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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dtbrfs (f07vh)

## Purpose

nag_lapack_dtbrfs (f07vh) returns error bounds for the solution of a real triangular band system of linear equations with multiple right-hand sides, $AX=B$ or ${A}^{\mathrm{T}}X=B$.

## Syntax

[ferr, berr, info] = f07vh(uplo, trans, diag, kd, ab, b, x, 'n', n, 'nrhs_p', nrhs_p)
[ferr, berr, info] = nag_lapack_dtbrfs(uplo, trans, diag, kd, ab, b, x, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_dtbrfs (f07vh) 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 ${A}^{\mathrm{T}}X=B$. The function handles each right-hand side vector (stored as a column of the matrix $B$) independently, so we describe the function of nag_lapack_dtbrfs (f07vh) 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
 $A+δAx=b+δb δaij≤βaij and δbi≤βbi .$
Then the function estimates a bound for the component-wise forward error in the computed solution, defined by:
 $maxixi-x^i/maxixi$
where $\stackrel{^}{x}$ is the true solution.
For details of the method, see the F07 Chapter Introduction.

## References

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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{uplo}$ – string (length ≥ 1)
Specifies whether $A$ is upper or lower triangular.
${\mathbf{uplo}}=\text{'U'}$
$A$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathrm{trans}$ – string (length ≥ 1)
Indicates the form of the equations.
${\mathbf{trans}}=\text{'N'}$
The equations are of the form $AX=B$.
${\mathbf{trans}}=\text{'T'}$ or $\text{'C'}$
The equations are of the form ${A}^{\mathrm{T}}X=B$.
Constraint: ${\mathbf{trans}}=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.
3:     $\mathrm{diag}$ – string (length ≥ 1)
Indicates whether $A$ is a nonunit or unit triangular matrix.
${\mathbf{diag}}=\text{'N'}$
$A$ is a nonunit triangular matrix.
${\mathbf{diag}}=\text{'U'}$
$A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$.
Constraint: ${\mathbf{diag}}=\text{'N'}$ or $\text{'U'}$.
4:     $\mathrm{kd}$int64int32nag_int scalar
${k}_{d}$, the number of superdiagonals of the matrix $A$ if ${\mathbf{uplo}}=\text{'U'}$, or the number of subdiagonals if ${\mathbf{uplo}}=\text{'L'}$.
Constraint: ${\mathbf{kd}}\ge 0$.
5:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – double array
The first dimension of the array ab must be at least ${\mathbf{kd}}+1$.
The second dimension of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $n$ by $n$ triangular band matrix $A$.
The matrix is stored in rows $1$ to ${k}_{d}+1$, more precisely,
• if ${\mathbf{uplo}}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{ab}}\left({k}_{d}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{d}\right)\le i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{ab}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)\text{.}$
If ${\mathbf{diag}}=\text{'U'}$, the diagonal elements of $A$ are assumed to be $1$, and are not referenced.
6:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double array
The first dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
The $n$ by $r$ right-hand side matrix $B$.
7:     $\mathrm{x}\left(\mathit{ldx},:\right)$ – double array
The first dimension of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
The $n$ by $r$ solution matrix $X$, as returned by nag_lapack_dtbtrs (f07ve).

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the second dimension of the array ab.
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathrm{nrhs_p}$int64int32nag_int scalar
Default: the second dimension of the arrays b, x.
$r$, the number of right-hand sides.
Constraint: ${\mathbf{nrhs_p}}\ge 0$.

### Output Parameters

1:     $\mathrm{ferr}\left({\mathbf{nrhs_p}}\right)$ – double array
${\mathbf{ferr}}\left(\mathit{j}\right)$ contains an estimated error bound for the $\mathit{j}$th solution vector, that is, the $\mathit{j}$th column of $X$, for $\mathit{j}=1,2,\dots ,r$.
2:     $\mathrm{berr}\left({\mathbf{nrhs_p}}\right)$ – double array
${\mathbf{berr}}\left(\mathit{j}\right)$ contains the component-wise backward error bound $\beta$ for the $\mathit{j}$th solution vector, that is, the $\mathit{j}$th column of $X$, for $\mathit{j}=1,2,\dots ,r$.
3:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## Accuracy

The bounds returned in ferr are not rigorous, because they are estimated, not computed exactly; but in practice they almost always overestimate the actual error.

A call to nag_lapack_dtbrfs (f07vh), for each right-hand side, involves solving a number of systems of linear equations of the form $Ax=b$ or ${A}^{\mathrm{T}}x=b$; the number is usually $4$ or $5$ and never more than $11$. Each solution involves approximately $2nk$ floating-point operations (assuming $n\gg k$).
The complex analogue of this function is nag_lapack_ztbrfs (f07vv).

## Example

This example solves the system of equations $AX=B$ and to compute forward and backward error bounds, where
 $A= -4.16 0.00 0.00 0.00 -2.25 4.78 0.00 0.00 0.00 5.86 6.32 0.00 0.00 0.00 -4.82 0.16 and B= -16.64 -4.16 -13.78 -16.59 13.10 -4.94 -14.14 -9.96 .$
```function f07vh_example

fprintf('f07vh example results\n\n');

% Solve AX=B and compute error bounds, where A is lower triangular banded
% and stored in triangular/symmetric banded format
kd = int64(1);
ab = [-4.16, 4.78,  6.32, 0.16;
-2.25, 5.86, -4.82, 0.00];
b = [-16.64,  -4.16;
-13.78, -16.59;
13.10,  -4.94;
-14.14,  -9.96];

% Solve
uplo  = 'L';
trans = 'N';
diag  = 'N';
[x, info] = f07ve( ...
uplo, trans, diag, kd, ab, b);

% Compute error bounds
[ferr, berr, info] = f07vh( ...
uplo, trans, diag, kd, ab, b, x);

% Display solution
[ifail] = x04ca( ...
'Gen', diag, x, 'Solution(s)');

fprintf('\nBackward errors (machine-dependent)\n   ')
fprintf('%11.1e', berr);
fprintf('\nEstimated forward error bounds (machine-dependent)\n   ')
fprintf('%11.1e', ferr);
fprintf('\n');

```
```f07vh example results

Solution(s)
1          2
1      4.0000     1.0000
2     -1.0000    -3.0000
3      3.0000     2.0000
4      2.0000    -2.0000

Backward errors (machine-dependent)
4.7e-17    2.5e-17
Estimated forward error bounds (machine-dependent)
5.4e-14    5.8e-14
```