ZPBTF2(3) computes the Cholesky factorization of a complex Hermitian positive definite band matrix A

## SYNOPSIS

SUBROUTINE ZPBTF2(
UPLO, N, KD, AB, LDAB, INFO )

CHARACTER UPLO

INTEGER INFO, KD, LDAB, N

COMPLEX*16 AB( LDAB, * )

## PURPOSE

ZPBTF2 computes the Cholesky factorization of a complex Hermitian positive definite band matrix A. The factorization has the form

A = U' * U ,  if UPLO = 'U', or

A = L  * L',  if UPLO = 'L',
where U is an upper triangular matrix, U' is the conjugate transpose of U, and L is lower triangular.
This is the unblocked version of the algorithm, calling Level 2 BLAS.

## ARGUMENTS

UPLO (input) CHARACTER*1
Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular
N (input) INTEGER
The order of the matrix A. N >= 0.
KD (input) INTEGER
The number of super-diagonals of the matrix A if UPLO = 'U', or the number of sub-diagonals if UPLO = 'L'. KD >= 0.
AB (input/output) COMPLEX*16 array, dimension (LDAB,N)
On entry, 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). On exit, if INFO = 0, the triangular factor U or L from the Cholesky factorization A = U'*U or A = L*L' 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 = -k, the k-th argument had an illegal value
> 0: if INFO = k, the leading minor of order k 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.