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# zpbtrf

```
NAME
ZPBTRF - compute the Cholesky factorization of a complex
Hermitian positive definite band matrix A

SYNOPSIS
SUBROUTINE ZPBTRF( UPLO, N, KD, AB, LDAB, INFO )

CHARACTER      UPLO

INTEGER        INFO, KD, LDAB, N

COMPLEX*16     AB( LDAB, * )

PURPOSE
ZPBTRF computes the Cholesky factorization of a complex Her-
mitian positive definite band matrix A.

The factorization has the form
A = U**H * U,  if UPLO = 'U', or
A = L  * L**H,  if UPLO = 'L',
where U is an upper triangular matrix and L is lower tri-
angular.

ARGUMENTS
UPLO    (input) CHARACTER*1
= 'U':  Upper triangle of A is stored;
= 'L':  Lower triangle 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/output) COMPLEX*16 array, dimension (LDAB,N)
On entry, the upper or lower triangle of the Hermi-
tian 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).

On exit, if INFO = 0, the triangular factor U or L
from the Cholesky factorization A = U**H*U or A =
L*L**H of the band matrix A, in the same storage
format as A.

LDAB    (input) INTEGER

The leading dimension of the array AB.  LDAB >=
KD+1.

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal
value
> 0:  if INFO = i, the leading minor of order i is
not positive definite, and the factorization could
not be completed.

FURTHER DETAILS
The band storage scheme is illustrated by the following
example, when N = 6, KD = 2, and UPLO = 'U':

On entry:                       On exit:

*    *   a13  a24  a35  a46      *    *   u13  u24  u35
u46
*   a12  a23  a34  a45  a56      *   u12  u23  u34  u45
u56
a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55
u66

Similarly, if UPLO = 'L' the format of A is as follows:

On entry:                       On exit:

a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55
l66
a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65
*
a31  a42  a53  a64   *    *      l31  l42  l53  l64   *
*

Array elements marked * are not used by the routine.

Contributed by
Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March
23, 1989
```