SLATEC Routines --- COMQR2 ---

*DECK COMQR2
      SUBROUTINE COMQR2 (NM, N, LOW, IGH, ORTR, ORTI, HR, HI, WR, WI,
     +   ZR, ZI, IERR)
C***BEGIN PROLOGUE  COMQR2
C***PURPOSE  Compute the eigenvalues and eigenvectors of a complex upper
C            Hessenberg matrix.
C***LIBRARY   SLATEC (EISPACK)
C***CATEGORY  D4C2B
C***TYPE      COMPLEX (HQR2-S, COMQR2-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  COMLR2, NUM. MATH. 16, 181-204(1970) by Peters
C     and Wilkinson.
C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 372-395(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 and eigenvectors
C     of a COMPLEX UPPER Hessenberg matrix by the QR
C     method.  The eigenvectors of a COMPLEX GENERAL matrix
C     can also be found if  CORTH  has been used to reduce
C     this general matrix to Hessenberg form.
C
C     On INPUT
C
C        NM must be set to the row dimension of the two-dimensional
C          array parameters, HR, HI, ZR, and ZI, as declared in the
C          calling program dimension statement.  NM is an INTEGER
C          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        ORTR and ORTI contain information about the unitary trans-
C          formations used in the reduction by  CORTH, if performed.
C          Only elements LOW through IGH are used.  If the eigenvectors
C          of the Hessenberg matrix are desired, set ORTR(J) and
C          ORTI(J) to 0.0E0 for these elements.  ORTR and ORTI are
C          one-dimensional REAL arrays, dimensioned ORTR(IGH) and
C          ORTI(IGH).
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.  If the eigenvectors of the Hessenberg matrix
C          are desired, these elements may be arbitrary.  HR and HI
C          are two-dimensional REAL arrays, dimensioned HR(NM,N) and
C          HI(NM,N).
C
C     On OUTPUT
C
C        ORTR, ORTI, and the upper Hessenberg portions of HR and HI
C          have been destroyed.
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        ZR and ZI contain the real and imaginary parts, respectively,
C          of the eigenvectors.  The eigenvectors are unnormalized.
C          If an error exit is made, none of the eigenvectors has been
C          found.  ZR and ZI are two-dimensional REAL arrays,
C          dimensioned ZR(NM,N) and ZI(NM,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, but no eigenvectors are
C                     computed.
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  COMQR2