# htribk.f

          SUBROUTINE HTRIBK (NM, N, AR, AI, TAU, M, ZR, ZI)
    C***BEGIN PROLOGUE  HTRIBK
    C***PURPOSE  Form the eigenvectors of a complex Hermitian matrix from
    C            the eigenvectors of a real symmetric tridiagonal matrix
    C            output from HTRIDI.
    C***LIBRARY   SLATEC (EISPACK)
    C***CATEGORY  D4C4
    C***TYPE      SINGLE PRECISION (HTRIBK-S)
    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 TRBAK1, NUM. MATH. 11, 181-195(1968)
    C     by Martin, Reinsch, and Wilkinson.
    C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 212-226(1971).
    C
    C     This subroutine forms the eigenvectors of a COMPLEX HERMITIAN
    C     matrix by back transforming those of the corresponding
    C     real symmetric tridiagonal matrix determined by  HTRIDI.
    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        N is the order of the matrix.  N is an INTEGER variable.
    C          N must be less than or equal to NM.
    C
    C        AR and AI contain some information about the unitary
    C          transformations used in the reduction by  HTRIDI  in the
    C          strict lower triangle of AR and the full lower triangle of
    C          AI.  The remaining upper parts of the matrices are arbitrary.
    C          AR and AI are two-dimensional REAL arrays, dimensioned
    C          AR(NM,N) and AI(NM,N).
    C
    C        TAU contains further information about the transformations.
    C          TAU is a one-dimensional REAL array, dimensioned TAU(2,N).
    C
    C        M is the number of eigenvectors to be back transformed.
    C          M is an INTEGER variable.
    C
    C       ZR contains the eigenvectors to be back transformed in its first
    C          M columns.  The contents of ZI are immaterial.  ZR and ZI are
    C          two-dimensional REAL arrays, dimensioned ZR(NM,M) and
    C          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     Note that the last component of each returned vector
    C     is real and that vector Euclidean norms are preserved.
    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  HTRIBK