NAG Library Routine Document
F08QGF (DTRSEN)
1 Purpose
F08QGF (DTRSEN) reorders the Schur factorization of a real general matrix so that a selected cluster of eigenvalues appears in the leading elements or blocks 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 F08QGF ( |
JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI, M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO) |
| INTEGER |
N, LDT, LDQ, M, LWORK, IWORK(max(1,LIWORK)), LIWORK, INFO |
| REAL (KIND=nag_wp) |
T(LDT,*), Q(LDQ,*), WR(*), WI(*), S, SEP, WORK(max(1,LWORK)) |
| LOGICAL |
SELECT(*) |
| CHARACTER(1) |
JOB, COMPQ |
|
The routine may be called by its
LAPACK
name dtrsen.
3 Description
F08QGF (DTRSEN) reorders the Schur factorization of a real general matrix A=QTQT, so that a selected cluster of eigenvalues appears in the leading diagonal elements or blocks of the Schur form.
The reordered Schur form T~ is computed by an orthogonal similarity transformation: T~=ZTTZ. Optionally the updated matrix Q~ of Schur vectors is computed as Q~=QZ, giving A=Q~T~Q~T.
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: the eigenvalues in the selected cluster. To select a real eigenvalue
λj,
SELECTj must be set .TRUE.. To select a complex conjugate pair of eigenvalues
λj and
λj+1 (corresponding to a
2 by
2 diagonal block),
SELECTj and/or
SELECTj+1 must be set to .TRUE.. A complex conjugate pair of eigenvalues
must be either both included in the cluster or both excluded. See also
Section 8.
- 4: N – INTEGERInput
-
On entry:
n, the order of the matrix T.
Constraint:
N≥0.
- 5: T(LDT,*) – REAL (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 quasi-triangular matrix
T in canonical Schur form, as returned by
F08PEF (DHSEQR). See also
Section 8.
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 F08QGF (DTRSEN) is called.
Constraint:
LDT≥max1,N.
- 7: Q(LDQ,*) – REAL (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 orthogonal matrix
Q of Schur vectors, as returned by
F08PEF (DHSEQR).
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 F08QGF (DTRSEN) is called.
Constraints:
- if COMPQ='V', LDQ≥max1,N;
- if COMPQ='N', LDQ≥1.
- 9: WR(*) – REAL (KIND=nag_wp) arrayOutput
- 10: WI(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the arrays
WR and
WI
must be at least
max1,N.
On exit: the real and imaginary parts, respectively, of the reordered eigenvalues of
T~. The eigenvalues are stored in the same order as on the diagonal of
T~; see
Section 8 for details. Note that if a complex eigenvalue is sufficiently ill-conditioned, then its value may differ significantly from its value before reordering.
- 11: M – INTEGEROutput
-
On exit:
m, the dimension of the specified invariant subspace. The value of
m is obtained by counting
1 for each selected real eigenvalue and
2 for each selected complex conjugate pair of eigenvalues (see
SELECT);
0≤m≤n.
- 12: 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
INFO=1 (see
Section 6),
S is set to zero.
If
JOB='N' or
'V',
S is not referenced.
- 13: 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
INFO=1 (see
Section 6),
SEP is set to zero.
If
JOB='N' or
'E',
SEP is not referenced.
- 14: WORK(max1,LWORK) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0,
WORK1 contains the minimum value of
LWORK required for optimal performance.
- 15: LWORK – INTEGERInput
-
On entry: the dimension of the array
WORK as declared in the (sub)program from which F08QGF (DTRSEN) 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≥max1,N 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'.
- 16: IWORK(max1,LIWORK) – INTEGER arrayWorkspace
On exit: if
INFO=0,
IWORK1 contains the required minimal size of
LIWORK.
- 17: LIWORK – INTEGERInput
-
On entry: the dimension of the array
IWORK as declared in the (sub)program from which F08QGF (DTRSEN) is called, unless
LIWORK=-1, in which case a workspace query is assumed and the routine only calculates the minimum dimension of
IWORK.
Constraints:
- if JOB='N' or 'E', LIWORK≥1 or LIWORK=-1;
- if JOB='V' or 'B', LIWORK≥max1,m×N-m or LIWORK=-1.
The actual amount of workspace required cannot exceed N2/2 if JOB='V' or 'B'.
- 18: 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
The reordering of
T failed because a selected eigenvalue was too close to an eigenvalue which was not selected; this error exit can only occur if at least one of the eigenvalues involved was complex. The problem is too ill-conditioned: consider modifying the selection of eigenvalues so that eigenvalues which are very close together are either all included in the cluster or all excluded. On exit,
T may have been partially reordered, but
WR,
WI and
Q (if requested) are updated consistently with
T;
S and
SEP (if requested) are both set to zero.
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
.
Note that if a 2 by 2 diagonal block is involved in the reordering, its off-diagonal elements are in general changed; the diagonal elements and the eigenvalues of the block are unchanged unless the block is sufficiently ill-conditioned, in which case they may be noticeably altered. It is possible for a 2 by 2 block to break into two 1 by 1 blocks, i.e., for a pair of complex eigenvalues to become purely real. The values of real eigenvalues however are never changed by the reordering.
8 Further Comments
The input matrix T must be in canonical Schur form, as is the output matrix T~. This has the following structure.
If all the computed eigenvalues are real, T~ is upper triangular, and the diagonal elements of T~ are the eigenvalues; WRi=t~ii, for i=1,2,…,n and WIi=0.0.
If some of the computed eigenvalues form complex conjugate pairs, then
T~ has
2 by
2 diagonal blocks. Each diagonal block has the form
where
βγ<0. The corresponding eigenvalues are
α±βγ;
WRi=WRi+1=α;
WIi=+βγ;
WIi+1=-WIi.
The complex analogue of this routine is
F08QUF (ZTRSEN).
9 Example
This example reorders the Schur factorization of the matrix
A=QTQT such that the two real eigenvalues appear as the leading elements on the diagonal of the reordered matrix
T~, where
and
The example program for F08QGF (DTRSEN) illustrates the computation of error bounds for the eigenvalues.
The original matrix
A is given in
Section 9 in F08NFF (DORGHR).
9.1 Program Text
Program Text (f08qgfe.f90)
9.2 Program Data
Program Data (f08qgfe.d)
9.3 Program Results
Program Results (f08qgfe.r)