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.