NAG Library Routine Document
F08QUF (ZTRSEN)
1 Purpose
F08QUF (ZTRSEN) reorders the Schur factorization of a complex general matrix so that a selected cluster of eigenvalues appears in the leading elements on the diagonal of the Schur form. The routine also optionally computes the reciprocal condition numbers of the cluster of eigenvalues and/or the invariant subspace.
2 Specification
| SUBROUTINE F08QUF ( |
JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W, M, S, SEP, WORK, LWORK, INFO) |
| INTEGER |
N, LDT, LDQ, M, LWORK, INFO |
| REAL (KIND=nag_wp) |
S, SEP |
| COMPLEX (KIND=nag_wp) |
T(LDT,*), Q(LDQ,*), W(*), WORK(max(1,LWORK)) |
| LOGICAL |
SELECT(*) |
| CHARACTER(1) |
JOB, COMPQ |
|
The routine may be called by its
LAPACK
name ztrsen.
3 Description
F08QUF (ZTRSEN) reorders the Schur factorization of a complex general matrix A=QTQH, so that a selected cluster of eigenvalues appears in the leading diagonal elements of the Schur form.
The reordered Schur form T~ is computed by a unitary similarity transformation: T~=ZHTZ. Optionally the updated matrix Q~ of Schur vectors is computed as Q~=QZ, giving A=Q~T~Q~H.
Let T~=
T11
T12
0
T22
, where the selected eigenvalues are precisely the eigenvalues of the leading m by m sub-matrix T11. Let Q~ be correspondingly partitioned as
Q1
Q2
where Q1 consists of the first m columns of Q. Then AQ1=Q1T11, and so the m columns of Q1 form an orthonormal basis for the invariant subspace corresponding to the selected cluster of eigenvalues.
Optionally the routine also computes estimates of the reciprocal condition numbers of the average of the cluster of eigenvalues and of the invariant subspace.
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 condition numbers are required for the cluster of eigenvalues and/or the invariant subspace.
- JOB='N'
- No condition numbers are required.
- JOB='E'
- Only the condition number for the cluster of eigenvalues is computed.
- JOB='V'
- Only the condition number for the invariant subspace is computed.
- JOB='B'
- Condition numbers for both the cluster of eigenvalues and the invariant subspace are computed.
Constraint:
JOB='N', 'E', 'V' or 'B'.
- 2: COMPQ – CHARACTER(1)Input
-
On entry: indicates whether the matrix
Q of Schur vectors is to be updated.
- COMPQ='V'
- The matrix Q of Schur vectors is updated.
- COMPQ='N'
- No Schur vectors are updated.
Constraint:
COMPQ='V' or 'N'.
- 3: SELECT(*) – LOGICAL arrayInput
-
Note: the dimension of the array
SELECT
must be at least
max1,N.
On entry: specifies the eigenvalues in the selected cluster. To select a complex eigenvalue λj, SELECTj must be set .TRUE..
- 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:
T is overwritten by the updated matrix
T~.
- 6: LDT – INTEGERInput
-
On entry: the first dimension of the array
T as declared in the (sub)program from which F08QUF (ZTRSEN) is called.
Constraint:
LDT≥max1,N.
- 7: Q(LDQ,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
Q
must be at least
max1,N if
COMPQ='V' and at least
1 if
COMPQ='N'.
On entry: if
COMPQ='V',
Q must contain the
n by
n unitary matrix
Q of Schur vectors, as returned by
F08PSF (ZHSEQR).
On exit: if
COMPQ='V',
Q contains the updated matrix of Schur vectors; the first
m columns of
Q form an orthonormal basis for the specified invariant subspace.
If
COMPQ='N',
Q is not referenced.
- 8: LDQ – INTEGERInput
-
On entry: the first dimension of the array
Q as declared in the (sub)program from which F08QUF (ZTRSEN) is called.
Constraints:
- if COMPQ='V', LDQ≥max1,N;
- if COMPQ='N', LDQ≥1.
- 9: W(*) – COMPLEX (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
W
must be at least
max1,N.
On exit: the reordered eigenvalues of T~. The eigenvalues are stored in the same order as on the diagonal of T~.
- 10: M – INTEGEROutput
-
On exit:
m, the dimension of the specified invariant subspace, which is the same as the number of selected eigenvalues (see
SELECT);
0≤m≤n.
- 11: S – REAL (KIND=nag_wp)Output
-
On exit: if
JOB='E' or
'B',
S is a lower bound on the reciprocal condition number of the average of the selected cluster of eigenvalues. If
M=0 or N,
S=1.
If
JOB='N' or
'V',
S is not referenced.
- 12: SEP – REAL (KIND=nag_wp)Output
-
On exit: if
JOB='V' or
'B',
SEP is the estimated reciprocal condition number of the specified invariant subspace. If
M=0 or N,
SEP=T.
If
JOB='N' or
'E',
SEP is not referenced.
- 13: WORK(max1,LWORK) – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0, the real part of
WORK1 contains the minimum value of
LWORK required for optimal performance.
- 14: LWORK – INTEGERInput
-
On entry: the dimension of the array
WORK as declared in the (sub)program from which F08QUF (ZTRSEN) is called, unless
LWORK=-1, in which case a workspace query is assumed and the routine only calculates the minimum dimension of
WORK.
Constraints:
- if JOB='N', LWORK≥1 or LWORK=-1;
- if JOB='E', LWORK≥max1,m×N-m or LWORK=-1;
- if JOB='V' or 'B', LWORK≥max1,2m×N-m or LWORK=-1.
The actual amount of workspace required cannot exceed N2/4 if JOB='E' or N2/2 if JOB='V' or 'B'.
- 15: 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.
7 Accuracy
The computed matrix
T~ is similar to a matrix
T+E, where
and
ε is the
machine precision.
S cannot underestimate the true reciprocal condition number by more than a factor of
minm,n-m.
SEP may differ from the true value by
mn-m. The angle between the computed invariant subspace and the true subspace is
OεA2sep
.
The values of the eigenvalues are never changed by the reordering.
8 Further Comments
The real analogue of this routine is
F08QGF (DTRSEN).
9 Example
This example reorders the Schur factorization of the matrix
A=QTQH such that the eigenvalues stored in elements
t11 and
t44 appear as the leading elements on the diagonal of the reordered matrix
T~, where
and
The original matrix
A is given in
Section 9 in F08NTF (ZUNGHR).
9.1 Program Text
Program Text (f08qufe.f90)
9.2 Program Data
Program Data (f08qufe.d)
9.3 Program Results
Program Results (f08qufe.r)