NAG Library Routine Document
F02GCF
1 Purpose
F02GCF computes selected eigenvalues and eigenvectors of a complex general matrix.
2 Specification
| SUBROUTINE F02GCF ( | CRIT, N, A, LDA, WL, WU, MEST, M, W, V, LDV, WORK, LWORK, RWORK, IWORK, BWORK, IFAIL) |
| INTEGER | N, LDA, MEST, M, LDV, LWORK, IWORK(N), IFAIL |
| double precision | WL, WU, RWORK(2*N) |
| complex*16 | A(LDA,*), W(N), V(LDV,MEST), WORK(LWORK) |
| LOGICAL | BWORK(N) |
| CHARACTER*1 | CRIT |
3 Description
F02GCF computes selected eigenvalues and the corresponding right eigenvectors of a complex general matrix
A:
Eigenvalues
λi may be selected either by
modulus, satisfying
or by
real part, satisfying
4 References
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5 Parameters
- 1: CRIT – CHARACTER*1Input
On entry: indicates the criterion for selecting eigenvalues.
- CRIT='M'
- Eigenvalues are selected according to their moduli: wl≤λi≤wu.
- CRIT='R'
- Eigenvalues are selected according to their real parts: wl≤Reλi≤wu.
Constraint:
CRIT='M' or 'R'.
- 2: N – INTEGERInput
On entry:
n, the order of the matrix A.
Constraint:
N≥0.
- 3: A(LDA,*) – complex*16 arrayInput/Output
Note: the second dimension of the array
A
must be at least
max1,N.
On entry: the n by n general matrix A.
On exit: contains the Hessenberg form of the balanced input matrix
A′ (see
Section 8).
- 4: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F02GCF is called.
Constraint:
LDA≥max1,N.
- 5: WL – double precisionInput
- 6: WU – double precisionInput
On entry:
wl and
wu, the lower and upper bounds on the criterion for the selected eigenvalues (see
CRIT).
Constraint:
WU>WL.
- 7: MEST – INTEGERInput
On entry: the second dimension of the array
V as declared in the (sub)program from which F02GCF is called.
MEST must be an upper bound on
m, the number of eigenvalues and eigenvectors selected. No eigenvectors are computed if
MEST<m.
Constraint:
MEST≥max1,m.
- 8: M – INTEGEROutput
On exit: m, the number of eigenvalues actually selected.
- 9: W(N) – complex*16 arrayOutput
On exit: the first
M elements of
W hold the selected eigenvalues; elements
M+1 to
N contain the other eigenvalues.
- 10: V(LDV,MEST) – complex*16 arrayOutput
On exit: contains the selected eigenvectors, with the ith column holding the eigenvector associated with the eigenvalue λi (stored in Wi).
- 11: LDV – INTEGERInput
On entry: the first dimension of the array
V as declared in the (sub)program from which F02GCF is called.
Constraint:
LDV≥max1,N.
- 12: WORK(LWORK) – complex*16 arrayWorkspace
- 13: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F02GCF is called.
Constraint:
LWORK≥max1,N×N+2.
- 14: RWORK(2×N) – double precision arrayWorkspace
- 15: IWORK(N) – INTEGER arrayWorkspace
- 16: BWORK(N) – LOGICAL arrayWorkspace
- 17: IFAIL – INTEGERInput/Output
-
On entry:
IFAIL must be set to
0,
-1 or 1. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
On exit:
IFAIL=0 unless the routine detects an error (see
Section 6).
For environments where it might be inappropriate to halt program execution when an error is detected, the value
-1 or 1 is recommended. If the output of error messages is undesirable, then the value
1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
0.
When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.
6 Error Indicators and Warnings
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
- IFAIL=1
-
| On entry, | CRIT≠'M' or 'R', |
| or | N<0, |
| or | LDA<max1,N, |
| or | WU≤WL, |
| or | MEST<1, |
| or | LDV<max1,N, |
| or | LWORK<max1,N×N+2. |
- IFAIL=2
-
The QR algorithm failed to compute all the eigenvalues. No eigenvectors have been computed.
- IFAIL=3
-
There are more than
MEST eigenvalues in the specified range. The actual number of eigenvalues in the range is returned in
M. No eigenvectors have been computed. Rerun with the second dimension of
V=MEST≥M.
- IFAIL=4
-
Inverse iteration failed to compute all the specified eigenvectors. If an eigenvector failed to converge, the corresponding column of
V is set to zero.
7 Accuracy
If
λi is an exact eigenvalue, and
λ~i is the corresponding computed value, then
where
cn is a modestly increasing function of
n,
ε is the
machine precision, and
si is the reciprocal condition number of
λi;
A′ is the balanced form of the original matrix
A (see
Section 8), and
A′≤A.
If
xi is the corresponding exact eigenvector, and
x~i is the corresponding computed eigenvector, then the angle
θx~i,xi between them is bounded as follows:
where
sepi is the reciprocal condition number of
xi.
The condition numbers
si and
sepi may be computed from the Hessenberg form of the balanced matrix
A′ which is returned in the array
A. This requires calling
F08PSF (ZHSEQR) with
JOB='S' to compute the Schur form of
A′, followed by
F08QYF (ZTRSNA).
8 Further Comments
F02GCF calls routines from LAPACK in
Chapter F08. It first balances the matrix, using a diagonal similarity transformation to reduce its norm; and then reduces the balanced matrix
A′ to upper Hessenberg form
H, using a unitary similarity transformation:
A′=QHQH. The routine uses the Hessenberg
QR algorithm to compute all the eigenvalues of
H, which are the same as the eigenvalues of
A. It computes the eigenvectors of
H which correspond to the selected eigenvalues, using inverse iteration. It premultiplies the eigenvectors by
Q to form the eigenvectors of
A′; and finally transforms the eigenvectors to those of the original matrix
A.
Each eigenvector x is normalized so that x2=1, and the element of largest absolute value is real and positive.
The inverse iteration routine may make a small perturbation to the real parts of close eigenvalues, and this may shift their moduli just outside the specified bounds. If you are relying on eigenvalues being within the bounds, you should test them on return from F02GCF.
The time taken by the routine is approximately proportional to n3.
The routine can be used to compute
all eigenvalues and eigenvectors, by setting
WL large and negative, and
WU large and positive.
9 Example
This example computes those eigenvalues of the matrix
A which lie in the range
-5.5,+5.5
, and their corresponding eigenvectors, where
9.1 Program Text
Program Text (f02gcfe.f)
9.2 Program Data
Program Data (f02gcfe.d)
9.3 Program Results
Program Results (f02gcfe.r)