# trbak3.f

          SUBROUTINE TRBAK3 (NM, N, NV, A, M, Z)
    C***BEGIN PROLOGUE  TRBAK3
    C***PURPOSE  Form the eigenvectors of a real symmetric matrix from the
    C            eigenvectors of a symmetric tridiagonal matrix formed
    C            by TRED3.
    C***LIBRARY   SLATEC (EISPACK)
    C***CATEGORY  D4C4
    C***TYPE      SINGLE PRECISION (TRBAK3-S)
    C***KEYWORDS  EIGENVECTORS OF A REAL SYMMETRIC MATRIX, EISPACK
    C***AUTHOR  Smith, B. T., et al.
    C***DESCRIPTION
    C
    C     This subroutine is a translation of the ALGOL procedure TRBAK3,
    C     NUM. MATH. 11, 181-195(1968) 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 REAL SYMMETRIC
    C     matrix by back transforming those of the corresponding
    C     symmetric tridiagonal matrix determined by  TRED3.
    C
    C     On Input
    C
    C        NM must be set to the row dimension of the two-dimensional
    C          array parameter, Z, as declared in the calling program
    C          dimension statement.  NM is an INTEGER 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        NV is an INTEGER variable set equal to the dimension of the
    C          array A as specified in the calling program.  NV must not
    C          be less than  N*(N+1)/2.
    C
    C        A contains information about the orthogonal transformations
    C          used in the reduction by  TRED3  in its first N*(N+1)/2
    C          positions.  A is a one-dimensional REAL array, dimensioned
    C          A(NV).
    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 eigenvectors to be back transformed in its
    C          first M columns.  Z is a two-dimensional REAL array,
    C          dimensioned Z(NM,M).
    C
    C     On Output
    C
    C        Z contains the transformed eigenvectors in its first M columns.
    C
    C     Note that TRBAK3 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  TRBAK3