naginterfaces.library.lapacklin.zgbrfs

naginterfaces.library.lapacklin.zgbrfs(trans, kl, ku, nrhs, ab, afb, ipiv, b, x)[source]

zgbrfs returns error bounds for the solution of a complex band system of linear equations with multiple right-hand sides, , or . 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 f07bv

https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/f07/f07bvf.html

Parameters
transstr, length 1

Indicates the form of the linear equations for which is the computed solution as follows:

The linear equations are of the form .

The linear equations are of the form .

The linear equations are of the form .

klint

, the number of subdiagonals within the band of the matrix .

kuint

, the number of superdiagonals within the band of the matrix .

nrhsint

, the number of right-hand sides.

abcomplex, array-like, shape

The original band matrix as supplied to zgbtrf() but with reduced requirements since the matrix is not factorized.

See Further Comments for further details.

afbcomplex, array-like, shape

The factorization of , as returned by zgbtrf().

ipivint, array-like, shape

The pivot indices, as returned by zgbtrf().

bcomplex, array-like, shape

The right-hand side matrix .

xcomplex, array-like, shape

The solution matrix , as returned by zgbtrs().

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: .

(errno )

On entry, error in parameter .

Constraint: .

(errno )

On entry, error in parameter .

Constraint: .

Notes

zgbrfs returns the backward errors and estimated bounds on the forward errors for the solution of a complex band system of linear equations with multiple right-hand sides , or . The function handles each right-hand side vector (stored as a column of the matrix ) independently, so we describe the function of zgbrfs 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