
ZPBEQU(l) ) ZPBEQU(l)
NAME
ZPBEQU  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 twonorm)
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 equilibrate a Hermitian positive defi
nite band matrix A and reduce its condition number (with respect to the twonorm). 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.
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 sub
diagonals 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 jth column of A is stored in the jth column of the
array AB as follows: if UPLO = 'U', AB(kd+1+ij,j) = A(i,j) for max(1,jkd)<=i<=j;
if UPLO = 'L', AB(1+ij,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 ith argument had an illegal value.
> 0: if INFO = i, the ith diagonal element is nonpositive.
LAPACK version 3.0 15 June 2000 ZPBEQU(l) 
