_QZVAL(3F) _QZVAL(3F)
QZVAL, SQZVAL  EISPACK routine. This subroutine is the third step of
the QZ algorithm for solving generalized matrix eigenvalue problems,
subroutine qzval(nm, n, a, b, alfr, alfi, beta, matz, z)
integer nm, n
double precision a(nm,n),b(nm,n),alfr(n),alfi(n),beta(n),z(nm,n)
logical matz
subroutine sqzval(nm, n, a, b, alfr, alfi, beta, matz, z)
integer nm, n
real a(nm,n),b(nm,n),alfr(n),alfi(n),beta(n),z(nm,n)
logical matz
On Input This subroutine accepts a pair of REAL matrices, one of them in
quasitriangular form and the other in upper triangular form. It reduces
the quasitriangular matrix further, so that any remaining 2by2 blocks
correspond to pairs of complex eigenvalues, and returns quantities whose
ratios give the generalized eigenvalues. It is usually preceded by
QZHES and QZIT and may be followed by QZVEC.
NM must be set to the row dimension of twodimensional array parameters
as declared in the calling program dimension statement.
N is the order of the matrices.
A contains a real upper quasitriangular matrix.
B contains a real upper triangular matrix. In addition, location B(N,1)
contains the tolerance quantity (EPSB) computed and saved in QZIT.
MATZ should be set to .TRUE. If the right hand transformations are to be
accumulated for later use in computing eigenvectors, and to .FALSE.
otherwise.
Z contains, if MATZ has been set to .TRUE., the transformation matrix
produced in the reductions by QZHES and QZIT, if performed, or else the
identity matrix. If MATZ has been set to .FALSE., Z is not referenced.
On Output
A has been reduced further to a quasitriangular matrix in which all
nonzero subdiagonal elements correspond to pairs of complex eigenvalues.
B is still in upper triangular form, although its elements have been
altered. B(N,1) is unaltered.
ALFR and ALFI contain the real and imaginary parts of the diagonal
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_QZVAL(3F) _QZVAL(3F)
elements of the triangular matrix that would be obtained if a were
reduced completely to triangular form by unitary transformations. Nonzero
values of ALFI occur in pairs, the first member positive and the
second negative.
BETA contains the diagonal elements of the corresponding B, normalized to
be real and nonnegative. The generalized eigenvalues are then the
ratios ((ALFR+I*ALFI)/BETA).
Z contains the product of the right hand transformations (for all three
steps) if MATZ has been set to .TRUE. Questions and comments should be
directed to B. S. Garbow, APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL
LABORATORY
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