SLATEC Routines --- COMQR ---

*DECK COMQR
      SUBROUTINE COMQR (NM, N, LOW, IGH, HR, HI, WR, WI, IERR)
C***BEGIN PROLOGUE  COMQR
C***PURPOSE  Compute the eigenvalues of complex upper Hessenberg matrix
C            using the QR method.
C***LIBRARY   SLATEC (EISPACK)
C***CATEGORY  D4C2B
C***TYPE      COMPLEX (HQR-S, COMQR-C)
C***KEYWORDS  EIGENVALUES, EIGENVECTORS, EISPACK
C***AUTHOR  Smith, B. T., et al.
C***DESCRIPTION
C
C     This subroutine is a translation of a unitary analogue of the
C     ALGOL procedure  COMLR, NUM. MATH. 12, 369-376(1968) by Martin
C     and Wilkinson.
C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 396-403(1971).
C     The unitary analogue substitutes the QR algorithm of Francis
C     (COMP. JOUR. 4, 332-345(1962)) for the LR algorithm.
C
C     This subroutine finds the eigenvalues of a COMPLEX
C     upper Hessenberg matrix by the QR method.
C
C     On INPUT
C
C        NM must be set to the row dimension of the two-dimensional
C          array parameters, HR and HI, as declared in the calling
C          program dimension statement.  NM is an INTEGER variable.
C
C        N is the order of the matrix H=(HR,HI).  N is an INTEGER
C          variable.  N must be less than or equal to NM.
C
C        LOW and IGH are two INTEGER variables determined by the
C          balancing subroutine  CBAL.  If  CBAL  has not been used,
C          set LOW=1 and IGH equal to the order of the matrix, N.
C
C        HR and HI contain the real and imaginary parts, respectively,
C          of the complex upper Hessenberg matrix.  Their lower
C          triangles below the subdiagonal contain information about
C          the unitary transformations used in the reduction by  CORTH,
C          if performed.  HR and HI are two-dimensional REAL arrays,
C          dimensioned HR(NM,N) and HI(NM,N).
C
C     On OUTPUT
C
C        The upper Hessenberg portions of HR and HI have been
C          destroyed.  Therefore, they must be saved before calling
C          COMQR  if subsequent calculation of eigenvectors is to
C          be performed.
C
C        WR and WI contain the real and imaginary parts, respectively,
C          of the eigenvalues of the upper Hessenberg matrix.  If an
C          error exit is made, the eigenvalues should be correct for
C          indices IERR+1, IERR+2, ..., N.  WR and WI are one-
C          dimensional REAL arrays, dimensioned WR(N) and WI(N).
C
C        IERR is an INTEGER flag set to
C          Zero       for normal return,
C          J          if the J-th eigenvalue has not been
C                     determined after a total of 30*N iterations.
C                     The eigenvalues should be correct for indices
C                     IERR+1, IERR+2, ..., N.
C
C     Calls CSROOT for complex square root.
C     Calls PYTHAG(A,B) for sqrt(A**2 + B**2).
C     Calls CDIV for complex division.
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  CDIV, CSROOT, PYTHAG
C***REVISION HISTORY  (YYMMDD)
C   760101  DATE WRITTEN
C   890531  Changed all specific intrinsics to generic.  (WRB)
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  COMQR