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# spoequ

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NAME
SPOEQU - compute row and column scalings intended to equili-
brate 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 equili-
brate 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 con-
dition 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 diago-
nal 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 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.
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