Structured Condition Numbers and Backward Errors in Scalar Product Spaces

Transcription

1 Structured Condition Numbers and Backward Errors in Scalar Product Spaces Françoise Tisseur Department of Mathematics University of Manchester Joint work with Stef Graillat (Univ. of Perpignan). p. 1/19

2 Motivations Condition numbers and backward errors play an important role in numerical linear algebra. f orward error condition number backward error. Growing interest in structured perturbation analysis. Substantial development of algorithms for structured problems. Backward error analysis of structure preserving algorithms may be difficult. p. 2/19

3 Motivations Cont. For symmetric linear systems and for distances measured in the 2 or Frobenius norm: It makes no difference whether perturbations are restricted to be symmetric or not. Same holds for skew-symmetric and persymmetric structures. [S. Rump, 03]. Our contribution: Extend and unify these results to Structured matrices in Lie and Jordan algebras, Several structured matrix problems. p. 3/19

4 Structured Problems Normwise structured condition numbers for Matrix inversion, Nearness to singularity, Linear systems, Eigenvalue problems. Normwise structured backward errors for Linear systems, Eigenvalue problems. p. 4/19

5 Scalar Products A scalar product, M is a non degenerate (M nonsingular) bilinear or sesquilinear form on K n (K = R or C). x,y M = { x T My, real or complex bilinear forms, x My, sesquilinear forms. Adjoint A of A K n n wrt, M : A = { M 1 A T M, for bilinear forms, M 1 A M, for sesquilinear forms., M orthosymmetric if { M T = ±M, (bilinear), M = αm, α = 1, (sesquilinear)., M is unitary if M = βu for some unitary U and β > 0. p. 5/19

6 Jordan and Lie algebras Two important classes of matrices associated with, M : Lie algebra: Jordan algebra: L = {A K n n : A = A}. J = {A K n n : A = A}. Recall that A = { M 1 A T M, for bilinear forms, M 1 A M, for sesquilinear forms. Concentrate on Jordan and Lie algebras of orthosymmetric and unitary scalar products, M. p. 6/19

7 Some Structured Matrices Space M Jordan Algebra Lie Algebra Bilinear forms R n I Symm. Skew-symm. C n I Complex symm. Complex skew-symm. R n R Persymmetric Perskew-symm. R n Σ p,q Pseudo symm. Pseudo skew-symm. R 2n J Skew-Hamiltonian. Hamiltonian Sesquilinear form C n I Hermitian Skew-Herm. C n Σ p,q Pseudo Hermitian Pseudo skew-herm. C 2n J J-skew-Hermitian J-Hermitian [ R= ], J= 0 I n, Σ p,q= I p 0 In 0 0 Iq p. 7/19

8 Matrix Inverse Structured condition number for matrix inverse (ν = 2,F): { (A + A) 1 A 1 ν κ ν (A; S) := lim sup : A } ν ɛ, A S ɛ 0 ɛ A 1 ν A ν. S: Jordan or Lie algebra of orthosymm. and unitary, M. For nonsingular A S, κ 2 (A; S) = κ 2 (A; C n n ) = A 2 A 1 2 κ F (A; S) = κ F (A; C n n ) = A F A 1 2 2/ A 1 F. p. 8/19

9 Nearness to Singularity Structured distance to singularity (ν = 2,F): { δ ν (A; S) = min ɛ : A ν A ν } ɛ,a + A singular, A S. S: Jordan or Lie algebra of, M orthosymm. and unitary. For nonsingular A S, δ 2 (A; S) = δ 2 (A; C n n ) = δ F (A; C n n ) δ F (A; S) 2 δ F (A; C n n ). 1 A 2 A 1 2, p. 9/19

10 Linear Systems Structured condition number for linear system Ax = b, x 0: { x 2 cond ν (A,x; S) = lim sup : (A + A)(x + x) = b + b, ɛ 0 ɛ x 2 A ν ɛ, b } 2 ɛ, A S, ν = 2,F. A ν b 2 S: Jordan or Lie algebra of, M orthosymm. and unitary. For nonsingular A S, x 0 and ν = 2,F, cond ν (A,x; C n n ) 2 cond ν (A,x; S) cond ν (A,x; C n n ). p. 10/19

11 Key Tools Define Sym(K) = {A K n n : A T = A}, K = R or C, Skew(K) = {A K n n : A T = A}. S: Lie algebra L or Jordan algebra J of orthosymm., M. Orthosymmetry K n n = J L and, { M = M T and S = J, Sym(K) if M = M T and S = L, M S = { (bilinear forms) M = M T and S = L, Skew(K) if M = M T and S = J. Left multiplication of S by M is a bijection from K n n to K n n taking J and L to Sym(K) and Skew(K). p. 11/19

12 Key Tools Cont. Define Sym(K) = {A K n n : A T = A}, K = R or C, Skew(K) = {A K n n : A T = A}, Herm(C) = {A C n n : A = A}. S: Lie algebra L or Jordan algebra J of orthosymm., M. { M = M T and S = J, Sym(K) if M = M T and S = L, M S = { (bilinear forms) M = M T and S = L, Skew(K) if M = M T and S = J. { Herm(C) if S = J, M S = (sesquilinear forms) i Herm(C) if S = L. p. 12/19

13 Distance to Singularity { Recall δ 2 (A; S) = min ɛ : A 2 A 2 } ɛ,a + A singular, A S. Want to show that δ 2 (A; S) = δ 2 (A; C n n ) ( ), M unitary { δ 2 (A; S) = δ 2 (MA;M S), δ 2 (MA; C n n ) = δ 2 (A; C n n ). Just need to prove ( ) for S = Sym(K), Skew(K), Herm(C). p. 13/19

14 Proof of δ 2 (A; S) = δ 2 (A; C n n ) Suppose S = Skew(K) = {A K n n : A T = A}. Clearly, δ 2 (A; Skew(K)) δ 2 (A; C n n ) = 1/( A 2 A 1 2 ). Assume A 2 = 1. Need to find A Skew(K) s.t. A 2 = σ min (A) = 1/ A 1 2 and A + A singular. Let u,v s.t. Av = σ min (A)u. A Skew(K) ū v = 0. Let Q unitary s.t. Q[e 1, e 2 ] = [v, ū]. Then, A = σ min (A)Q(e 1 e T 2 e 2e T 1 )QT Skew(K), A 2 = σ min (A), (A + A)v = 0. p. 14/19

15 Eigenvalue Condition Number { λ κ(a,λ; S) = lim sup : λ + λ Sp(A + A), ɛ 0 ɛ } A ɛ, A S. S: Jordan or Lie algebra of orthosymm. and unitary, M. For sesquilinear forms: κ(a,λ; S) = κ(a,λ, C n n ). For bilinear forms: if M S = Sym(C), κ(a,λ; S) = κ(a,λ, C n n ). if M L = Skew(C), 1 κ(a,λ; S) κ(a,λ; C n n ). Still incomplete. p. 15/19

16 Structured Backward Errors µ ν (y,r, S) = min{ A ν : Ay = r, A S}, ν = 2,F. For linear systems: y 0 is the approx. sol. to Ax = b and r = b Ay. For eigenproblems: (y, λ) approx. eigenpair of A, r = (λi A)y. S: Jordan or Lie algebra of, M orthosymm. and unitary. µ ν (y,r, S) iff y,r satisfies the conditions: M S Condition Sym(K) none Skew(K) r T y = 0 Herm(C) r y R. p. 16/19

17 Structured Backward Errors Cont. µ ν (y,r, S) = min{ A ν : Ay = r, A S}, ν = 2,F. Recall µ ν (y,r; C n n ) = r 2 / y 2. S: Jordan or Lie algebra of, M orthosymm. and unitary. If µ ν (y,r, S) (ν = 2,F), µ ν (y,r; C n n ) µ ν (y,r; S) 2 µ ν (y,r; C n n ). In particular for ν = F, µ F (y,r; S) = 1 y 2 2 r 2 2 y,r M 2 β 2 y 2 2. p. 17/19

18 Example Take S = [ Skew(R) ] = {A R n n : A = A T [ }. 0 α 1 Let A = Skew(R) and b = α α 0 1 True solution x = [1, 1] T satisfies b T x = 0. ]. Let y = [1 + ɛ, 1 ɛ] T be an approximate solution. Then r := b Ay = αɛx and r T y = 2αɛ 0 µ F (y,r; Skew(R)) =. Using a structure [ preserving ] algorithm backward error 0 ɛ matrix A = Skew(R) and y = (α/(ɛ + α))x. ɛ 0 Hence, r = b Ay = (ɛ/(ɛ + α))b satisfies r T y = 0 and µ F (y,r; Skew(R)) = 2 r 2 / y 2. p. 18/19

19 Conclusion For matrices in Jordan or Lie algebras of orthosymmetric and unitary scalar products, This includes symmetric, complex symmetric, skew-symmetric, pseudo symmetric, persymmetric, Hamiltonian, skew-hamiltonian, Hermitian and J-Hermitian matrices. p. 19/19

20 Conclusion For matrices in Jordan or Lie algebras of orthosymmetric and unitary scalar products, Usual unstructured perturbation analysis sufficient for matrix inversion condition number, distance to singularity, linear system condition number. p. 19/19

21 Conclusion For matrices in Jordan or Lie algebras of orthosymmetric and unitary scalar products, Usual unstructured perturbation analysis sufficient for matrix inversion condition number, distance to singularity, linear system condition number. Partial answer for eigenvalue condition numbers. p. 19/19

22 Conclusion For matrices in Jordan or Lie algebras of orthosymmetric and unitary scalar products, Usual unstructured perturbation analysis sufficient for matrix inversion condition number, distance to singularity, linear system condition number. Partial answer for eigenvalue condition numbers. Structured backward error: may be when using non structure preserving algorithm, when finite, is within a small factor of the unstructured one. p. 19/19