# naginterfaces.library.lapacklin.zpbtrf¶

naginterfaces.library.lapacklin.zpbtrf(uplo, kd, ab)[source]

zpbtrf computes the Cholesky factorization of a complex Hermitian positive definite band matrix.

For full information please refer to the NAG Library document for f07hr

https://www.nag.com/numeric/nl/nagdoc_29/flhtml/f07/f07hrf.html

Parameters
uplostr, length 1

Specifies whether the upper or lower triangular part of is stored and how is to be factorized.

The upper triangular part of is stored and is factorized as , where is upper triangular.

The lower triangular part of is stored and is factorized as , where is lower triangular.

kdint

, the number of superdiagonals or subdiagonals of the matrix .

abcomplex, array-like, shape

The Hermitian positive definite band matrix .

Returns
abcomplex, ndarray, shape

The upper or lower triangle of is overwritten by the Cholesky factor or as specified by , using the same storage format as described above.

Raises
NagValueError
(errno )

On entry, error in parameter .

Constraint: or .

(errno )

On entry, error in parameter .

Constraint: .

(errno )

On entry, error in parameter .

Constraint: .

(errno )

The leading minor of order is not positive definite and the factorization could not be completed. Hence itself is not positive definite. This may indicate an error in forming the matrix . There is no function specifically designed to factorize a Hermitian band matrix which is not positive definite; the matrix must be treated either as a nonsymmetric band matrix, by calling zgbtrf() or as a full Hermitian matrix, by calling zhetrf().

Notes

zpbtrf forms the Cholesky factorization of a complex Hermitian positive definite band matrix either as if or if , where (or ) is an upper (or lower) triangular band matrix with the same number of superdiagonals (or subdiagonals) as .

References

Demmel, J W, 1989, On floating-point errors in Cholesky, LAPACK Working Note No. 14, University of Tennessee, Knoxville, https://www.netlib.org/lapack/lawnspdf/lawn14.pdf

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore