SLATEC Routines --- CORTH ---

*DECK CORTH
      SUBROUTINE CORTH (NM, N, LOW, IGH, AR, AI, ORTR, ORTI)
C***BEGIN PROLOGUE  CORTH
C***PURPOSE  Reduce a complex general matrix to complex upper Hessenberg
C            form using unitary similarity transformations.
C***LIBRARY   SLATEC (EISPACK)
C***CATEGORY  D4C1B2
C***TYPE      COMPLEX (ORTHES-S, CORTH-C)
C***KEYWORDS  EIGENVALUES, EIGENVECTORS, EISPACK
C***AUTHOR  Smith, B. T., et al.
C***DESCRIPTION
C
C     This subroutine is a translation of a complex analogue of
C     the ALGOL procedure ORTHES, NUM. MATH. 12, 349-368(1968)
C     by Martin and Wilkinson.
C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 339-358(1971).
C
C     Given a COMPLEX GENERAL matrix, this subroutine
C     reduces a submatrix situated in rows and columns
C     LOW through IGH to upper Hessenberg form by
C     unitary similarity transformations.
C
C     On INPUT
C
C        NM must be set to the row dimension of the two-dimensional
C          array parameters, AR and AI, as declared in the calling
C          program dimension statement.  NM is an INTEGER variable.
C
C        N is the order of the matrix A=(AR,AI).  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        AR and AI contain the real and imaginary parts, respectively,
C          of the complex input matrix.  AR and AI are two-dimensional
C          REAL arrays, dimensioned AR(NM,N) and AI(NM,N).
C
C     On OUTPUT
C
C        AR and AI contain the real and imaginary parts, respectively,
C          of the Hessenberg matrix.  Information about the unitary
C          transformations used in the reduction is stored in the
C          remaining triangles under the Hessenberg matrix.
C
C        ORTR and ORTI contain further information about the unitary
C          transformations.  Only elements LOW through IGH are used.
C          ORTR and ORTI are one-dimensional REAL arrays, dimensioned
C          ORTR(IGH) and ORTI(IGH).
C
C     Calls PYTHAG(A,B) for sqrt(A**2 + B**2).
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  PYTHAG
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  CORTH