Provided by: liblapack-doc_3.3.1-1_all

#### NAME

```       LAPACK-3  -  computes row and column scalings intended to equilibrate a Hermitian positive
definite band matrix A and reduce its condition number (with respect to the two-norm)

```

#### SYNOPSIS

```       SUBROUTINE CPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )

CHARACTER      UPLO

INTEGER        INFO, KD, LDAB, N

REAL           AMAX, SCOND

REAL           S( * )

COMPLEX        AB( LDAB, * )

```

#### PURPOSE

```       CPBEQU computes row and column scalings  intended  to  equilibrate  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 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 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 underflow, 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 element is nonpositive.

LAPACK routine (version 3.2)               April 2011                            CPBEQU(3lapack)
```