naginterfaces.library.lapacklin.dpbtrf

naginterfaces.library.lapacklin.dpbtrf(uplo, kd, ab)

dpbtrf computes the Cholesky factorization of a real symmetric positive definite band matrix.

For full information please refer to the NAG Library document for f07hd

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f07/f07hdf.html

Parameters

uplostr, length 1

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

$uplo=\text{'U'}$

The upper triangular part of $A$ is stored and $A$ is factorized as $U^{T}U$, where $U$ is upper triangular.

$uplo=\text{'L'}$

The lower triangular part of $A$ is stored and $A$ is factorized as $L{L}^T$, where $L$ is lower triangular.

kdint

$k_d$, the number of superdiagonals or subdiagonals of the matrix $A$.

abfloat, array-like, shape $(kd+1,n)$

The $n\times n$ symmetric positive definite band matrix $A$.

Returns

abfloat, ndarray, shape $(kd+1,n)$

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

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $uplo$.

Constraint: $uplo=\text{'U'}$ or $\text{'L'}$.

(errno $-2$)

On entry, error in parameter $\text{n}$.

Constraint: $n\ge 0$.

(errno $-3$)

On entry, error in parameter $kd$.

Constraint: $kd\ge 0$.

(errno $i>0$)

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

Notes

dpbtrf forms the Cholesky factorization of a real symmetric positive definite band matrix $A$ either as $A=U^{T}U$ if $uplo=\text{'U'}$ or $A=L{L}^T$ if $uplo=\text{'L'}$, where $U$ (or $L$) is an upper (or lower) triangular band matrix with the same number of superdiagonals (or subdiagonals) as $A$.

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