naginterfaces.library.matop.real_​symm_​posdef_​inv

naginterfaces.library.matop.real_symm_posdef_inv(a)[source]

real_symm_posdef_inv calculates the accurate inverse of a real symmetric positive definite matrix, using a Cholesky factorization and iterative refinement.

For full information please refer to the NAG Library document for f01ab

https://www.nag.com/numeric/nl/nagdoc_28.4/flhtml/f01/f01abf.html

Parameters
afloat, array-like, shape

The upper triangle of the positive definite symmetric matrix . The elements of the array below the diagonal need not be set.

Returns
afloat, ndarray, shape

The lower triangle of the inverse matrix is stored in the elements of the array below the diagonal, in rows to ; is stored in for . The upper triangle of the original matrix is unchanged.

bfloat, ndarray, shape

The lower triangle of the inverse matrix , with stored in , for .

Raises
NagValueError
(errno )

The matrix is not positive definite, possibly due to rounding errors.

(errno )

The refinement process failed to converge. The matrix is ill-conditioned.

(errno )

On entry, .

Constraint: .

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

To compute the inverse of a real symmetric positive definite matrix , real_symm_posdef_inv first computes a Cholesky factorization of as , where is lower triangular. An approximation to is found by computing and then the product . The residual matrix is calculated using additional precision, and a correction to is found by solving . is replaced by , and this iterative refinement of the inverse is repeated until full machine accuracy has been obtained.

References

Wilkinson, J H and Reinsch, C, 1971, Handbook for Automatic Computation II, Linear Algebra, Springer–Verlag