DPBEQU - 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 DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX,
INFO )
CHARACTER UPLO
INTEGER INFO, KD, LDAB, N
DOUBLE PRECISION AMAX, SCOND
DOUBLE PRECISION AB( LDAB, * ), S( * )
PURPOSE
DPBEQU 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) DOUBLE PRECISION 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) DOUBLE PRECISION
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) 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, the i-th diagonal element is
nonpositive.
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