F08YKF (DTGEVC) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08YKF (DTGEVC)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

F08YKF (DTGEVC) computes some or all of the right and/or left generalized eigenvectors of a pair of real matrices A,B which are in generalized real Schur form.

2  Specification

SUBROUTINE F08YKF ( SIDE, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, MM, M, WORK, INFO)
INTEGER  N, LDA, LDB, LDVL, LDVR, MM, M, INFO
REAL (KIND=nag_wp)  A(LDA,*), B(LDB,*), VL(LDVL,*), VR(LDVR,*), WORK(6*N)
LOGICAL  SELECT(*)
CHARACTER(1)  SIDE, HOWMNY
The routine may be called by its LAPACK name dtgevc.

3  Description

F08YKF (DTGEVC) computes some or all of the right and/or left generalized eigenvectors of the matrix pair A,B which is assumed to be in generalized upper Schur form. If the matrix pair A,B is not in the generalized upper Schur form, then F08XEF (DHGEQZ) should be called before invoking F08YKF (DTGEVC).
The right generalized eigenvector x and the left generalized eigenvector y of A,B corresponding to a generalized eigenvalue λ are defined by
A-λBx=0
and
yH A-λ B=0.
If a generalized eigenvalue is determined as 0/0, which is due to zero diagonal elements at the same locations in both A and B, a unit vector is returned as the corresponding eigenvector.
Note that the generalized eigenvalues are computed using F08XEF (DHGEQZ) but F08YKF (DTGEVC) does not explicitly require the generalized eigenvalues to compute eigenvectors. The ordering of the eigenvectors is based on the ordering of the eigenvalues as computed by F08YKF (DTGEVC).
If all eigenvectors are requested, the routine may either return the matrices X and/or Y of right or left eigenvectors of A,B, or the products ZX and/or QY, where Z and Q are two matrices supplied by you. Usually, Q and Z are chosen as the orthogonal matrices returned by F08XEF (DHGEQZ). Equivalently, Q and Z are the left and right Schur vectors of the matrix pair supplied to F08XEF (DHGEQZ). In that case, QY and ZX are the left and right generalized eigenvectors, respectively, of the matrix pair supplied to F08XEF (DHGEQZ).
A must be block upper triangular; with 1 by 1 and 2 by 2 diagonal blocks. Corresponding to each 2 by 2 diagonal block is a complex conjugate pair of eigenvalues and eigenvectors; only one eigenvector of the pair is computed, namely the one corresponding to the eigenvalue with positive imaginary part. Each 1 by 1 block gives a real generalized eigenvalue and a corresponding eigenvector.

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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Moler C B and Stewart G W (1973) An algorithm for generalized matrix eigenproblems SIAM J. Numer. Anal. 10 241–256
Stewart G W and Sun J-G (1990) Matrix Perturbation Theory Academic Press, London

5  Parameters

1:     SIDE – CHARACTER(1)Input
On entry: specifies the required sets of generalized eigenvectors.
SIDE='R'
Only right eigenvectors are computed.
SIDE='L'
Only left eigenvectors are computed.
SIDE='B'
Both left and right eigenvectors are computed.
Constraint: SIDE='B', 'L' or 'R'.
2:     HOWMNY – CHARACTER(1)Input
Constraint: HOWMNY='A', 'B' or 'S'.
3:     SELECT(*) – LOGICAL arrayInput
Note: the dimension of the array SELECT must be at least max1,N if HOWMNY='S', and at least 1 otherwise.
On entry: specifies the eigenvectors to be computed if HOWMNY='S'. To select the generalized eigenvector corresponding to the jth generalized eigenvalue, the jth element of SELECT should be set to .TRUE.; if the eigenvalue corresponds to a complex conjugate pair, then real and imaginary parts of eigenvectors corresponding to the complex conjugate eigenvalue pair will be computed.
Constraint: SELECTj=.TRUE. or .FALSE., for j=1,2,,n.
4:     N – INTEGERInput
On entry: n, the order of the matrices A and B.
Constraint: N0.
5:     A(LDA,*) – REAL (KIND=nag_wp) arrayInput
Note: the second dimension of the array A must be at least max1,N.
On entry: the matrix pair A,B must be in the generalized Schur form. Usually, this is the matrix A returned by F08XEF (DHGEQZ).
6:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08YKF (DTGEVC) is called.
Constraint: LDAmax1,N.
7:     B(LDB,*) – REAL (KIND=nag_wp) arrayInput
Note: the second dimension of the array B must be at least max1,N.
On entry: the matrix pair A,B must be in the generalized Schur form. If A has a 2 by 2 diagonal block then the corresponding 2 by 2 block of B must be diagonal with positive elements. Usually, this is the matrix B returned by F08XEF (DHGEQZ).
8:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F08YKF (DTGEVC) is called.
Constraint: LDBmax1,N.
9:     VL(LDVL,*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array VL must be at least max1,MM if SIDE='L' or 'B' and at least 1 if SIDE='R'.
On entry: if HOWMNY='B' and SIDE='L' or 'B', VL must be initialized to an n by n matrix Q. Usually, this is the orthogonal matrix Q of left Schur vectors returned by F08XEF (DHGEQZ).
On exit: if SIDE='L' or 'B', VL contains:
  • if HOWMNY='A', the matrix Y of left eigenvectors of A,B;
  • if HOWMNY='B', the matrix QY;
  • if HOWMNY='S', the left eigenvectors of A,B specified by SELECT, stored consecutively in the columns of the array VL, in the same order as their corresponding eigenvalues.
A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive columns, the first holding the real part, and the second the imaginary part.
10:   LDVL – INTEGERInput
On entry: the first dimension of the array VL as declared in the (sub)program from which F08YKF (DTGEVC) is called.
Constraints:
  • if SIDE='L' or 'B', LDVLmax1,N;
  • if SIDE='R', LDVL1.
11:   VR(LDVR,*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array VR must be at least max1,MM if SIDE='R' or 'B' and at least 1 if SIDE='L'.
On entry: if HOWMNY='B' and SIDE='R' or 'B', VR must be initialized to an n by n matrix Z. Usually, this is the orthogonal matrix Z of right Schur vectors returned by F08XEF (DHGEQZ).
On exit: if SIDE='R' or 'B', VR contains:
  • if HOWMNY='A', the matrix X of right eigenvectors of A,B;
  • if HOWMNY='B', the matrix ZX;
  • if HOWMNY='S', the right eigenvectors of A,B specified by SELECT, stored consecutively in the columns of the array VR, in the same order as their corresponding eigenvalues.
A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive columns, the first holding the real part, and the second the imaginary part.
12:   LDVR – INTEGERInput
On entry: the first dimension of the array VR as declared in the (sub)program from which F08YKF (DTGEVC) is called.
Constraints:
  • if SIDE='R' or 'B', LDVRmax1,N;
  • if SIDE='L', LDVR1.
13:   MM – INTEGERInput
Constraints:
  • if HOWMNY='A' or 'B', MMN;
  • if HOWMNY='S', MM must not be less than the number of requested eigenvectors.
14:   M – INTEGEROutput
On exit: the number of columns in the arrays VL and/or VR actually used to store the eigenvectors. If HOWMNY='A' or 'B', M is set to N. Each selected real eigenvector occupies one column and each selected complex eigenvector occupies two columns.
15:   WORK(6×N) – REAL (KIND=nag_wp) arrayWorkspace
16:   INFO – INTEGEROutput

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>0
If INFO=i, the 2 by 2 block INFO:INFO+1 does not have complex eigenvalues.

7  Accuracy

It is beyond the scope of this manual to summarize the accuracy of the solution of the generalized eigenvalue problem. Interested readers should consult Section 4.11 of the LAPACK Users' Guide (see Anderson et al. (1999)) and Chapter 6 of Stewart and Sun (1990).

8  Further Comments

F08YKF (DTGEVC) is the sixth step in the solution of the real generalized eigenvalue problem and is called after F08XEF (DHGEQZ).
The complex analogue of this routine is F08YXF (ZTGEVC).

9  Example

This example computes the α and β parameters, which defines the generalized eigenvalues and the corresponding left and right eigenvectors, of the matrix pair A,B given by
A = 1.0 1.0 1.0 1.0 1.0 2.0 4.0 8.0 16.0 32.0 3.0 9.0 27.0 81.0 243.0 4.0 16.0 64.0 256.0 1024.0 5.0 25.0 125.0 625.0 3125.0   and   B= 1.0 2.0 3.0 4.0 5.0 1.0 4.0 9.0 16.0 25.0 1.0 8.0 27.0 64.0 125.0 1.0 16.0 81.0 256.0 625.0 1.0 32.0 243.0 1024.0 3125.0 .
To compute generalized eigenvalues, it is required to call five routines:
F08WHF (DGGBAL) to balance the matrix, F08AEF (DGEQRF) to perform the QR factorization of B, F08AGF (DORMQR) to apply Q to A, F08WEF (DGGHRD) to reduce the matrix pair to the generalized Hessenberg form and F08XEF (DHGEQZ) to compute the eigenvalues via the QZ algorithm.
The computation of generalized eigenvectors is done by calling F08YKF (DTGEVC) to compute the eigenvectors of the balanced matrix pair. The routine F08WJF (DGGBAK) is called to backward transform the eigenvectors to the user-supplied matrix pair. If both left and right eigenvectors are required then F08WJF (DGGBAK) must be called twice.

9.1  Program Text

Program Text (f08ykfe.f90)

9.2  Program Data

Program Data (f08ykfe.d)

9.3  Program Results

Program Results (f08ykfe.r)


F08YKF (DTGEVC) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2011