SPBEQU - compute row and column scalings intended to equi
librate a symmetric positive definite band matrix A and
reduce its condition number (with respect to the two-norm)
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 equi
librate 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 diago
nal. This choice of S puts the condition number of B
within a factor N of the smallest possible condition num
ber 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.
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 ille
gal value.
> 0: if INFO = i, the i-th diagonal element is
nonpositive.
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