naginterfaces.library.lapacklin.dpftri

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

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

For full information please refer to the NAG Library document for f07wj

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f07/f07wjf.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 how $A$ has been factorized.

$uplo=\text{'U'}$

$A=U^{T}U$, where $U$ is upper triangular.

$uplo=\text{'L'}$

$A=L{L}^T$, where $L$ is lower triangular.

nint

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

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

The Cholesky factorization of $A$ stored in RFP format, as returned by dpftrf().

Returns

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

The factorization is overwritten by the $n\times n$ matrix $A^{-1}$ stored in RFP format.

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

Notes

dpftri is used to compute the inverse of a real symmetric positive definite matrix $A$, stored in RFP format. The RFP storage format is described in the F07 Introduction. The function must be preceded by a call to dpftrf(), which computes the Cholesky factorization of $A$.

If $uplo=\text{'U'}$, $A=U^{T}U$ and $A^{-1}$ is computed by first inverting $U$ and then forming $\left(U^{-1}\right)U^{-T}$.

If $uplo=\text{'L'}$, $A=L{L}^T$ and $A^{-1}$ is computed by first inverting $L$ and then forming $L^{-T}\left(L^{-1}\right)$.

References

Du Croz, J J and Higham, N J, 1992, Stability of methods for matrix inversion, IMA J. Numer. Anal. (12), 1–19

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)