F02BJF calculates all the eigenvalues and, if required, all the eigenvectors of the generalized eigenproblem Ax=λBx, where A and B are real, square matrices, using the QZ algorithm.
| SUBROUTINE F02BJF ( | N, A, LDA, B, LDB, EPS1, ALFR, ALFI, BETA, MATV, V, LDV, ITER, IFAIL) |
| INTEGER | N, LDA, LDB, LDV, ITER(N), IFAIL |
| double precision | A(LDA,N), B(LDB,N), EPS1, ALFR(N), ALFI(N), BETA(N), V(LDV,N) |
| LOGICAL | MATV |
All the eigenvalues and, if required, all the eigenvectors of the generalized eigenproblem
Ax=λ Bx, where
A and
B are real, square matrices, are determined using the
QZ algorithm. The
QZ algorithm consists of four stages:
| (i) |
A is reduced to upper Hessenberg form and at the same time B is reduced to upper triangular form. |
| (ii) |
A is further reduced to quasi-triangular form while the triangular form of B is maintained. |
| (iii) |
The quasi-triangular form of A is reduced to triangular form and the eigenvalues extracted. F02BJF 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. Pairs of complex eigenvalues occur with αj/βj and αj+1/βj+1 complex conjugates, even though αj and αj+1 are not conjugate. |
| (iv) |
If the eigenvectors are required (MATV=.TRUE.), they are obtained from the triangular matrices and then transformed 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.
- 2: A(LDA,N) – double precision arrayInput/Output
On entry: the n by n matrix A.
On exit:
A is overwritten.
- 3: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F02BJF is called.
Constraint:
LDA≥N.
- 4: B(LDB,N) – double precision arrayInput/Output
On entry: the n by n matrix B.
On exit:
B is overwritten.
- 5: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F02BJF is called.
Constraint:
LDB≥N.
- 6: EPS1 – double precisionInput
On entry: the 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 in place of EPS1.
A positive value of
EPS1 may result in faster execution but less accurate results.
- 7: ALFR(N) – double precision arrayOutput
- 8: ALFI(N) – double precision arrayOutput
On exit: the real and imaginary parts of αj, for j=1,2,…,n.
- 9: BETA(N) – double precision arrayOutput
On exit: βj, for j=1,2,…,n.
- 10: MATV – LOGICALInput
On entry: must be set .TRUE. if the eigenvectors are required, otherwise .FALSE..
- 11: V(LDV,N) – double precision arrayOutput
On exit: if
MATV=.TRUE.,
| – |
if the jth eigenvalue is real, the jth column of V contains its eigenvector; |
| – |
if the jth and j+1th eigenvalues form a complex pair, the jth and j+1th columns of V contain the real and imaginary parts of the eigenvector associated with the first eigenvalue of the pair. The conjugate of this vector is the eigenvector for the conjugate eigenvalue. |
Each eigenvector is normalized so that the component of largest modulus is real and the sum of squares of the moduli equal one.
If
MATV=.FALSE.,
V is not used.
- 12: LDV – INTEGERInput
On entry: the first dimension of the array
V as declared in the (sub)program from which F02BJF is called.
Constraint:
LDV≥N.
- 13: 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.
- 14: 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).
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 F02BJF 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