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dppequ

 NAME
      DPPEQU - compute row and column scalings intended to equili-
      brate a symmetric positive definite matrix A in packed
      storage and reduce its condition number (with respect to the
      two-norm)

 SYNOPSIS
      SUBROUTINE DPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFO )

          CHARACTER      UPLO

          INTEGER        INFO, N

          DOUBLE         PRECISION AMAX, SCOND

          DOUBLE         PRECISION AP( * ), S( * )

 PURPOSE
      DPPEQU computes row and column scalings intended to equili-
      brate a symmetric positive definite matrix A in packed
      storage and reduce its condition number (with respect to the
      two-norm).  S contains the scale factors,
      S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix B with
      elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal.
      This choice of S puts the condition number of B within a
      factor N of the smallest possible condition number over all
      possible diagonal scalings.

 ARGUMENTS
      UPLO    (input) CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

      N       (input) INTEGER
              The order of the matrix A.  N >= 0.

      AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
              The upper or lower triangle of the symmetric matrix
              A, packed columnwise in a linear array.  The j-th
              column of A is stored in the array AP as follows: if
              UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for
              j<=i<=n.

      S       (output) DOUBLE PRECISION array, dimension (N)
              If INFO = 0, S contains the scale factors for A.

      SCOND   (output) DOUBLE PRECISION
              If INFO = 0, S contains the ratio of the smallest
              S(i) to the largest S(i).  If SCOND >= 0.1 and AMAX
              is neither too large nor too small, it is not worth

              scaling by S.

      AMAX    (output) DOUBLE PRECISION
              Absolute value of largest matrix element.  If AMAX
              is very close to overflow or very close to under-
              flow, the matrix should be scaled.

      INFO    (output) INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal
              value
              > 0:  if INFO = i, the i-th diagonal entry is nonpo-
              sitive.