# comqr.f

          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