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# spbequ

```
NAME
SPBEQU - compute row and column scalings intended to equili-
brate a symmetric positive definite band matrix A and reduce
its condition number (with respect to the two-norm)

SYNOPSIS
SUBROUTINE SPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX,
INFO )

CHARACTER      UPLO

INTEGER        INFO, KD, LDAB, N

REAL           AMAX, SCOND

REAL           AB( LDAB, * ), S( * )

PURPOSE
SPBEQU computes row and column scalings intended to equili-
brate a symmetric positive definite 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 diago-
nal 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) REAL array, dimension (LDAB,N)
The upper or lower triangle of the symmetric 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 under-
flow, 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 entry is nonpo-
sitive.
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