SUBROUTINE QZVAL(NM,N,A,B,ALFR,ALFI,BETA,MATZ,Z)
C***BEGIN PROLOGUE  QZVAL
C***DATE WRITTEN   760101   (YYMMDD)
C***REVISION DATE  830518   (YYMMDD)
C***CATEGORY NO.  D4C2C
C***KEYWORDS  EIGENVALUES,EIGENVECTORS,EISPACK
C***AUTHOR  SMITH, B. T., ET AL.
C***PURPOSE  The third step of the QZ algorithm for generalized
C            eigenproblems. Accepts a pair of real matrices, one
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***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 two-dimensional
C          array parameters as declared in the calling program
C          dimension statement.
C
C        N is the order of the matrices.
C
C        A contains a real upper quasi-triangular matrix.
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.
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.
C
C        Z contains, if MATZ has been set to .TRUE., the
C          transformation matrix produced in the reductions by QZHES
C          and QZIT, if performed, or else the identity matrix.
C          If MATZ has been set to .FALSE., Z is not referenced.
C
C     On Output
C
C        A has been reduced further to a quasi-triangular matrix
C          in which all nonzero subdiagonal elements correspond to
C          pairs 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
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
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***REFERENCES  B. T. SMITH, J. M. BOYLE, J. J. DONGARRA, B. S. GARBOW,
C                 Y. IKEBE, V. C. KLEMA, C. B. MOLER, *MATRIX EIGEN-
C                 SYSTEM ROUTINES - EISPACK GUIDE*, SPRINGER-VERLAG,
C                 1976.
C***ROUTINES CALLED  (NONE)
C***END PROLOGUE  QZVAL