# naginterfaces.library.lapacklin.zpprfs¶

naginterfaces.library.lapacklin.zpprfs(uplo, n, ap, afp, b, x)[source]

zpprfs returns error bounds for the solution of a complex Hermitian positive definite system of linear equations with multiple right-hand sides, , using packed storage. It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.

For full information please refer to the NAG Library document for f07gv

https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/f07/f07gvf.html

Parameters
uplostr, length 1

Specifies whether the upper or lower triangular part of is stored and how is to be factorized.

The upper triangular part of is stored and is factorized as , where is upper triangular.

The lower triangular part of is stored and is factorized as , where is lower triangular.

nint

, the order of the matrix .

apcomplex, array-like, shape

The original Hermitian positive definite matrix as supplied to zpptrf().

afpcomplex, array-like, shape

The Cholesky factor of stored in packed form, as returned by zpptrf().

bcomplex, array-like, shape

The right-hand side matrix .

xcomplex, array-like, shape

The solution matrix , as returned by zpptrs().

Returns
xcomplex, ndarray, shape

The improved solution matrix .

ferrfloat, ndarray, shape

contains an estimated error bound for the th solution vector, that is, the th column of , for .

berrfloat, ndarray, shape

contains the component-wise backward error bound for the th solution vector, that is, the th column of , for .

Raises
NagValueError
(errno )

On entry, error in parameter .

Constraint: or .

(errno )

On entry, error in parameter .

Constraint: .

(errno )

On entry, error in parameter .

Constraint: .

Notes

zpprfs returns the backward errors and estimated bounds on the forward errors for the solution of a complex Hermitian positive definite system of linear equations with multiple right-hand sides , using packed storage. The function handles each right-hand side vector (stored as a column of the matrix ) independently, so we describe the function of zpprfs in terms of a single right-hand side and solution .

Given a computed solution , the function computes the component-wise backward error . This is the size of the smallest relative perturbation in each element of and such that is the exact solution of a perturbed system

Then the function estimates a bound for the component-wise forward error in the computed solution, defined by:

where is the true solution.

For details of the method, see the F07 Introduction.

References

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore