zpbequ(3P) Sun Performance Library zpbequ(3P)NAMEzpbequ - compute 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)
SYNOPSIS
SUBROUTINE ZPBEQU(UPLO, N, KD, A, LDA, SCALE, SCOND, AMAX, INFO)
CHARACTER * 1 UPLO
DOUBLE COMPLEX A(LDA,*)
INTEGER N, KD, LDA, INFO
DOUBLE PRECISION SCOND, AMAX
DOUBLE PRECISION SCALE(*)
SUBROUTINE ZPBEQU_64(UPLO, N, KD, A, LDA, SCALE, SCOND, AMAX,
INFO)
CHARACTER * 1 UPLO
DOUBLE COMPLEX A(LDA,*)
INTEGER*8 N, KD, LDA, INFO
DOUBLE PRECISION SCOND, AMAX
DOUBLE PRECISION SCALE(*)
F95 INTERFACE
SUBROUTINE PBEQU(UPLO, [N], KD, A, [LDA], SCALE, SCOND, AMAX,
[INFO])
CHARACTER(LEN=1) :: UPLO
COMPLEX(8), DIMENSION(:,:) :: A
INTEGER :: N, KD, LDA, INFO
REAL(8) :: SCOND, AMAX
REAL(8), DIMENSION(:) :: SCALE
SUBROUTINE PBEQU_64(UPLO, [N], KD, A, [LDA], SCALE, SCOND, AMAX,
[INFO])
CHARACTER(LEN=1) :: UPLO
COMPLEX(8), DIMENSION(:,:) :: A
INTEGER(8) :: N, KD, LDA, INFO
REAL(8) :: SCOND, AMAX
REAL(8), DIMENSION(:) :: SCALE
C INTERFACE
#include <sunperf.h>
void zpbequ(char uplo, int n, int kd, doublecomplex *a, int lda, double
*scale, double *scond, double *amax, int *info);
void zpbequ_64(char uplo, long n, long kd, doublecomplex *a, long lda,
double *scale, double *scond, double *amax, long *info);
PURPOSEzpbequ 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.
ARGUMENTS
UPLO (input)
= 'U': Upper triangular of A is stored;
= 'L': Lower triangular of A is stored.
N (input) The order of the matrix A. N >= 0.
KD (input)
The number of superdiagonals of the matrix A if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'. KD >= 0.
A (input) 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 A as follows:
if UPLO = 'U', A(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', A(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
LDA (input)
The leading dimension of the array A. LDA >= KD+1.
SCALE (output)
If INFO = 0, SCALE contains the scale factors for A.
SCOND (output)
If INFO = 0, SCALE contains the ratio of the smallest
SCALE(i) to the largest SCALE(i). If SCOND >= 0.1 and AMAX
is neither too large nor too small, it is not worth scaling
by SCALE.
AMAX (output)
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)
= 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.
6 Mar 2009 zpbequ(3P)