DPBEQU(l) ) DPBEQU(l)
DPBEQU - compute row and column scalings intended to equilibrate a symmetric positive def-
inite band matrix A and reduce its condition number (with respect to the two-norm)
SUBROUTINE DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
INTEGER INFO, KD, LDAB, N
DOUBLE PRECISION AMAX, SCOND
DOUBLE PRECISION AB( LDAB, * ), S( * )
DPBEQU computes row and column scalings intended to equilibrate a symmetric positive defi-
nite band matrix A and reduce its condition number (with respect to the two-norm). S con-
tains 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.
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 sub-
diagonals 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.
S (output) DOUBLE PRECISION array, dimension (N)
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
AMAX (output) DOUBLE PRECISION
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.
LAPACK version 3.0 15 June 2000 DPBEQU(l)