Structured Condition Numbers and Backward Errors in Scalar Product - PowerPoint PPT Presentation

Structured Condition Numbers and Backward Errors in Scalar Product Spaces Françoise Tisseur Department of Mathematics University of Manchester ftisseur@ma.man.ac.uk http://www.ma.man.ac.uk/~ftisseur/ Joint work with Stef Graillat (Univ. of Perpignan). – p. 1/19

Motivations ◮ Condition numbers and backward errors play an important role in numerical linear algebra. forward 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

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

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

Scalar Products A scalar product �· , ·� M is a nondegenerate ( M nonsingular) bilinear or sesquilinear form on K n ( K = R or C ). � x T My, real or complex bilinear forms, � x, y � M = sesquilinear forms. x ∗ My, Adjoint A⋆ of A ∈ K n × n wrt �· , ·� M : � M − 1 A T M, for bilinear forms, A⋆ = M − 1 A ∗ M, for sesquilinear forms. M T = ± M, � (bilinear), �· , ·� M orthosymmetric if M ∗ = αM, | α | = 1 , (sesquilinear). �· , ·� M is unitary if M = βU for some unitary U and β > 0 . – p. 5/19

Matrix Groups, Jordan and Lie Algebras Three important classes of matrices associated with �· , ·� M : G = { A ∈ K n × n : A⋆ = A − 1 } Automorphism group: L = { A ∈ K n × n : A⋆ = − A } . Lie algebra: J = { A ∈ K n × n : A⋆ = A } . Jordan algebra: Recall that � M − 1 A T M, for bilinear forms, A⋆ = M − 1 A ∗ M, for sesquilinear forms. Concentrate on Jordan and Lie algebras of orthosymmetric and unitary scalar products �· , ·� M . – p. 6/19

Some Structured Matrices Space M Jordan Algebra Lie Algebra Bilinear forms R n Symm. Skew-symm. I C n Complex symm. Complex skew-symm. I R n Persymmetric Perskew-symm. R R n Σ p,q Pseudo symm. Pseudo skew-symm. R 2 n J Skew-Hamiltonian. Hamiltonian Sesquilinear form C n Hermitian Skew-Herm. I C n Σ p,q Pseudo Hermitian Pseudo skew-Herm. C 2 n J -skew-Hermitian J -Hermitian J � � ...     1 0 In  Ip 0 R = , J =  , Σp,q =   − In − Iq 0 0 1 – p. 7/19

Matrix Inverse Structured condition number for matrix inverse ( ν = 2 , F ): � � ( A + ∆A ) − 1 − A − 1 � ν : � ∆A � ν � κ ν ( A ; S ) := lim ǫ → 0 sup ≤ ǫ, ∆A ∈ S . ǫ � 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

Nearness to Singularity Structured distance to singularity ( ν = 2 , F ): ǫ : � ∆A � ν � � ≤ ǫ, A + ∆A singular , ∆A ∈ S δ ν ( A ; S ) = min . � A � ν S : Jordan or Lie algebra of �· , ·� M orthosymm. and unitary. For nonsingular A ∈ S , 1 δ 2 ( A ; S ) = δ 2 ( A ; C n × n ) = , � A � 2 � A − 1 � 2 √ δ F ( A ; C n × n ) ≤ δ F ( A ; S ) ≤ 2 δ F ( A ; C n × n ) . – p. 9/19

Linear Systems Structured condition number for linear system Ax = b , x � = 0 : � � ∆x � 2 cond ν ( A, x ; S ) = lim ǫ → 0 sup : ( A + ∆A )( x + ∆x ) = b + ∆b, ǫ � x � 2 � � ∆A � ν ≤ ǫ, � ∆b � 2 ≤ ǫ, ∆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 ) ≤ cond ν ( A, x ; S ) ≤ cond ν ( A, x ; C n × n ) . √ 2 – p. 10/19

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

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

Distance to Singularity ǫ : � ∆A � 2 � � Recall δ 2 ( A ; S ) = min ≤ ǫ, A + ∆A singular , ∆A ∈ S . � A � 2 Want to show that δ 2 ( A ; S ) = δ 2 ( A ; C n × n ) ( ⋆ ) � δ 2 ( A ; S ) = δ 2 ( MA ; M · S ) , �· , ·� M unitary ⇒ δ 2 ( MA ; C n × n ) = δ 2 ( A ; C n × n ) . ⇒ Just need to prove ( ⋆ ) for S = Sym( K ) , Skew( K ) , Herm( C ) , K = R or C . – p. 13/19

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 ) ⇒ ¯ u ∗ v = 0 . Let Q unitary s.t. Q [ e 1 , − e 2 ] = [ v, ¯ u ] . Then, 1 ) Q T ∈ Skew( K ) , ∆A = − σ min ( A ) Q ( e 1 e T 2 − e 2 e T � ∆A � 2 = σ min ( A ) , ( A + ∆A ) v = 0 . – p. 14/19

Eigenvalue Condition Number λ : simple eigenvalue of A . � | ∆λ | κ ( A, λ ; S ) = lim ǫ → 0 sup : λ + ∆λ ∈ Sp ( A + ∆A ) , ǫ � � ∆A � ≤ ǫ, ∆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: κ ( A, λ ; S ) = κ ( A, λ, C n × n ) . ◮ if M · S = Sym( C ) , 1 ≤ κ ( A, λ ; S ) ≤ κ ( A, λ ; C n × n ) . ◮ if M · L = Skew( C ) , Still incomplete. – p. 15/19

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 r T y = 0 Skew( K ) Herm( C ) r ∗ y ∈ R . – p. 16/19

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 , � 2 − |� y, r � M | 2 1 2 � r � 2 µ F ( y, r ; S ) = . β 2 � y � 2 � y � 2 2 – p. 17/19

Example Take S = Skew( R ) = { A ∈ R n × n : A = − A T } . � 0 � 1 α � � Let A = ∈ Skew( R ) and b = α . − α 0 − 1 True solution x = [1 , 1] T satisfies b T x = 0 . ◮ Let y = [1 + ǫ, 1 − ǫ ] T be an approximate solution. Then r := b − Ay = αǫx and r T y = 2 αǫ � = 0 ⇒ µ F ( y, r ; Skew( R )) = ∞ . ◮ Using a structure preserving algorithm ⇒ backward error � 0 ǫ � matrix ∆A = ∈ Skew( R ) and y = ( α/ ( ǫ + α )) x . − ǫ 0 Hence, r = b − Ay = ( ǫ/ ( ǫ + α )) b satisfies r T y = 0 and √ µ F ( y, r ; Skew( R )) = 2 � r � 2 / � y � 2 � = ∞ . – p. 18/19

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

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

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