SLATEC Routines --- QZVAL ---
*DECK QZVAL
SUBROUTINE QZVAL (NM, N, A, B, ALFR, ALFI, BETA, MATZ, Z)
C***BEGIN PROLOGUE QZVAL
C***PURPOSE The third step of the QZ algorithm for generalized
C eigenproblems. Accepts a pair of real matrices, one in
C quasi-triangular form and the other in upper triangular
C form and computes the eigenvalues of the associated
C eigenproblem. Usually preceded by QZHES, QZIT, and
C followed by QZVEC.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4C2C
C***TYPE SINGLE PRECISION (QZVAL-S)
C***KEYWORDS EIGENVALUES, EIGENVECTORS, EISPACK
C***AUTHOR Smith, B. T., et al.
C***DESCRIPTION
C
C This subroutine is the third step of the QZ algorithm
C for solving generalized matrix eigenvalue problems,
C SIAM J. NUMER. ANAL. 10, 241-256(1973) by MOLER and STEWART.
C
C This subroutine accepts a pair of REAL matrices, one of them
C in quasi-triangular form and the other in upper triangular form.
C It reduces the quasi-triangular matrix further, so that any
C remaining 2-by-2 blocks correspond to pairs of complex
C eigenvalues, and returns quantities whose ratios give the
C generalized eigenvalues. It is usually preceded by QZHES
C and QZIT and may be followed by QZVEC.
C
C On Input
C
C NM must be set to the row dimension of the two-dimensional
C array parameters, A, B, and Z, as declared in the calling
C program dimension statement. NM is an INTEGER variable.
C
C N is the order of the matrices A and B. N is an INTEGER
C variable. N must be less than or equal to NM.
C
C A contains a real upper quasi-triangular matrix. A is a two-
C dimensional REAL array, dimensioned A(NM,N).
C
C B contains a real upper triangular matrix. In addition,
C location B(N,1) contains the tolerance quantity (EPSB)
C computed and saved in QZIT. B is a two-dimensional REAL
C array, dimensioned B(NM,N).
C
C MATZ should be set to .TRUE. if the right hand transformations
C are to be accumulated for later use in computing
C eigenvectors, and to .FALSE. otherwise. MATZ is a LOGICAL
C variable.
C
C Z contains, if MATZ has been set to .TRUE., the transformation
C matrix produced in the reductions by QZHES and QZIT, if
C performed, or else the identity matrix. If MATZ has been set
C to .FALSE., Z is not referenced. Z is a two-dimensional REAL
C array, dimensioned Z(NM,N).
C
C On Output
C
C A has been reduced further to a quasi-triangular matrix in
C which all nonzero subdiagonal elements correspond to pairs
C of complex eigenvalues.
C
C B is still in upper triangular form, although its elements
C have been altered. B(N,1) is unaltered.
C
C ALFR and ALFI contain the real and imaginary parts of the
C diagonal elements of the triangular matrix that would be
C obtained if A were reduced completely to triangular form
C by unitary transformations. Non-zero values of ALFI occur
C in pairs, the first member positive and the second negative.
C ALFR and ALFI are one-dimensional REAL arrays, dimensioned
C ALFR(N) and ALFI(N).
C
C BETA contains the diagonal elements of the corresponding B,
C normalized to be real and non-negative. The generalized
C eigenvalues are then the ratios ((ALFR+I*ALFI)/BETA).
C BETA is a one-dimensional REAL array, dimensioned BETA(N).
C
C Z contains the product of the right hand transformations
C (for all three steps) if MATZ has been set to .TRUE.
C
C Questions and comments should be directed to B. S. Garbow,
C APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
C ------------------------------------------------------------------
C
C***REFERENCES B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow,
C Y. Ikebe, V. C. Klema and C. B. Moler, Matrix Eigen-
C system Routines - EISPACK Guide, Springer-Verlag,
C 1976.
C***ROUTINES CALLED (NONE)
C***REVISION HISTORY (YYMMDD)
C 760101 DATE WRITTEN
C 890831 Modified array declarations. (WRB)
C 890831 REVISION DATE from Version 3.2
C 891214 Prologue converted to Version 4.0 format. (BAB)
C 920501 Reformatted the REFERENCES section. (WRB)
C***END PROLOGUE QZVAL