NAG Library Routine Document

F08QXF (ZTREVC)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

1 Purpose

F08QXF (ZTREVC) computes selected left and/or right eigenvectors of a complex upper triangular matrix.

2 Specification

SUBROUTINE F08QXF (

JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, MM, M, WORK, RWORK, INFO)

INTEGER	N, LDT, LDVL, LDVR, MM, M, INFO
REAL (KIND=nag_wp)	RWORK(N)
COMPLEX (KIND=nag_wp)	T(LDT,), VL(LDVL,), VR(LDVR,), WORK(2N)
LOGICAL	SELECT(*)
CHARACTER(1)	JOB, HOWMNY

The routine may be called by its LAPACK name ztrevc.

3 Description

F08QXF (ZTREVC) computes left and/or right eigenvectors of a complex upper triangular matrix T. Such a matrix arises from the Schur factorization of a complex general matrix, as computed by F08PSF (ZHSEQR), for example.

The right eigenvector x, and the left eigenvector y, corresponding to an eigenvalue λ, are defined by:

Tx = λx and yHT = λyH or THy = λ-y .

The routine can compute the eigenvectors corresponding to selected eigenvalues, or it can compute all the eigenvectors. In the latter case the eigenvectors may optionally be pre-multiplied by an input matrix Q. Normally Q is a unitary matrix from the Schur factorization of a matrix A as A=QTQH; if x is a (left or right) eigenvector of T, then Qx is an eigenvector of A.

The eigenvectors are computed by forward or backward substitution. They are scaled so that max⁡ Rexi + Im⁡xi =1.

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Parameters

1: JOB – CHARACTER(1)Input

On entry: indicates whether left and/or right eigenvectors are to be computed.

JOB='R': Only right eigenvectors are computed.
JOB='L': Only left eigenvectors are computed.
JOB='B': Both left and right eigenvectors are computed.

Constraint: JOB='R', 'L' or 'B'.

2: HOWMNY – CHARACTER(1)Input

On entry: indicates how many eigenvectors are to be computed.

HOWMNY='A'

Constraint: HOWMNY='A', 'B', 'O' 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 which eigenvectors are to be computed if HOWMNY='S'. To obtain the eigenvector corresponding to the eigenvalue λj, SELECTj must be set .TRUE..

If HOWMNY='A', 'O' or 'B', SELECT is not referenced.

4: N – INTEGERInput

On entry: n, the order of the matrix T.

Constraint: N≥0.

5: T(LDT,*) – COMPLEX (KIND=nag_wp) arrayInput/Output

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

On entry: the n by n upper triangular matrix T, as returned by F08PSF (ZHSEQR).

On exit: is used as internal workspace prior to being restored and hence is unchanged.

6: LDT – INTEGERInput

On entry: the first dimension of the array T as declared in the (sub)program from which F08QXF (ZTREVC) is called.

Constraint: LDT≥max1,N.

7: VL(LDVL,*) – COMPLEX (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array VL must be at least max1,MM if JOB='L' or 'B' and at least 1 if JOB='R'.

On entry: if HOWMNY='O' or 'B' and JOB='L' or 'B', VL must contain an n by n matrix Q (usually the matrix of Schur vectors returned by F08PSF (ZHSEQR)).

If HOWMNY='A' or 'S', VL need not be set.

On exit: if JOB='L' or 'B', VL contains the computed left eigenvectors (as specified by HOWMNY and SELECT). The eigenvectors are stored consecutively in the columns of the array, in the same order as their eigenvalues.

If JOB='R', VL is not referenced.

8: LDVL – INTEGERInput

On entry: the first dimension of the array VL as declared in the (sub)program from which F08QXF (ZTREVC) is called.

Constraints:

if JOB='L' or 'B', LDVL≥max1,N;
if JOB='R', LDVL≥1.

9: VR(LDVR,*) – COMPLEX (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array VR must be at least max1,MM if JOB='R' or 'B' and at least 1 if JOB='L'.

On entry: if HOWMNY='O' or 'B' and JOB='R' or 'B', VR must contain an n by n matrix Q (usually the matrix of Schur vectors returned by F08PSF (ZHSEQR)).

If HOWMNY='A' or 'S', VR need not be set.

On exit: if JOB='R' or 'B', VR contains the computed right eigenvectors (as specified by HOWMNY and SELECT). The eigenvectors are stored consecutively in the columns of the array, in the same order as their eigenvalues.

If JOB='L', VR is not referenced.

10: LDVR – INTEGERInput

On entry: the first dimension of the array VR as declared in the (sub)program from which F08QXF (ZTREVC) is called.

Constraints:

if JOB='R' or 'B', LDVR≥max1,N;
if JOB='L', LDVR≥1.

11: MM – INTEGERInput

On entry: the number of columns in the arrays VL and/or VR. The precise number of columns required, m, is n if HOWMNY='A', 'O' or 'B'; if HOWMNY='S', m is the number of selected eigenvectors (see SELECT), in which case 0≤m≤n.

Constraint: MM≥m.

12: M – INTEGEROutput