naginterfaces.library.lapacklin.zpftri

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

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://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/f07/f07wwf.html

Parameters
transrstr, length 1

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

The matrix is stored in normal RFP format.

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

uplostr, length 1

Specifies how has been factorized.

, where is upper triangular.

, where is lower triangular.

nint

, the order of the matrix .

arcomplex, array-like, shape

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

Returns
arcomplex, ndarray, shape

The factorization is overwritten by the matrix stored in RFP format.

Raises
NagValueError
(errno )

On entry, error in parameter .

Constraint: or .

(errno )

On entry, error in parameter .

Constraint: or .

(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 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 , 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 .

If , and is computed by first inverting and then forming .

If , and is computed by first inverting and then forming .

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)