NAME¶

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

SYNOPSIS¶

SUBROUTINE SPOEQU(

N, A, LDA, S, SCOND, AMAX, INFO )

INTEGER INFO, LDA, N REAL AMAX, SCOND REAL A( LDA, * ), S( * )

PURPOSE¶

SPOEQU computes row and column scalings intended to equilibrate a symmetric 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) REAL array, dimension (LDA,N)

The N-by-N symmetric 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.