CPBEQU(1) LAPACK routine (version 3.2) CPBEQU(1)NAME
CPBEQU - computes row and column scalings intended to equilibrate a
Hermitian positive definite band matrix A and reduce its condition num‐
ber (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 Her‐
mitian 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 condi‐
tion 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 subdiagonals 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 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
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 routine (version 3.2) November 2008 CPBEQU(1)