NAG Library Routine Document

F08YGF (DTGSEN)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F08YGF (DTGSEN) reorders the generalized Schur factorization of a matrix pair in real generalized Schur form, so that a selected cluster of eigenvalues appears in the leading elements, or blocks on the diagonal of the generalized Schur form. The routine also, optionally, computes the reciprocal condition numbers of the cluster of eigenvalues and/or corresponding deflating subspaces.

2 Specification

SUBROUTINE F08YGF (

IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO)

INTEGER	IJOB, N, LDA, LDB, LDQ, LDZ, M, LWORK, IWORK(*), LIWORK, INFO
REAL (KIND=nag_wp)	A(LDA,), B(LDB,), ALPHAR(N), ALPHAI(N), BETA(N), Q(LDQ,), Z(LDZ,), PL, PR, DIF(*), WORK(max(1,LWORK))
LOGICAL	WANTQ, WANTZ, SELECT(N)

The routine may be called by its LAPACK name dtgsen.

3 Description

F08YGF (DTGSEN) factorizes the generalized real n by n matrix pair S,T in real generalized Schur form, using an orthogonal equivalence transformation as

S = Q^ S^ Z^T , T= Q^ T^ Z^T ,

where S^,T^ are also in real generalized Schur form and have the selected eigenvalues as the leading diagonal elements, or diagonal blocks. The leading columns of Q and Z are the generalized Schur vectors corresponding to the selected eigenvalues and form orthonormal subspaces for the left and right eigenspaces (deflating subspaces) of the pair S,T.

The pair S,T are in real generalized Schur form if S is block upper triangular with 1 by 1 and 2 by 2 diagonal blocks and T is upper triangular as returned, for example, by F08XAF (DGGES), or F08XEF (DHGEQZ) with JOB='S'. The diagonal elements, or blocks, define the generalized eigenvalues αi,βi, for i=1,2,…,n, of the pair S,T. The eigenvalues are given by

λi = αi / βi ,

but are returned as the pair αi,βi in order to avoid possible overflow in computing λi. Optionally, the routine returns reciprocals of condition number estimates for the selected eigenvalue cluster, p and q, the right and left projection norms, and of deflating subspaces, Difu and Difl. For more information see Sections 2.4.8 and 4.11 of Anderson et al. (1999).

If S and T are the result of a generalized Schur factorization of a matrix pair A,B

A = QSZT , B= QTZT

then, optionally, the matrices Q and Z can be updated as QQ^ and ZZ^. Note that the condition numbers of the pair S,T are the same as those of the pair A,B.

4 References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

5 Parameters

1: IJOB – INTEGERInput

On entry: specifies whether condition numbers are required for the cluster of eigenvalues (p and q) or the deflating subspaces (Difu and Difl).

IJOB=0

2: WANTQ – LOGICALInput

On entry: if WANTQ=.TRUE., update the left transformation matrix Q.

If WANTQ=.FALSE., do not update Q.

3: WANTZ – LOGICALInput

On entry: if WANTZ=.TRUE., update the right transformation matrix Z.

If WANTZ=.FALSE., do not update Z.

4: SELECT(N) – LOGICAL arrayInput

On entry: specifies the eigenvalues in the selected cluster. To select a real eigenvalue λj, SELECTj must be set to .TRUE..

To select a complex conjugate pair of eigenvalues λj and λj+1, corresponding to a 2 by 2 diagonal block, either SELECTj or SELECTj+1 or both must be set to .TRUE.; a complex conjugate pair of eigenvalues must be either both included in the cluster or both excluded.

5: N – INTEGERInput

On entry: n, the order of the matrices S and T.

Constraint: N≥0.

6: A(LDA,*) – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array A must be at least max1,N.

On entry: the matrix S in the pair S,T.

On exit: the updated matrix S^.

7: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F08YGF (DTGSEN) is called.

Constraint: LDA≥max1,N.

8: B(LDB,*) – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array B must be at least max1,N.

On entry: the matrix T, in the pair S,T.

On exit: the updated matrix T^

9: LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F08YGF (DTGSEN) is called.

Constraint: LDB≥max1,N.

10: ALPHAR(N) – REAL (KIND=nag_wp) arrayOutput

On exit: see the description of BETA.

11: ALPHAI(N) – REAL (KIND=nag_wp) arrayOutput

On exit: see the description of BETA.

12: BETA(N) – REAL (KIND=nag_wp) arrayOutput

On exit: ALPHARj / BETAj and ALPHAIj / BETAj are the real and imaginary parts respectively of the jth eigenvalue, for j=1,2,…,N.

If ALPHAIj is zero, then the jth eigenvalue is real; if positive then ALPHAIj+1 is negative, and the jth and j+1st eigenvalues are a complex conjugate pair.

Conjugate pairs of eigenvalues correspond to the 2 by 2 diagonal blocks of S^. These 2 by 2 blocks can be reduced by applying complex unitary transformations to S^,T^ to obtain the complex Schur form S~,T~ , where S~ is triangular (and complex). In this form ALPHAR+iALPHAI and BETA are the diagonals of S~ and T~ respectively.

13: Q(LDQ,*) – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array Q must be at least max1,N if WANTQ=.TRUE., and at least 1 otherwise.

On entry: if WANTQ=.TRUE., the n by n matrix Q.

On exit: if WANTQ=.TRUE., the updated matrix QQ^.

If WANTQ=.FALSE., Q is not referenced.

14: LDQ – INTEGERInput

On entry: the first dimension of the array Q as declared in the (sub)program from which F08YGF (DTGSEN) is called.

Constraints:

if WANTQ=.TRUE., LDQ≥max1,N;
otherwise LDQ≥1.

15: Z(LDZ,*) – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array Z must be at least max1,N if WANTZ=.TRUE., and at least 1 otherwise.

On entry: if WANTZ=.TRUE., the n by n matrix Z.

On exit: if WANTZ=.TRUE., the updated matrix ZZ^.

If WANTZ=.FALSE., Z is not referenced.

16: LDZ – INTEGERInput

On entry: the first dimension of the array Z as declared in the (sub)program from which F08YGF (DTGSEN) is called.

Constraints:

if WANTZ=.TRUE., LDZ≥max1,N;
otherwise LDZ≥1.

17: M – INTEGEROutput

On exit: the dimension of the specified pair of left and right eigenspaces (deflating subspaces).

18: PL – REAL (KIND=nag_wp)Output

19: PR – REAL (KIND=nag_wp)Output

On exit: if IJOB=1, 4 or 5, PL and PR are lower bounds on the reciprocal of the norm of ‘projections’ p and q onto left and right eigenspaces with respect to the selected cluster. 0<PL, PR≤1.

If M=0 or M=N, PL=PR=1.

If IJOB=0, 2 or 3, PL and PR are not referenced.

20: DIF(*) – REAL (KIND=nag_wp) arrayOutput

Note: the dimension of the array DIF must be at least 2.

On exit: if IJOB≥2, DIF1:2 store the estimates of Difu and Difl.

If IJOB=2 or 4, DIF1:2 are F-norm-based upper bounds on Difu and Difl.

If IJOB=3 or 5, DIF1:2 are 1-norm-based estimates of Difu and Difl.

If M=0 or n, DIF1:2 =A,BF.

If IJOB=0 or 1, DIF is not referenced.

21: WORK(max1,LWORK) – REAL (KIND=nag_wp) arrayWorkspace

On exit: if INFO=0, WORK1 returns the minimum LWORK.

If IJOB=0, WORK is not referenced.

22: LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F08YGF (DTGSEN) is called.

If LWORK=-1, a workspace query is assumed; the routine only calculates the minimum sizes of the WORK and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued.

Constraints:

if LWORK≠-1,

if N=0, LWORK≥1;
if IJOB=1, 2 or 4, LWORK≥max4×N+16,2×M×N-M;
if IJOB=3 or 5, LWORK≥max4×N+16,4×M×N-M;
otherwise LWORK≥4×N+16.

23: IWORK(*) – INTEGER arrayWorkspace

Note: the dimension of the array IWORK must be at least max1,LIWORK.

On exit: if INFO=0, IWORK1 returns the minimum LIWORK.

If IJOB=0, IWORK is not referenced.

24: LIWORK – INTEGERInput

On entry: the dimension of the array IWORK as declared in the (sub)program from which F08YGF (DTGSEN) is called.

If LIWORK=-1, a workspace query is assumed; the routine only calculates the minimum sizes of the WORK and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued.

Constraints:

if LIWORK≠-1,

if IJOB=1, 2 or 4, LIWORK≥N+6;
if IJOB=3 or 5, LIWORK≥ max2×M× N-M ,N+6;
otherwise LIWORK≥1.

25: INFO – INTEGEROutput

On exit: INFO=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

Errors or warnings detected by the routine:

INFO<0: If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.

INFO=1: Reordering of S,T failed because the transformed matrix pair S^,T^ would be too far from generalized Schur form; the problem is very ill-conditioned. S,T may have been partially reordered. If requested, 0 is returned in DIF1:2, PL and PR.

7 Accuracy

The computed generalized Schur form is nearly the exact generalized Schur form for nearby matrices S+E and T+F, where

E2 = O⁡ε S2 and F2= O⁡ε T2 ,

and ε is the machine precision. See Section 4.11 of Anderson et al. (1999) for further details of error bounds for the generalized nonsymmetric eigenproblem, and for information on the condition numbers returned.