DGBEQU - compute row and column scalings intended to equi
librate an M-by-N band matrix A and reduce its condition
number
SYNOPSIS
SUBROUTINE DGBEQU( M, N, KL, KU, AB, LDAB, R, C, ROWCND,
COLCND, AMAX, INFO )
INTEGER INFO, KL, KU, LDAB, M, N
DOUBLE PRECISION AMAX, COLCND, ROWCND
DOUBLE PRECISION AB( LDAB, * ), C( * ), R( * )
PURPOSE
DGBEQU computes row and column scalings intended to equi
librate an M-by-N band matrix A and reduce its condition
number. R returns the row scale factors and C the column
scale factors, chosen to try to make the largest element
in each row and column of the matrix B with elements
B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.
R(i) and C(j) are restricted to be between SMLNUM = small
est safe number and BIGNUM = largest safe number. Use of
these scaling factors is not guaranteed to reduce the con
dition number of A but works well in practice.
ARGUMENTS
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
KL (input) INTEGER
The number of subdiagonals within the band of A.
KL >= 0.
KU (input) INTEGER
The number of superdiagonals within the band of A.
KU >= 0.
AB (input) DOUBLE PRECISION array, dimension (LDAB,N)
The band matrix A, stored in rows 1 to KL+KU+1.
The j-th column of A is stored in the j-th column
of the array AB as follows: AB(ku+1+i-j,j) =
A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >=
KL+KU+1.
If INFO = 0, or INFO > M, R contains the row scale
factors for A.
C (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, C contains the column scale factors
for A.
ROWCND (output) DOUBLE PRECISION
If INFO = 0 or INFO > M, ROWCND contains the ratio
of the smallest R(i) to the largest R(i). If
ROWCND >= 0.1 and AMAX is neither too large nor
too small, it is not worth scaling by R.
COLCND (output) DOUBLE PRECISION
If INFO = 0, COLCND contains the ratio of the
smallest C(i) to the largest C(i). If COLCND >=
0.1, it is not worth scaling by C.
AMAX (output) DOUBLE PRECISION
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, and i is
<= M: the i-th row of A is exactly zero
> M: the (i-M)-th column of A is exactly zero
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