naginterfaces.library.lapacklin.zpftri

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

zpftri computes the inverse of a complex Hermitian positive definite matrix using the Cholesky factorization computed by zpftrf() stored in Rectangular Full Packed (RFP) format.

For full information please refer to the NAG Library document for f07ww

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f07/f07wwf.html

Parameters

transrstr, length 1

Specifies whether the normal RFP representation of $A$ or its conjugate transpose is stored.

$transr=\text{'N'}$

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

$transr=\text{'C'}$

The conjugate transpose of the RFP representation of the matrix $A$ is stored.

uplostr, length 1

Specifies how $A$ has been factorized.

$uplo=\text{'U'}$

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

$uplo=\text{'L'}$

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

nint

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

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

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

Returns

arcomplex, 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 $\text{'C'}$.

(errno $-2$)

On entry, error in parameter $uplo$.

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

(errno $-3$)

On entry, error in parameter $n$.

Constraint: $n\ge 0$.

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

Notes

zpftri is used to compute the inverse of a complex Hermitian 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 zpftrf(), which computes the Cholesky factorization of $A$.

If $uplo=\text{'U'}$, $A=U^{H}U$ and $A^{-1}$ is computed by first inverting $U$ and then forming $\left(U^{-1}\right)U^{-H}$.

If $uplo=\text{'L'}$, $A=L{L}^H$ and $A^{-1}$ is computed by first inverting $L$ and then forming $L^{-H}\left(L^{-1}\right)$.

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)