Preconditioned inverse iteration and shift-invert Arnoldi method

Transcription

1 Preconditioned inverse iteration and shift-invert Arnoldi method Melina Freitag Department of Mathematical Sciences University of Bath CSC Seminar Max-Planck-Institute for Dynamics of Complex Technical Systems Magdeburg 3rd May 2011 joint work with Alastair Spence (Bath)

2 1 Introduction 2 Inexact inverse iteration 3 Inexact Shift-invert Arnoldi method 4 Inexact Shift-invert Arnoldi method with implicit restarts 5 Conclusions

3 Outline 1 Introduction 2 Inexact inverse iteration 3 Inexact Shift-invert Arnoldi method 4 Inexact Shift-invert Arnoldi method with implicit restarts 5 Conclusions

4 Problem and iterative methods Find a small number of eigenvalues and eigenvectors of: Ax = λx, λ C,x C n A is large, sparse, nonsymmetric

5 Problem and iterative methods Find a small number of eigenvalues and eigenvectors of: Ax = λx, λ C,x C n A is large, sparse, nonsymmetric Iterative solves Power method Simultaneous iteration Arnoldi method Jacobi-Davidson method repeated application of the matrix A to a vector Generally convergence to largest/outlying eigenvector

6 Problem and iterative methods Find a small number of eigenvalues and eigenvectors of: Ax = λx, λ C,x C n A is large, sparse, nonsymmetric Iterative solves Power method Simultaneous iteration Arnoldi method Jacobi-Davidson method The first three of these involve repeated application of the matrix A to a vector Generally convergence to largest/outlying eigenvector

7 Shift-invert strategy Wish to find a few eigenvalues close to a shift σ σ λ3 λ1 λ2 λ4 λ n

8 Shift-invert strategy Wish to find a few eigenvalues close to a shift σ σ Problem becomes λ3 λ1 λ2 λ4 λ n (A σi) 1 x = 1 λ σ x each step of the iterative method involves repeated application of A = (A σi) 1 to a vector Inner iterative solve: (A σi)y = x using Krylov method for linear systems. leading to inner-outer iterative method.

9 Shift-invert strategy This talk: Inner iteration and preconditioning Fixed shifts only Inverse iteration and Arnoldi method

10 Outline 1 Introduction 2 Inexact inverse iteration 3 Inexact Shift-invert Arnoldi method 4 Inexact Shift-invert Arnoldi method with implicit restarts 5 Conclusions

11 The algorithm Inexact inverse iteration for i = 1 to... do choose τ (i) solve Rescale x (i+1) = y(i) y (i), (A σi)y (i) x (i) = d (i) τ (i), Update λ (i+1) = x (i+1)h Ax (i+1), Test: eigenvalue residual r (i+1) = (A λ (i+1) I)x (i+1). end for

12 The algorithm Inexact inverse iteration for i = 1 to... do choose τ (i) solve Rescale x (i+1) = y(i) y (i), (A σi)y (i) x (i) = d (i) τ (i), Update λ (i+1) = x (i+1)h Ax (i+1), Test: eigenvalue residual r (i+1) = (A λ (i+1) I)x (i+1). end for Convergence rates If τ (i) = C r (i) then convergence rate is linear (same convergence rate as for exact solves).

13 The inner iteration for (A σi)y = x Standard GMRES theory for y 0 = 0 and A diagonalisable x (A σi)y k κ(w) min max p(λj) x j=1,...,n p P k where λ j are eigenvalues of A σi and (A σi) = WΛW 1.

14 The inner iteration for (A σi)y = x Standard GMRES theory for y 0 = 0 and A diagonalisable x (A σi)y k κ(w) min max p(λj) x j=1,...,n p P k where λ j are eigenvalues of A σi and (A σi) = WΛW 1. Number of inner iterations for x (A σi)y k τ. k C 1 +C 2log x τ

15 The inner iteration for (A σi)y = x More detailed GMRES theory for y 0 = 0 λ2 λ1 x (A σi)y k κ(w) λ 1 where λ j are eigenvalues of A σi. Number of inner iterations min p P k 1 max p(λj) Qx j=2,...,n k C 1 +C 2log Qx, τ where Q projects onto the space not spanned by the eigenvector.

16 The inner iteration for (A σi)y = x Good news! k (i) C 1 +C 2log C3 r(i) τ (i), where τ (i) = C r (i). Iteration number approximately constant!

17 The inner iteration for (A σi)y = x Good news! k (i) C 1 +C 2log C3 r(i) τ (i), where τ (i) = C r (i). Iteration number approximately constant! Bad news :-( For a standard preconditioner P (A σi)p 1 ỹ (i) = x (i) P 1 ỹ (i) = y (i) k (i) C 1 +C 2 log Qx (i) τ (i) = C 1 +C 2 log C τ (i), where τ (i) = C r (i). Iteration number increases!

18 Convection-Diffusion operator Finite difference discretisation on a grid of the convection-diffusion operator u+5u x +5u y = λu on (0,1) 2, with homogeneous Dirichlet boundary conditions ( matrix). smallest eigenvalue: λ , Preconditioned GMRES with tolerance τ (i) = 0.01 r (i), standard and tuned preconditioner (incomplete LU).

19 Convection-Diffusion operator 45 right preconditioning, 569 iterations 10 0 right preconditioning, 569 iterations inner iterations residual norms \ r^{(i)}\ outer iterations sum of inner iterations Figure: Inner iterations vs outer iterations Figure: Eigenvalue residual norms vs total number of inner iterations

20 The inner iteration for (A σi)p 1 ỹ = x How to overcome this problem Use a different preconditioner, namely one that satisfies P ix (i) = Ax (i), P i := P +(A P)x (i) x (i)h minor modification and minor extra computational cost, [AP 1 i ]Ax (i) = Ax (i).

21 The inner iteration for (A σi)p 1 ỹ = x How to overcome this problem Use a different preconditioner, namely one that satisfies P ix (i) = Ax (i), P i := P +(A P)x (i) x (i)h minor modification and minor extra computational cost, [AP 1 i ]Ax (i) = Ax (i). Why does that work? Assume we have found eigenvector x 1 Ax 1 = Px 1 = λ 1x 1 (A σi)p 1 x 1 = λ1 σ λ 1 x 1 and convergence of Krylov method applied to (A σi)p 1 ỹ = x 1 in one iteration. For general x (i) k (i) C 1 +C 2 log C3 r(i), where τ (i) = C r (i). τ (i)

22 Convection-Diffusion operator Finite difference discretisation on a grid of the convection-diffusion operator u+5u x +5u y = λu on (0,1) 2, with homogeneous Dirichlet boundary conditions ( matrix). smallest eigenvalue: λ , Preconditioned GMRES with tolerance τ (i) = 0.01 r (i), standard and tuned preconditioner (incomplete LU).

23 Convection-Diffusion operator 45 right preconditioning, 569 iterations 10 0 right preconditioning, 569 iterations inner iterations residual norms \ r^{(i)}\ outer iterations sum of inner iterations Figure: Inner iterations vs outer iterations Figure: Eigenvalue residual norms vs total number of inner iterations

24 Convection-Diffusion operator 45 right preconditioning, 569 iterations tuned right preconditioning, 115 iterations 10 0 right preconditioning, 569 iterations tuned right preconditioning, 115 iterations inner iterations residual norms \ r^{(i)}\ outer iterations sum of inner iterations Figure: Inner iterations vs outer iterations Figure: Eigenvalue residual norms vs total number of inner iterations

25 Outline 1 Introduction 2 Inexact inverse iteration 3 Inexact Shift-invert Arnoldi method 4 Inexact Shift-invert Arnoldi method with implicit restarts 5 Conclusions

26 The algorithm Arnoldi s method Arnoldi method constructs an orthogonal basis of k-dimensional Krylov subspace K k (A,q (1) ) = span{q (1),Aq (1),A 2 q (1),...,A k 1 q (1) }, [ ] AQ k = Q k H k +q k+1 h k+1,k e H H k k = Q k+1 h k+1,k e H k Q H k Q k = I.

27 The algorithm Arnoldi s method Arnoldi method constructs an orthogonal basis of k-dimensional Krylov subspace K k (A,q (1) ) = span{q (1),Aq (1),A 2 q (1),...,A k 1 q (1) }, [ ] AQ k = Q k H k +q k+1 h k+1,k e H H k k = Q k+1 h k+1,k e H k Q H k Q k = I. Eigenvalues of H k are eigenvalue approximations of (outlying) eigenvalues of A r k = Ax θx = (AQ k Q k H k )u = h k+1,k e H k u,

28 The algorithm Arnoldi s method Arnoldi method constructs an orthogonal basis of k-dimensional Krylov subspace K k (A,q (1) ) = span{q (1),Aq (1),A 2 q (1),...,A k 1 q (1) }, [ ] AQ k = Q k H k +q k+1 h k+1,k e H H k k = Q k+1 h k+1,k e H k Q H k Q k = I. Eigenvalues of H k are eigenvalue approximations of (outlying) eigenvalues of A r k = Ax θx = (AQ k Q k H k )u = h k+1,k e H k u, at each step, application of A to q k : Aq k = q k+1

29 Example random complex matrix of dimension n = 144 generated in Matlab: G=numgrid( N,14);B=delsq(G);A=sprandn(B)+i*sprandn(B) eigenvalues of A

30 after 5 Arnoldi steps Arnoldi after 5 steps

31 after 10 Arnoldi steps Arnoldi after 10 steps

32 after 15 Arnoldi steps Arnoldi after 15 steps

33 after 20 Arnoldi steps Arnoldi after 20 steps

34 after 25 Arnoldi steps Arnoldi after 25 steps

35 after 30 Arnoldi steps Arnoldi after 30 steps

36 The algorithm: take σ = 0 Shift-Invert Arnoldi s method A := A 1 Arnoldi method constructs an orthogonal basis of k-dimensional Krylov subspace K k (A 1,q (1) ) = span{q (1),A 1 q (1),(A 1 ) 2 q (1),...,(A 1 ) k 1 q (1) }, [ ] A 1 Q k = Q k H k +q k+1 h k+1,k e H H k k = Q k+1 h k+1,k e H k Q H k Q k = I. Eigenvalues of H k are eigenvalue approximations of (outlying) eigenvalues of A 1 r k = A 1 x θx = (A 1 Q k Q k H k )u = h k+1,k e H k u, at each step, application of A 1 to q k : A 1 q k = q k+1

37 Inexact solves Inexact solves (Simoncini 2005), Bouras and Frayssé (2000) Wish to solve q k A q k+1 = d k τ k

38 Inexact solves Inexact solves (Simoncini 2005), Bouras and Frayssé (2000) Wish to solve q k A q k+1 = d k τ k leads to inexact Arnoldi relation [ [ A 1 H k H k Q k = Q k+1 ]+D h k+1,k e H k = Q k+1 k h k+1,k e H k ] +[d 1... d k ]

39 Inexact solves Inexact solves (Simoncini 2005), Bouras and Frayssé (2000) Wish to solve q k A q k+1 = d k τ k leads to inexact Arnoldi relation [ [ A 1 H k H k Q k = Q k+1 ]+D h k+1,k e H k = Q k+1 k h k+1,k e H k ] +[d 1... d k ] u eigenvector of H k : r k = (A 1 Q k Q k H k )u = h k+1,k e H k u +D k u,

40 Inexact solves Inexact solves (Simoncini 2005), Bouras and Frayssé (2000) Wish to solve q k A q k+1 = d k τ k leads to inexact Arnoldi relation [ [ A 1 H k H k Q k = Q k+1 ]+D h k+1,k e H k = Q k+1 k h k+1,k e H k ] +[d 1... d k ] u eigenvector of H k : r k = (A 1 Q k Q k H k )u = h k+1,k e H k u +D k u, Linear combination of the columns of D k k D k u = d l u l, if u l l=1 small, then d l allowed to be large!

41 Inexact solves Inexact solves (Simoncini 2005), Bouras and Frayssé (2000) Linear combination of the columns of D k k D k u = d l u l, if u l l=1 small, then d l allowed to be large! and leads to Solve tolerance can be relaxed. d l u l 1 k ε D ku < ε u l C(l,k) r l 1 q k A q k+1 = d k d k = C 1 r k 1

42 The inner iteration for AP 1 q k+1 = q k Preconditioning GMRES convergence bound depending on q k AP 1 q l k+1 = κ min p Π l max i=1,...,n p(µi) q k

43 The inner iteration for AP 1 q k+1 = q k Preconditioning GMRES convergence bound depending on q k AP 1 q l k+1 = κ min p Π l max i=1,...,n p(µi) q k the eigenvalue clustering of AP 1 the condition number the right hand side (initial guess)

44 The inner iteration for AP 1 q k+1 = q k Preconditioning Introduce preconditioner P and solve AP 1 q k+1 = q k, P 1 q k+1 = q k+1 using GMRES

45 The inner iteration for AP 1 q k+1 = q k Preconditioning Introduce preconditioner P and solve using GMRES AP 1 q k+1 = q k, P 1 q k+1 = q k+1 Tuned Preconditioner using a tuned preconditioner for Arnoldi s method P k Q k = AQ k ; given by P k = P +(A P)Q k Q H k

46 The inner iteration for A q = q Theorem (Properties of the tuned preconditioner) Let P with P = A+E be a preconditioner for A and assume k steps of Arnoldi s method have been carried out; then k eigenvalues of AP 1 k are equal to one: [AP 1 k ]AQ k = AQ k and n k eigenvalues are close to the corresponding eigenvalues of AP 1.

47 The inner iteration for A q = q Theorem (Properties of the tuned preconditioner) Let P with P = A+E be a preconditioner for A and assume k steps of Arnoldi s method have been carried out; then k eigenvalues of AP 1 k are equal to one: [AP 1 k ]AQ k = AQ k and n k eigenvalues are close to the corresponding eigenvalues of AP 1. Implementation Sherman-Morrison-Woodbury. Only minor extra costs (one back substitution per outer iteration)

48 Numerical Example sherman5.mtx nonsymmetric matrix from the Matrix Market library ( ). smallest eigenvalue: λ , Preconditioned GMRES as inner solver (both fixed tolerance and relaxation strategy), standard and tuned preconditioner (incomplete LU).

49 No tuning and standard preconditioner Arnoldi tolerance 10e 14/m inner itertaions residual norms r (i) Arnoldi tolerance 10e 14/m outer iterations sum of inner iterations Figure: Inner iterations vs outer iterations Figure: Eigenvalue residual norms vs total number of inner iterations

50 Tuning the preconditioner Arnoldi tolerance 10e 14/m Arnoldi tolerance 10e 14 tuned inner itertaions residual norms r (i) Arnoldi tolerance 10e 14/m Arnoldi tolerance 10e 14 tuned outer iterations sum of inner iterations Figure: Inner iterations vs outer iterations Figure: Eigenvalue residual norms vs total number of inner iterations

51 Relaxation Arnoldi tolerance 10e 14/m Arnoldi relaxed tolerance Arnoldi tolerance 10e 14 tuned inner itertaions residual norms r (i) Arnoldi tolerance 10e 14/m Arnoldi relaxed tolerance Arnoldi tolerance 10e 14 tuned outer iterations sum of inner iterations Figure: Inner iterations vs outer iterations Figure: Eigenvalue residual norms vs total number of inner iterations

52 Tuning and relaxation strategy Arnoldi tolerance 10e 14/m Arnoldi relaxed tolerance Arnoldi tolerance 10e 14/m tuned Arnoldi relaxed tolerance tuned inner itertaions Arnoldi tolerance 10e 14/m Arnoldi relaxed tolerance Arnoldi tolerance 10e 14/m tuned Arnoldi relaxed tolerance tuned residual norms r (i) outer iterations sum of inner iterations Figure: Inner iterations vs outer iterations Figure: Eigenvalue residual norms vs total number of inner iterations

53 Ritz values of exact and inexact Arnoldi Exact eigenvalues Ritz values (exact Arnoldi) Ritz values (inexact Arnoldi, tuning) e e e e e e e e e e e e e e e-01 Table: Ritz values of exact Arnoldi s method and inexact Arnoldi s method with the tuning strategy compared to exact eigenvalues closest to zero after 14 shift-invert Arnoldi steps.

54 Outline 1 Introduction 2 Inexact inverse iteration 3 Inexact Shift-invert Arnoldi method 4 Inexact Shift-invert Arnoldi method with implicit restarts 5 Conclusions

55 Implicitly restarted Arnoldi (Sorensen (1992)) Exact shifts take an k +p step Arnoldi factorisation AQ k+p = Q k+p H k+p +q k+p+1 h k+p+1,k+p e H k+p Compute Λ(H k+p ) and select p shifts for an implicit QR iteration implicit restart with new starting vector ˆq (1) = p(a)q(1) p(a)q (1)

56 Implicitly restarted Arnoldi (Sorensen (1992)) Exact shifts take an k +p step Arnoldi factorisation AQ k+p = Q k+p H k+p +q k+p+1 h k+p+1,k+p e H k+p Compute Λ(H k+p ) and select p shifts for an implicit QR iteration implicit restart with new starting vector ˆq (1) = p(a)q(1) p(a)q (1) Aim of IRA AQ k = Q k H k +q k+1 h k+1,k }{{} 0 e H k

57 Relaxation strategy for IRA Theorem For any given ε R with ε > 0 assume that ε C if l > k, d l R k ε otherwise. Then AQ mu Q muθ R m ε. Very technical Relaxation strategy also works for IRA!

58 Tuning Tuning for implicitly restarted Arnoldi s method Introduce preconditioner P and solve AP 1 k q k+1 = q k, P 1 k q k+1 = q k+1 using GMRES and a tuned preconditioner P k Q k = AQ k ; given by P k = P +(A P)Q k Q H k

59 Tuning Why does tuning help? Assume we have found an approximate invariant subspace, that is A 1 Q k = Q k H k +q k+1 h k+1,k e H k }{{} 0

60 Tuning Why does tuning help? Assume we have found an approximate invariant subspace, that is A 1 Q k = Q k H k +q k+1 h k+1,k e H k }{{} 0 let A 1 have the upper Hessenberg form [ Qk Q ] H k A 1 [ Q k Q ] [ k = H k T 12 h k+1,k e 1e H k T 22 where [ Q k Q ] k is unitary and H k C k,k and T 22 C n k,n k are upper Hessenberg. ],

61 Tuning Why does tuning help? Assume we have found an approximate invariant subspace, that is A 1 Q k = Q k H k +q k+1 h k+1,k e H k }{{} 0 let A 1 have the upper Hessenberg form [ Qk Q ] H k A 1 [ Q k Q ] [ k = H k T 12 h k+1,k e 1e H k T 22 where [ Q k Q ] k is unitary and H k C k,k and T 22 C n k,n k are upper Hessenberg. ], If h k+1,k 0 then [ Qk Q ] H [ k AP 1 k Qk Q ] k = [ I + ] Q H k AP 1 k Q k T 1 22 (Q H k PQ k ) 1 +

62 Tuning Why does tuning help? Assume we have found an approximate invariant subspace, that is A 1 Q k = Q k H k +q k+1 h k+1,k e H k }{{} 0 let A 1 have the upper Hessenberg form [ Qk Q ] H k A 1 [ Q k Q ] [ k = H k T 12 h k+1,k e 1e H k T 22 where [ Q k Q ] k is unitary and H k C k,k and T 22 C n k,n k are upper Hessenberg. ], If h k+1,k = 0 then [ Qk Q ] H [ k AP 1 k Qk Q ] k = [ ] I Q H k AP 1 k Q k 0 T 1 22 (Q H k PQ k ) 1

63 Tuning Another advantage of tuning System to be solved at each step of Arnoldi s method is AP 1 k q k+1 = q k, P 1 k q k+1 = q k

64 Tuning Another advantage of tuning System to be solved at each step of Arnoldi s method is AP 1 k q k+1 = q k, P 1 k q k+1 = q k Assuming invariant subspace found then (A 1 Q k = Q k H k ): AP 1 k q k = q k

65 Tuning Another advantage of tuning System to be solved at each step of Arnoldi s method is AP 1 k q k+1 = q k, P 1 k q k+1 = q k Assuming invariant subspace found then (A 1 Q k = Q k H k ): AP 1 k q k = q k the right hand side of the system matrix is an eigenvector of the system!

66 Tuning Another advantage of tuning System to be solved at each step of Arnoldi s method is AP 1 k q k+1 = q k, P 1 k q k+1 = q k Assuming invariant subspace found then (A 1 Q k = Q k H k ): AP 1 k q k = q k the right hand side of the system matrix is an eigenvector of the system! Krylov methods converge in one iteration

67 Tuning Another advantage of tuning In practice: and A 1 Q k = Q k H k +q k+1 h k+1,k e H k AP 1 k q k q k = O( h k+1,k )

68 Tuning Another advantage of tuning In practice: and A 1 Q k = Q k H k +q k+1 h k+1,k e H k AP 1 k q k q k = O( h k+1,k ) number of iterations decreases as the outer iteration proceeds

69 Tuning Another advantage of tuning In practice: and A 1 Q k = Q k H k +q k+1 h k+1,k e H k AP 1 k q k q k = O( h k+1,k ) number of iterations decreases as the outer iteration proceeds Rigorous analysis quite technical.

70 Numerical Example sherman5.mtx nonsymmetric matrix from the Matrix Market library ( ). k = 8 eigenvalues closest to zero IRA with exact shifts p = 4 Preconditioned GMRES as inner solver (fixed tolerance and relaxation strategy), standard and tuned preconditioner (incomplete LU).

71 No tuning and standard preconditioner Arnoldi tolerance 10e 12 inner iterations residual norms r (i) Arnoldi tolerance 10e outer iterations sum of inner iterations Figure: Inner iterations vs outer iterations Figure: Eigenvalue residual norms vs total number of inner iterations

72 Tuning Arnoldi tolerance 10e 12 Arnoldi 10e 12 tuned inner iterations residual norms r (i) Arnoldi tolerance 10e 12 Arnoldi 10e 12 tuned outer iterations sum of inner iterations Figure: Inner iterations vs outer iterations Figure: Eigenvalue residual norms vs total number of inner iterations

73 Relaxation Arnoldi tolerance 10e 12 Arnold relaxed tolerance Arnoldi 10e 12 tuned inner iterations residual norms r (i) Arnoldi tolerance 10e 12 Arnold relaxed tolerance Arnoldi 10e 12 tuned outer iterations sum of inner iterations Figure: Inner iterations vs outer iterations Figure: Eigenvalue residual norms vs total number of inner iterations

74 Tuning and relaxation strategy inner iterations Arnoldi tolerance 10e Arnold relaxed tolerance Arnoldi 10e 12 tuned Arnold relaxed tolerance tuned outer iterations residual norms r (i) sum of inner iterations Arnoldi tolerance 10e 12 Arnold relaxed tolerance Arnoldi 10e 12 tuned Arnold relaxed tolerance tuned Figure: Inner iterations vs outer iterations Figure: Eigenvalue residual norms vs total number of inner iterations

75 Numerical Example qc2534.mtx matrix from the Matrix Market library. k = 6 eigenvalues closest to zero IRA with exact shifts p = 4 Preconditioned GMRES as inner solver (fixed tolerance and relaxation strategy), standard and tuned preconditioner (incomplete LU).

76 Tuning and relaxation strategy inner iterations Arnoldi tolerance 10e 12 residual norms r (i) Arnoldi tolerance 10e 12 Arnold relaxed tolerance Arnoldi 10e 12 tuned Arnold relaxed tolerance tuned 20 Arnold relaxed tolerance 10 Arnoldi 10e 12 tuned Arnold relaxed tolerance tuned outer iterations sum of inner iterations Figure: Inner iterations vs outer iterations Figure: Eigenvalue residual norms vs total number of inner iterations

77 Outline 1 Introduction 2 Inexact inverse iteration 3 Inexact Shift-invert Arnoldi method 4 Inexact Shift-invert Arnoldi method with implicit restarts 5 Conclusions

78 Conclusions For eigenvalue computations it is advantageous to consider small rank changes to the standard preconditioners Works for any preconditioner Works for SI versions of Power method, Simultaneous iteration, Arnoldi method Inexact inverse iteration with a special tuned preconditioner is equivalent to the Jacobi-Davidson method (without subspace expansion) For Arnoldi method best results are obtained when relaxation and tuning are combined

79 M. A. Freitag and A. Spence, Rayleigh quotient iteration and simplified Jacobi-Davidson method with preconditioned iterative solves., Convergence rates for inexact inverse iteration with application to preconditioned iterative solves, BIT, 47 (2007), pp , A tuned preconditioner for inexact inverse iteration applied to Hermitian eigenvalue problems, IMA journal of numerical analysis, 28 (2008), p. 522., Shift-invert Arnoldi s method with preconditioned iterative solves, SIAM Journal on Matrix Analysis and Applications, 31 (2009), pp F. Xue and H. C. Elman, Convergence analysis of iterative solvers in inexact rayleigh quotient iteration, SIAM J. Matrix Anal. Appl., 31 (2009), pp , Fast inexact subspace iteration for generalized eigenvalue problems with spectral transformation, Linear Algebra and its Applications, (2010).