naginterfaces.library.lapacklin.dpbrfs

naginterfaces.library.lapacklin.dpbrfs(uplo, kd, nrhs, ab, afb, b, x)[source]

dpbrfs returns error bounds for the solution of a real symmetric positive definite band system of linear equations with multiple right-hand sides, . 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 f07hh

https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/f07/f07hhf.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.

kdint

, the number of superdiagonals or subdiagonals of the matrix .

nrhsint

, the number of right-hand sides.

abfloat, array-like, shape

The original symmetric positive definite band matrix as supplied to dpbtrf().

afbfloat, array-like, shape

The Cholesky factor of , as returned by dpbtrf().

bfloat, array-like, shape

The right-hand side matrix .

xfloat, array-like, shape

The solution matrix , as returned by dpbtrs().

Returns
xfloat, 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: .

Notes

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