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)
- SUBROUTINE SPBEQU(
UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
INFO, KD, LDAB, N
AB( LDAB, * ), S( * )
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
- 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.