# ortbak.f

          SUBROUTINE ORTBAK (NM, LOW, IGH, A, ORT, M, Z)
    C***BEGIN PROLOGUE  ORTBAK
    C***PURPOSE  Form the eigenvectors of a general real matrix from the
    C            eigenvectors of the upper Hessenberg matrix output from
    C            ORTHES.
    C***LIBRARY   SLATEC (EISPACK)
    C***CATEGORY  D4C4
    C***TYPE      SINGLE PRECISION (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 the ALGOL procedure ORTBAK,
    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  ORTHES.
    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 some information about the orthogonal trans-
    C          formations used in the reduction to Hessenberg form by
    C          ORTHES  in its strict lower triangle.  A is a two-dimensional
    C          REAL array, dimensioned A(NM,IGH).
    C
    C        ORT contains further information about the orthogonal trans-
    C          formations used in the reduction by  ORTHES.  Only elements
    C          LOW through IGH are used.  ORT is a one-dimensional REAL
    C          array, dimensioned ORT(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 to
    C          be back transformed in its first M columns.  Z is a two-
    C          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        ORT has been used for temporary storage as is not restored.
    C
    C     NOTE that ORTBAK 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  ORTBAK