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sppequ

 NAME
      SPPEQU - 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 SPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFO )

          CHARACTER      UPLO

          INTEGER        INFO, N

          REAL           AMAX, SCOND

          REAL           AP( * ), S( * )

 PURPOSE
      SPPEQU 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) REAL 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) REAL array, dimension (N)
              If INFO = 0, S contains the scale factors for A.

      SCOND   (output) REAL
              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) REAL
              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.