SLATEC Routines --- QZVEC ---

*DECK QZVEC
      SUBROUTINE QZVEC (NM, N, A, B, ALFR, ALFI, BETA, Z)
C***BEGIN PROLOGUE  QZVEC
C***PURPOSE  The optional fourth step of the QZ algorithm for
C            generalized eigenproblems.  Accepts a matrix in
C            quasi-triangular form and another in upper triangular
C            and computes the eigenvectors of the triangular problem
C            and transforms them back to the original coordinates
C            Usually preceded by QZHES, QZIT, and QZVAL.
C***LIBRARY   SLATEC (EISPACK)
C***CATEGORY  D4C3
C***TYPE      SINGLE PRECISION (QZVEC-S)
C***KEYWORDS  EIGENVALUES, EIGENVECTORS, EISPACK
C***AUTHOR  Smith, B. T., et al.
C***DESCRIPTION
C
C     This subroutine is the optional fourth 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 in
C     quasi-triangular form (in which each 2-by-2 block corresponds to
C     a pair of complex eigenvalues) and the other in upper triangular
C     form.  It computes the eigenvectors of the triangular problem and
C     transforms the results back to the original coordinate system.
C     It is usually preceded by  QZHES,  QZIT, and  QZVAL.
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        ALFR, ALFI, and BETA are one-dimensional REAL arrays with
C          components whose ratios ((ALFR+I*ALFI)/BETA) are the
C          generalized eigenvalues.  They are usually obtained from
C          QZVAL.  They are dimensioned ALFR(N), ALFI(N), and BETA(N).
C
C        Z contains the transformation matrix produced in the reductions
C          by  QZHES,  QZIT, and  QZVAL,  if performed.  If the
C          eigenvectors of the triangular problem are desired, Z must
C          contain the identity matrix.  Z is a two-dimensional REAL
C          array, dimensioned Z(NM,N).
C
C     On Output
C
C        A is unaltered.  Its subdiagonal elements provide information
C           about the storage of the complex eigenvectors.
C
C        B has been destroyed.
C
C        ALFR, ALFI, and BETA are unaltered.
C
C        Z contains the real and imaginary parts of the eigenvectors.
C          If ALFI(J) .EQ. 0.0, the J-th eigenvalue is real and
C            the J-th column of Z contains its eigenvector.
C          If ALFI(J) .NE. 0.0, the J-th eigenvalue is complex.
C            If ALFI(J) .GT. 0.0, the eigenvalue is the first of
C              a complex pair and the J-th and (J+1)-th columns
C              of Z contain its eigenvector.
C            If ALFI(J) .LT. 0.0, the eigenvalue is the second of
C              a complex pair and the (J-1)-th and J-th columns
C              of Z contain the conjugate of its eigenvector.
C          Each eigenvector is normalized so that the modulus
C          of its largest component is 1.0 .
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  QZVEC