# cortb.f

          SUBROUTINE CORTB (NM, LOW, IGH, AR, AI, ORTR, ORTI, M, ZR, ZI)
    C***BEGIN PROLOGUE  CORTB
    C***PURPOSE  Form the eigenvectors of a complex general matrix from
    C            eigenvectors of upper Hessenberg matrix output from
    C            CORTH.
    C***LIBRARY   SLATEC (EISPACK)
    C***CATEGORY  D4C4
    C***TYPE      COMPLEX (ORTBAK-S, CORTB-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 ORTBAK, 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     This subroutine forms the eigenvectors of a COMPLEX GENERAL
    C     matrix by back transforming those of the corresponding
    C     upper Hessenberg matrix determined by  CORTH.
    C
    C     On INPUT
    C
    C        NM must be set to the row dimension of the two-dimensional
    C          array parameters, AR, AI, ZR, and ZI, as declared in the
    C          calling program dimension statement.  NM is an INTEGER
    C          variable.
    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.
    C
    C        AR and AI contain information about the unitary trans-
    C          formations used in the reduction by  CORTH  in their
    C          strict lower triangles.  AR and AI are two-dimensional
    C          REAL arrays, dimensioned AR(NM,IGH) and AI(NM,IGH).
    C
    C        ORTR and ORTI contain further information about the unitary
    C          transformations used in the reduction by  CORTH.  Only
    C          elements LOW through IGH are used.  ORTR and ORTI are
    C          one-dimensional REAL arrays, dimensioned ORTR(IGH) and
    C          ORTI(IGH).
    C
    C        M is the number of columns of Z=(ZR,ZI) to be back transformed.
    C          M is an INTEGER variable.
    C
    C        ZR and ZI contain the real and imaginary parts, respectively,
    C          of the eigenvectors to be back transformed in their first
    C          M columns.  ZR and ZI are two-dimensional REAL arrays,
    C          dimensioned ZR(NM,M) and ZI(NM,M).
    C
    C     On OUTPUT
    C
    C        ZR and ZI contain the real and imaginary parts, respectively,
    C          of the transformed eigenvectors in their first M columns.
    C
    C        ORTR and ORTI have been altered.
    C
    C     Note that CORTB preserves vector Euclidean norms.
    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  CORTB