SLATEC Routines --- REDUC ---

*DECK REDUC
      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