F02GJF calculates all the eigenvalues and, if required, all the eigenvectors of the complex generalized eigenproblem Ax=λBx where A and B are complex, square matrices, using the QZ algorithm.
| SUBROUTINE F02GJF ( | N, AR, LDAR, AI, LDAI, BR, LDBR, BI, LDBI, EPS1, ALFR, ALFI, BETA, MATV, VR, LDVR, VI, LDVI, ITER, IFAIL) |
| INTEGER | N, LDAR, LDAI, LDBR, LDBI, LDVR, LDVI, ITER(N), IFAIL |
| double precision | AR(LDAR,N), AI(LDAI,N), BR(LDBR,N), BI(LDBI,N), EPS1, ALFR(N), ALFI(N), BETA(N), VR(LDVR,N), VI(LDVI,N) |
| LOGICAL | MATV |
All the eigenvalues and, if required, all the eigenvectors of the complex generalized eigenproblem
Ax=λBx where
A and
B are complex, square matrices, are determined using the
QZ algorithm. The complex
QZ algorithm consists of three stages:
- A is reduced to upper Hessenberg form (with real, non-negative subdiagonal elements) and at the same time B is reduced to upper triangular form.
- A is further reduced to triangular form while the triangular form of B is maintained and the diagonal elements of B are made real and non-negative.
F02GJF does not actually produce the eigenvalues
λj, but instead returns
αj and
βj such that
The division by
βj becomes the responsibility of your program, since
βj may be zero, indicating an infinite eigenvalue.
- If the eigenvectors are required (MATV=.TRUE.), they are obtained from the triangular matrices and then transferred back into the original co-ordinate system.
Moler C B and Stewart G W (1973) An algorithm for generalized matrix eigenproblems
SIAM J. Numer. Anal. 10 241–256
Ward R C (1975) The combination shift
QZ algorithm
SIAM J. Numer. Anal. 12 835–853
Wilkinson J H (1979) Kronecker's canonical form and the
QZ algorithm
Linear Algebra Appl. 28 285–303
- 1: N – INTEGERInput
On entry:
n, the order of the matrices A and B.
Constraint:
N≥1.
- 2: AR(LDAR,N) – double precision arrayInput/Output
On entry: the real parts of the elements of the n by n complex matrix A.
On exit:
AR is overwritten.
- 3: LDAR – INTEGERInput
On entry: the first dimension of the array
AR as declared in the (sub)program from which F02GJF is called.
Constraint:
LDAR≥N.
- 4: AI(LDAI,N) – double precision arrayInput/Output
On entry: the imaginary parts of the elements of the n by n complex matrix A.
On exit:
AI is overwritten.
- 5: LDAI – INTEGERInput
On entry: the first dimension of the array
AI as declared in the (sub)program from which F02GJF is called.
Constraint:
LDAI≥N.
- 6: BR(LDBR,N) – double precision arrayInput/Output
On entry: the real parts of the elements of the n by n complex matrix B.
On exit:
BR is overwritten.
- 7: LDBR – INTEGERInput
On entry: the first dimension of the array
BR as declared in the (sub)program from which F02GJF is called.
Constraint:
LDBR≥N.
- 8: BI(LDBI,N) – double precision arrayInput/Output
On entry: the imaginary parts of the elements of the n by n complex matrix B.
On exit:
BI is overwritten.
- 9: LDBI – INTEGERInput
On entry: the first dimension of the array
BI as declared in the (sub)program from which F02GJF is called.
Constraint:
LDBI≥N.
- 10: EPS1 – double precisionInput
On entry: a tolerance used to determine negligible elements.
- EPS1>0.0
- An element will be considered negligible if it is less than EPS1×the norm of its matrix.
- EPS1≤0.0
- machine precision is used for EPS1.
A positive value of
EPS1 may result in faster execution but less accurate results.
- 11: ALFR(N) – double precision arrayOutput
- 12: ALFI(N) – double precision arrayOutput
On exit: the real and imaginary parts of αj, for j=1,2,…,n.
- 13: BETA(N) – double precision arrayOutput
On exit: βj, for j=1,2,…,n.
- 14: MATV – LOGICALInput
On entry: must be set .TRUE. if the eigenvectors are required, otherwise .FALSE..
- 15: VR(LDVR,N) – double precision arrayOutput
On exit: if
MATV=.TRUE., the
jth column of
VR contains the real parts of the eigenvector corresponding to the
jth eigenvalue. The eigenvectors are normalized so that the sum of squares of the moduli of the components is equal to
1.0 and the component of largest modulus is real.
If
MATV=.FALSE.,
VR is not used.
- 16: LDVR – INTEGERInput
On entry: the first dimension of the array
VR as declared in the (sub)program from which F02GJF is called.
Constraint:
LDVR≥N.
- 17: VI(LDVI,N) – double precision arrayOutput
On exit: if
MATV=.TRUE., the
jth column of
VI contains the imaginary parts of the eigenvector corresponding to the
jth eigenvalue.
If
MATV=.FALSE.,
VI is not used.
- 18: LDVI – INTEGERInput
On entry: the first dimension of the array
VI as declared in the (sub)program from which F02GJF is called.
Constraint:
LDVI≥N.
- 19: ITER(N) – INTEGER arrayOutput
On exit: ITERj contains the number of iterations needed to obtain the jth eigenvalue. Note that the eigenvalues are obtained in reverse order, starting with the nth.
- 20: 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.
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
The computed eigenvalues are always exact for a problem
A+Ex=λB+Fx where
E/A and
F/B are both of the order of
maxEPS1,ε,
EPS1 being defined as in
Section 5 and
ε being the
machine precision.
Note: interpretation of results obtained with the
QZ algorithm often requires a clear understanding of the effects of small changes in the original data. These effects are reviewed in
Wilkinson (1979), in relation to the significance of small values of
αj and
βj. It should be noted that if
αj and
βj are
both small for any
j, it may be that no reliance can be placed on
any of the computed eigenvalues
λi=αi/βi. You are recommended to study
Wilkinson (1979) and, if in difficulty, to seek expert advice on determining the sensitivity of the eigenvalues to perturbations in the data.
The time taken by F02GJF is approximately proportional to
n3 and also depends on the value chosen for parameter
EPS1.
This example finds all the eigenvalues and eigenvectors of
Ax=λBx where
and