CPBEQU(l) ) CPBEQU(l)
CPBEQU - compute row and column scalings intended to equilibrate a Hermitian positive def-
inite band matrix A and reduce its condition number (with respect to the two-norm)
SUBROUTINE CPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
INTEGER INFO, KD, LDAB, N
REAL AMAX, SCOND
REAL S( * )
COMPLEX AB( LDAB, * )
CPBEQU computes row and column scalings intended to equilibrate a Hermitian 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) COMPLEX array, dimension (LDAB,N)
The upper or lower triangle of the Hermitian 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
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.
LAPACK version 3.0 15 June 2000 CPBEQU(l)