# elmbak.f

          SUBROUTINE ELMBAK (NM, LOW, IGH, A, INT, M, Z)
    C***BEGIN PROLOGUE  ELMBAK
    C***PURPOSE  Form the eigenvectors of a real general matrix from the
    C            eigenvectors of the upper Hessenberg matrix output from
    C            ELMHES.
    C***LIBRARY   SLATEC (EISPACK)
    C***CATEGORY  D4C4
    C***TYPE      SINGLE PRECISION (ELMBAK-S, COMBAK-C)
    C***KEYWORDS  EIGENVALUES, EIGENVECTORS, EISPACK
    C***AUTHOR  Smith, B. T., et al.
    C***DESCRIPTION
    C
    C     This subroutine is a translation of the ALGOL procedure ELMBAK,
    C     NUM. MATH. 12, 349-368(1968) 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 REAL GENERAL
    C     matrix by back transforming those of the corresponding
    C     upper Hessenberg matrix determined by  ELMHES.
    C
    C     On INPUT
    C
    C        NM must be set to the row dimension of the two-dimensional
    C          array parameters, A and Z, as declared in the calling
    C          program dimension statement.  NM is an INTEGER variable.
    C
    C        LOW and IGH are two INTEGER variables determined by the
    C          balancing subroutine  BALANC.  If  BALANC  has not been
    C          used, set LOW=1 and IGH equal to the order of the matrix.
    C
    C        A contains the multipliers which were used in the reduction
    C          by  ELMHES  in its lower triangle below the subdiagonal.
    C          A is a two-dimensional REAL array, dimensioned A(NM,IGH).
    C
    C        INT contains information on the rows and columns interchanged
    C          in the reduction by  ELMHES.  Only elements LOW through IGH
    C          are used.  INT is a one-dimensional INTEGER array,
    C          dimensioned INT(IGH).
    C
    C        M is the number of columns of Z to be back transformed.
    C          M is an INTEGER variable.
    C
    C        Z contains the real and imaginary parts of the eigenvectors
    C          to be back transformed in its first M columns.  Z is a
    C          two-dimensional REAL array, dimensioned Z(NM,M).
    C
    C     On OUTPUT
    C
    C        Z contains the real and imaginary parts of the transformed
    C          eigenvectors in its first M columns.
    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  ELMBAK