CPOEQU (3lapack)

compute row and column scalings intended to equilibrate a Hermitian positive definite matrix A and reduce its condition number (with respect to the two-norm)

SYNOPSIS

  SUBROUTINE CPOEQU( N, A, LDA, S, SCOND, AMAX, INFO )

      INTEGER        INFO, LDA, N

      REAL           AMAX, SCOND

      REAL           S( * )

      COMPLEX        A( LDA, * )

PURPOSE

  CPOEQU computes row and column scalings intended to equilibrate a Hermitian
  positive definite matrix A 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

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

  A       (input) COMPLEX array, dimension (LDA,N)
          The N-by-N Hermitian positive definite matrix whose scaling factors
          are to be computed.  Only the diagonal elements of A are
          referenced.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,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.