naginterfaces.library.lapacklin.dtbrfs

naginterfaces.library.lapacklin.dtbrfs(uplo, trans, diag, kd, nrhs, ab, b, x)[source]

dtbrfs returns error bounds for the solution of a real triangular band system of linear equations with multiple right-hand sides, or .

For full information please refer to the NAG Library document for f07vh

https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/f07/f07vhf.html

Parameters
uplostr, length 1

Specifies whether is upper or lower triangular.

is upper triangular.

is lower triangular.

transstr, length 1

Indicates the form of the equations.

The equations are of the form .

or

The equations are of the form .

diagstr, length 1

Indicates whether is a nonunit or unit triangular matrix.

is a nonunit triangular matrix.

is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be .

kdint

, the number of superdiagonals of the matrix if , or the number of subdiagonals if .

nrhsint

, the number of right-hand sides.

abfloat, array-like, shape

The triangular band matrix .

bfloat, array-like, shape

The right-hand side matrix .

xfloat, array-like, shape

The solution matrix , as returned by dtbtrs().

Returns
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: , or .

(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

dtbrfs returns the backward errors and estimated bounds on the forward errors for the solution of a real triangular 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 dtbrfs 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