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
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