NAME¶

ZPBTF2 - 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.