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