# reduc.f

          SUBROUTINE REDUC (NM, N, A, B, DL, IERR)
    C***BEGIN PROLOGUE  REDUC
    C***PURPOSE  Reduce a generalized symmetric eigenproblem to a standard
    C            symmetric eigenproblem using Cholesky factorization.
    C***LIBRARY   SLATEC (EISPACK)
    C***CATEGORY  D4C1C
    C***TYPE      SINGLE PRECISION (REDUC-S)
    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 REDUC1,
    C     NUM. MATH. 11, 99-110(1968) by Martin and Wilkinson.
    C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 303-314(1971).
    C
    C     This subroutine reduces the generalized SYMMETRIC eigenproblem
    C     Ax=(LAMBDA)Bx, where B is POSITIVE DEFINITE, to the standard
    C     symmetric eigenproblem using the Cholesky factorization of B.
    C
    C     On Input
    C
    C        NM must be set to the row dimension of the two-dimensional
    C          array parameters, A and B, as declared in the calling
    C          program dimension statement.  NM is an INTEGER variable.
    C
    C        N is the order of the matrices A and B.  If the Cholesky
    C          factor L of B is already available, N should be prefixed
    C          with a minus sign.  N is an INTEGER variable.
    C
    C        A and B contain the real symmetric input matrices.  Only
    C          the full upper triangles of the matrices need be supplied.
    C          If N is negative, the strict lower triangle of B contains,
    C          instead, the strict lower triangle of its Cholesky factor L.
    C          A and B are two-dimensional REAL arrays, dimensioned A(NM,N)
    C          and B(NM,N).
    C
    C       DL contains, if N is negative, the diagonal elements of L.
    C          DL is a one-dimensional REAL array, dimensioned DL(N).
    C
    C     On Output
    C
    C        A contains in its full lower triangle the full lower triangle
    C          of the symmetric matrix derived from the reduction to the
    C          standard form.  The strict upper triangle of A is unaltered.
    C
    C        B contains in its strict lower triangle the strict lower
    C          triangle of its Cholesky factor L.  The full upper triangle
    C          of B is unaltered.
    C
    C        DL contains the diagonal elements of L.
    C
    C        IERR is an INTEGER flag set to
    C          Zero       for normal return,
    C          7*N+1      if B is not positive definite.
    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   890531  Changed all specific intrinsics to generic.  (WRB)
    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  REDUC