_QZVEC(3F) _QZVEC(3F)
QZVEC, ZQZVEC - EISPACK routine. This subroutine is the optional
fourth step of the QZ algorithm for solving generalized matrix eigenvalue
problems,
subroutine qzvec(nm, n, a, b, alfr, alfi, beta, z)
integer nm, n
double precision a(nm,n),b(nm,n),alfr(n),alfi(n),beta(n),z(nm,n)
subroutine sqzvec(nm, n, a, b, alfr, alfi, beta, z)
integer nm, n
real a(nm,n),b(nm,n),alfr(n),alfi(n),beta(n),z(nm,n)
This subroutine accepts a pair of REAL matrices, one of them in quasitriangular
form (in which each 2-by-2 block corresponds to a pair of
complex eigenvalues) and the other in upper triangular form. It computes
the eigenvectors of the triangular problem and transforms the results
back to the original coordinate system. It is usually preceded by
QZHES, QZIT, and QZVAL.
On Input
NM must be set to the row dimension of two-dimensional array parameters
as declared in the calling program dimension statement.
N is the order of the matrices.
A contains a real upper quasi-triangular matrix.
B contains a real upper triangular matrix. In addition, location B(N,1)
contains the tolerance quantity (EPSB) computed and saved in QZIT.
ALFR , ALFI, and BETA are vectors with components whose ratios
((ALFR+I*ALFI)/BETA) are the generalized eigenvalues. They are usually
obtained from QZVAL.
Z contains the transformation matrix produced in the reductions by
QZHES, QZIT, and QZVAL, if performed. If the eigenvectors of the
triangular problem are desired, Z must contain the identity matrix. On
Output
A is unaltered. Its subdiagonal elements provide information
about the storage of the complex eigenvectors.
B has been destroyed.
ALFR , ALFI, and BETA are unaltered.
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_QZVEC(3F) _QZVEC(3F)
Z contains the real and imaginary parts of the eigenvectors. If ALFI(I)
.EQ. 0.0, the I-th eigenvalue is real and
the I-th column of Z contains its eigenvector. If ALFI(I) .NE. 0.0,
the I-th eigenvalue is complex.
If ALFI(I) .GT. 0.0, the eigenvalue is the first of
a complex pair and the I-th and (I+1)-th columns
of Z contain its eigenvector.
If ALFI(I) .LT. 0.0, the eigenvalue is the second of
a complex pair and the (I-1)-th and I-th columns
of Z contain the conjugate of its eigenvector. Each eigenvector is
normalized so that the modulus of its largest component is 1.0 .
Questions and comments should be directed to B. S. Garbow, APPLIED
MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
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