naginterfaces.library.lapacklin.dpftrf

naginterfaces.library.lapacklin.dpftrf(transr, uplo, n, ar)

dpftrf computes the Cholesky factorization of a real symmetric positive definite matrix stored in Rectangular Full Packed (RFP) format.

For full information please refer to the NAG Library document for f07wd

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f07/f07wdf.html

Parameters

transrstr, length 1

Specifies whether the RFP representation of $A$ is normal or transposed.

$transr=\text{'N'}$

The matrix $A$ is stored in normal RFP format.

$transr=\text{'T'}$

The matrix $A$ is stored in transposed RFP format.

uplostr, length 1

Specifies whether the upper or lower triangular part of $A$ is stored.

$uplo=\text{'U'}$

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.

nint

$n$, the order of the matrix $A$.

arfloat, array-like, shape $(n\times (n+1)/2)$

The upper or lower triangular part (as specified by $uplo$) of the $n\times n$ symmetric matrix $A$, in either normal or transposed RFP format (as specified by $transr$). The storage format is described in detail in the F07 Introduction.

Returns

arfloat, ndarray, shape $(n\times (n+1)/2)$

If no exception or warning is raised, the factor $U$ or $L$ from the Cholesky factorization $A=U^{T}U$ or $A=L{L}^T$, in the same storage format as $A$.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $transr$.

Constraint: $transr=\text{'N'}$ or $\text{'T'}$.

(errno $-2$)

On entry, error in parameter $uplo$.

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

(errno $-3$)

On entry, error in parameter $n$.

Constraint: $n\ge 0$.

Warns

NagAlgorithmicWarning

(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 matrix stored in RFP format which is not positive definite; the matrix must be treated as a full symmetric matrix, by calling dsytrf().

Notes

dpftrf forms the Cholesky factorization of a real symmetric positive definite matrix $A$ either as $A=U^{T}U$ if $uplo=\text{'U'}$ or $A=L{L}^T$ if $uplo=\text{'L'}$, where $U$ is an upper triangular matrix and $L$ is a lower triangular, stored in RFP format. The RFP storage format is described in the F07 Introduction.

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

Gustavson, F G, Waśniewski, J, Dongarra, J J and Langou, J, 2010, Rectangular full packed format for Cholesky’s algorithm: factorization, solution, and inversion, ACM Trans. Math. Software (37, 2)