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# cgbequ

```
NAME
CGBEQU - compute row and column scalings intended to equili-
brate an M by N band matrix A and reduce its condition
number

SYNOPSIS
SUBROUTINE CGBEQU( M, N, KL, KU, AB, LDAB, R, C, ROWCND,
COLCND, AMAX, INFO )

INTEGER        INFO, KL, KU, LDAB, M, N

REAL           AMAX, COLCND, ROWCND

REAL           C( * ), R( * )

COMPLEX        AB( LDAB, * )

PURPOSE
CGBEQU computes row and column scalings intended to equili-
brate an M by N band matrix A and reduce its condition
number.  R returns the row scale factors and C the column
scale factors, chosen to try to make the largest element in
each row and column of the matrix B with elements
B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.

R(i) and C(j) *  are restricted to be between SMLNUM = smal-
lest safe number and BIGNUM = largest safe number.  Use of
these scaling factors is not guaranteed to reduce the condi-
tion number of A but works well in practice.

ARGUMENTS
M       (input) INTEGER
The number of rows of the matrix A.  M >= 0.

N       (input) INTEGER
The number of columns of the matrix A.  N >= 0.

KL      (input) INTEGER
The number of subdiagonals within the band of A.  KL
>= 0.

KU      (input) INTEGER
The number of superdiagonals within the band of A.
KU >= 0.

AB      (input) COMPLEX array, dimension (LDAB,N)
The band matrix A, stored in rows 1 to KL+KU+1.  The
j-th column of A is stored in the j-th column of the
array AB as follows: AB(ku+1+i-j,j) = A(i,j) for
max(1,j-ku)<=i<=min(m,j+kl).

LDAB    (input) INTEGER
The leading dimension of the array AB.  LDAB >=
KL+KU+1.

R       (output) REAL array, dimension (M)
If INFO = 0, or INFO > M, R contains the row scale
factors for A.

C       (output) REAL array, dimension (N)
If INFO = 0, C contains the column scale factors for
A.

ROWCND  (output) REAL
If INFO = 0 or INFO > M, ROWCND contains the ratio
of the smallest R(i) to the largest R(i).  If ROWCND
>= 0.1 and AMAX is neither too large nor too small,
it is not worth scaling by R.

COLCND  (output) REAL
If INFO = 0, COLCND contains the ratio of the smal-
lest C(i) to the largest C(i).  If COLCND >= 0.1, it
is not worth scaling by C.

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, and i is
<= M:  the i-th row of A is exactly zero
>  M:  the (i-M)-th column of A is exactly zero
```