SYNOPSIS

SUBROUTINE SPBEQU(

UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )

    

CHARACTER UPLO

    

INTEGER INFO, KD, LDAB, N

    

REAL AMAX, SCOND

    

REAL AB( LDAB, * ), S( * )

PURPOSE

SPBEQU computes row and column scalings intended to equilibrate a symmetric positive definite band 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

UPLO (input) CHARACTER*1

= 'U': Upper triangular of A is stored;
= 'L': Lower triangular of A is stored.

N (input) INTEGER

The order of the matrix A. N >= 0.

KD (input) INTEGER

The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.

AB (input) REAL array, dimension (LDAB,N)

The upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

LDAB (input) INTEGER

The leading dimension of the array A. LDAB >= KD+1.

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.