zpbequ(3)

NAME

ZPBEQU - compute row and column scalings intended to equi

librate a Hermitian positive definite band matrix A and reduce

its condition number (with respect to the two-norm)

SYNOPSIS

SUBROUTINE ZPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND,  AMAX,
INFO )
    CHARACTER      UPLO
    INTEGER        INFO, KD, LDAB, N
    DOUBLE         PRECISION AMAX, SCOND
    DOUBLE         PRECISION S( * )
    COMPLEX*16     AB( LDAB, * )

PURPOSE

ZPBEQU computes row and column scalings intended to equi

librate a Hermitian 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 condition number 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) COMPLEX*16 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 fol

lows: 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 by S.

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 ille

gal value.
> 0: if INFO = i, the i-th diagonal element is

nonpositive.

LAPACK version 3.0 15 June 2000

ZPBEQU(3)