SPPEQU (3lapack)

compute row and column scalings intended to equilibrate 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 equilibrate 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 underflow, 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 element is nonpositive.