naginterfaces.library.lapacklin.dtbrfs

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

dtbrfs returns error bounds for the solution of a real triangular band system of linear equations with multiple right-hand sides, $AX=B$ or $A^{T}X=B$.

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

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f07/f07vhf.html

Parameters

uplostr, length 1

Specifies whether $A$ is upper or lower triangular.

$uplo=\text{'U'}$

$A$ is upper triangular.

$uplo=\text{'L'}$

$A$ is lower triangular.

transstr, length 1

Indicates the form of the equations.

$trans=\text{'N'}$

The equations are of the form $AX=B$.

$trans=\text{'T'}$ or $\text{'C'}$

The equations are of the form $A^{T}X=B$.

diagstr, length 1

Indicates whether $A$ is a nonunit or unit triangular matrix.

$diag=\text{'N'}$

$A$ is a nonunit triangular matrix.

$diag=\text{'U'}$

$A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$.

kdint

$k_d$, the number of superdiagonals of the matrix $A$ if $uplo=\text{'U'}$, or the number of subdiagonals if $uplo=\text{'L'}$.

nrhsint

$r$, the number of right-hand sides.

abfloat, array-like, shape $(kd+1,n)$

The $n\times n$ triangular band matrix $A$.

bfloat, array-like, shape $(n,nrhs)$

The $n\times r$ right-hand side matrix $B$.

xfloat, array-like, shape $(n,nrhs)$

The $n\times r$ solution matrix $X$, as returned by dtbtrs().

Returns

ferrfloat, ndarray, shape $\left(nrhs\right)$

$ferr[\text{j}-1]$ contains an estimated error bound for the $\text{j}$th solution vector, that is, the $\text{j}$th column of $X$, for $\text{j}=1,2,\dots ,r$.

berrfloat, ndarray, shape $\left(nrhs\right)$

$berr[\text{j}-1]$ contains the component-wise backward error bound $\beta$ for the $\text{j}$th solution vector, that is, the $\text{j}$th column of $X$, for $\text{j}=1,2,\dots ,r$.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $uplo$.

Constraint: $uplo=\text{'U'}$ or $\text{'L'}$.

(errno $-2$)

On entry, error in parameter $trans$.

Constraint: $trans=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.

(errno $-3$)

On entry, error in parameter $diag$.

Constraint: $diag=\text{'N'}$ or $\text{'U'}$.

(errno $-4$)

On entry, error in parameter $\text{n}$.

Constraint: $n\ge 0$.

(errno $-5$)

On entry, error in parameter $kd$.

Constraint: $kd\ge 0$.

(errno $-6$)

On entry, error in parameter $nrhs$.

Constraint: $nrhs\ge 0$.

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 $AX=B$ or $A^{T}X=B$. The function handles each right-hand side vector (stored as a column of the matrix $B$) independently, so we describe the function of dtbrfs in terms of a single right-hand side $b$ and solution $x$.

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

$$\begin{array}{c}\begin{array}{c}(A+\delta A)x=b+\delta b\\ \left|\delta a_{ij}\right|\le \beta \left|a_{ij}\right|\text{and}\left|\delta b_i\right|\le \beta \left|b_i\right|\text{.}\end{array}\end{array}$$

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

$${max}_i|x_i-{\hat{x}}_i|/{max}_i\left|x_i\right|$$

where $\hat{x}$ 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