naginterfaces.library.lapacklin.dpftri

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

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://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/f07/f07wjf.html

Parameters
transrstr, length 1

Specifies whether the RFP representation of is normal or transposed.

The matrix is stored in normal RFP format.

The matrix is stored in transposed RFP format.

uplostr, length 1

Specifies how has been factorized.

, where is upper triangular.

, where is lower triangular.

nint

, the order of the matrix .

arfloat, array-like, shape

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

Returns
arfloat, 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: .

Warns
NagAlgorithmicWarning
(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 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 , 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 .

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)