ Będlewo, /28 λ 1

34 Example 2 We discretize the two-parameter Sturm-Liouville problem from Binding, Browne, Ji (1993) W 1 (λ)x 1 (t 1 ) = x 1 (t 1) (λ 1 + λ 2 cos 2t 1 )x 1 (t 1 ), W 2 (λ)x 2 (t 2 ) = x 2 (t 2) λ 2 x 2 (t 2 ) with boundary conditions x i (0) = x i (π) = 0 for i = 1, 2 Będlewo, /28

35 Example 2 We discretize the two-parameter Sturm-Liouville problem from Binding, Browne, Ji (1993) W 1 (λ)x 1 (t 1 ) = x 1 (t 1) (λ 1 + λ 2 cos 2t 1 )x 1 (t 1 ), W 2 (λ)x 2 (t 2 ) = x 2 (t 2) λ 2 x 2 (t 2 ) with boundary conditions x i (0) = x i (π) = 0 for i = 1, 2 = V 10 = V 20 = 1 h 2 tridiag(1, 2, 1), V 11 = I, V 21 = 0, V 22 = I, V 12 = diag cos n+1 2π, cos 4π n+1,, cos 2nπ n λ 2 30 λ λ Będlewo, /28 λ 1

36 Overview Introduction Backward error and condition numbers Pseudospectrum Distance to the nearest singular problem Będlewo, /28

37 Distance to the nearest singular problem MEP is singular 0 is singular Będlewo, /28

38 Distance to the nearest singular problem MEP is singular 0 is singular Let k = 2 We define the distance to the nearest singular MEP as B1 δ(w ) = min F C 1 F F : W + W is singular B 2 F C 2 F Będlewo, /28

39 Distance to the nearest singular problem MEP is singular 0 is singular Let k = 2 We define the distance to the nearest singular MEP as B1 δ(w ) = min F C 1 F F : W + W is singular B 2 F C 2 F If W + W is singular then S 0 σ min ( 0 ), where S 0 := (B 1 + B 1 ) (C 2 + C 2 ) (C 1 + C 1 ) (B 2 + B 2 ) 0 Będlewo, /28

40 Distance to the nearest singular problem MEP is singular 0 is singular Let k = 2 We define the distance to the nearest singular MEP as B1 δ(w ) = min F C 1 F F : W + W is singular B 2 F C 2 F If W + W is singular then S 0 σ min ( 0 ), where S 0 := (B 1 + B 1 ) (C 2 + C 2 ) (C 1 + C 1 ) (B 2 + B 2 ) 0 If we neglect second order terms, it follows from X Y = X Y that S 0 B 1 C 2 + B 1 C 2 + C 1 B 2 + C 1 B 2 B1 F B1 C 1 F B 2 F C 2 F F C 1 F B 2 C 2 and it follows δ(w ) σ min ( 0 ) B1 C 1 F B 2 C 2 Będlewo, /28

41 Distance to the nearest singular problem for a right definite MEP Atkinson (1972): For a Hermitian MEP, 0 is positive definite (x y) 0 (x y) > 0 for all x, y 0 Będlewo, /28

42 Distance to the nearest singular problem for a right definite MEP Atkinson (1972): For a Hermitian MEP, 0 is positive definite (x y) 0 (x y) > 0 for all x, y 0 Corollary: A Hermitian MEP is not definite there exist x, y 0 such that (x y) 0 (x y) = 0 Będlewo, /28

43 Distance to the nearest singular problem for a right definite MEP Atkinson (1972): For a Hermitian MEP, 0 is positive definite (x y) 0 (x y) > 0 for all x, y 0 Corollary: A Hermitian MEP is not definite there exist x, y 0 such that (x y) 0 (x y) = 0 x B1 x x C Let us define M(x, y) = 1 x x B1 x x C y B 2 y y and M(x, y) = 1 x C 2 y y B 2 y y C 2 y Since (x y) 0 (x y) = det M(x, y) we want to make M(x, y) singular M(x, y) F B1 B1 F and a necessary condition is B1 F B 2 F C 1 F B 2 C 2 C 1 F C 2 F F σ 2 (M(x, y)) C 1 F B 2 F C 2 F F, Będlewo, /28

44 Distance to the nearest singular problem for a right definite MEP Atkinson (1972): For a Hermitian MEP, 0 is positive definite (x y) 0 (x y) > 0 for all x, y 0 Corollary: A Hermitian MEP is not definite there exist x, y 0 such that (x y) 0 (x y) = 0 x B1 x x C Let us define M(x, y) = 1 x x B1 x x C y B 2 y y and M(x, y) = 1 x C 2 y y B 2 y y C 2 y Since (x y) 0 (x y) = det M(x, y) we want to make M(x, y) singular M(x, y) F B1 B1 F and a necessary condition is B1 F B 2 F C 1 F B 2 C 2 C 1 F C 2 F F σ 2 (M(x, y)) C 1 F B 2 F C 2 F F, The smallest perturbation that makes M(x, y) singular is M(x, y) = σ 2 uv T, where σ 2 is the smallest singular value with the corresponding left and right singular vectors u and v If we take B 1 = σ 2 u 1 v 1 xx, C 1 = σ 2 u 1 v 2 xx, B 2 = σ 2 u 2 v 1 yy, C 2 = σ 2 u 2 v 2 yy, then M(x, y) + M(x, y) is singular and B1 F C 1 F B 2 F C 2 F F = σ 2 (M(x, y)) Będlewo, /28

45 Distance to the nearest singular problem for a right definite MEP Atkinson (1972): For a Hermitian MEP, 0 is positive definite (x y) 0 (x y) > 0 for all x, y 0 Corollary: A Hermitian MEP is not definite there exist x, y 0 such that (x y) 0 (x y) = 0 x B1 x x C Let us define M(x, y) = 1 x x B1 x x C y B 2 y y and M(x, y) = 1 x C 2 y y B 2 y y C 2 y Since (x y) 0 (x y) = det M(x, y) we want to make M(x, y) singular M(x, y) F B1 B1 F and a necessary condition is B1 F B 2 F C 1 F B 2 C 2 C 1 F C 2 F F σ 2 (M(x, y)) C 1 F B 2 F C 2 F F, The smallest perturbation that makes M(x, y) singular is M(x, y) = σ 2 uv T, where σ 2 is the smallest singular value with the corresponding left and right singular vectors u and v If we take B 1 = σ 2 u 1 v 1 xx, C 1 = σ 2 u 1 v 2 xx, B 2 = σ 2 u 2 v 1 yy, C 2 = σ 2 u 2 v 2 yy, then M(x, y) + M(x, y) is singular and B1 F C 1 F B 2 F C 2 F F = σ 2 (M(x, y)) Thus, δ(w ) = min σ 2(M(x, y)) x = y =1 Będlewo, /28

46 Distance to singular problems for the right definite MEP α := B1 C 1 B 2 C 2 F δ(w ) = min σ 2(M(x, y)) = min M(x, x = y =1 x = y =1 y) 1 1 x B M(x, y) = 1 x x C 1 x y B 2 y y C 2 y M(x, y) 1 1 y C = 2 y x C 1 x (x B 1 x)(y C 2 y) (x C 1 x)(y B 2 y) y B 2 y x B 1 x M(x, y) 1 α (x y) 0 (x y) Thus, δ(w ) min (x x = y =1 y) 0 (x y) α min z =1 z 0 z α = σ min( 0 ) α Będlewo, /28

47 Bibliography M E Hochstenbach, B Plestenjak: Backward error, condition numbers, and pseudospectrum for the multiparameter eigenvalue problem, Linear Algebra Appl 375 (2003) Będlewo, /28

HARMONIC RAYLEIGH RITZ EXTRACTION FOR THE MULTIPARAMETER EIGENVALUE PROBLEM

HARMONIC RAYLEIGH RITZ EXTRACTION FOR THE MULTIPARAMETER EIGENVALUE PROBLEM MICHIEL E. HOCHSTENBACH AND BOR PLESTENJAK Abstract. We study harmonic and refined extraction methods for the multiparameter