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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_ztrrfs (f07tv)

## Purpose

nag_lapack_ztrrfs (f07tv) returns error bounds for the solution of a complex triangular system of linear equations with multiple right-hand sides, $AX=B$, ${A}^{\mathrm{T}}X=B$ or ${A}^{\mathrm{H}}X=B$.

## Syntax

[ferr, berr, info] = f07tv(uplo, trans, diag, a, b, x, 'n', n, 'nrhs_p', nrhs_p)
[ferr, berr, info] = nag_lapack_ztrrfs(uplo, trans, diag, a, b, x, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_ztrrfs (f07tv) returns the backward errors and estimated bounds on the forward errors for the solution of a complex triangular system of linear equations with multiple right-hand sides $AX=B$, ${A}^{\mathrm{T}}X=B$ or ${A}^{\mathrm{H}}X=B$. 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_ztrrfs (f07tv) 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 $AX=B$.
${\mathbf{trans}}=\text{'T'}$
The equations are of the form ${A}^{\mathrm{T}}X=B$.
${\mathbf{trans}}=\text{'C'}$
The equations are of the form ${A}^{\mathrm{H}}X=B$.
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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $n$ by $n$ triangular matrix $A$.
• If ${\mathbf{uplo}}=\text{'U'}$, $a$ is upper triangular and the elements of the array below the diagonal are not referenced.
• If ${\mathbf{uplo}}=\text{'L'}$, $a$ is lower triangular and the elements of the array above the diagonal are not referenced.
• If ${\mathbf{diag}}=\text{'U'}$, the diagonal elements of $a$ are assumed to be $1$, and are not referenced.
5:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex 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 $B$.
6:     $\mathrm{x}\left(\mathit{ldx},:\right)$ – complex 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_ztrtrs (f07ts).

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the arrays a, b, x and the second dimension of the array a.
$n$, 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. (An error is raised if these dimensions are not equal.)
$r$, 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_ztrrfs (f07tv), for each right-hand side, involves solving a number of systems of linear equations of the form $Ax=b$ or ${A}^{\mathrm{H}}x=b$; the number is usually $5$ and never more than $11$. Each solution involves approximately $4{n}^{2}$ real floating-point operations.
The real analogue of this function is nag_lapack_dtrrfs (f07th).

## Example

This example solves the system of equations $AX=B$ and to compute forward and backward error bounds, where
 $A= 4.78+4.56i 0.00+0.00i 0.00+0.00i 0.00+0.00i 2.00-0.30i -4.11+1.25i 0.00+0.00i 0.00+0.00i 2.89-1.34i 2.36-4.25i 4.15+0.80i 0.00+0.00i -1.89+1.15i 0.04-3.69i -0.02+0.46i 0.33-0.26i$
and
 $B= -14.78-32.36i -18.02+28.46i 2.98-02.14i 14.22+15.42i -20.96+17.06i 5.62+35.89i 9.54+09.91i -16.46-01.73i .$
```function f07tv_example

fprintf('f07tv example results\n\n');

% Solve AX=B  and get error bounds, where A is Lower triangular
a = [  4.78 +  4.56i,   0    +  0i,     0    + 0i,    0    + 0i;
2.00 -  0.30i,  -4.11 +  1.25i,  0    + 0i,    0    + 0i;
2.89 -  1.34i,   2.36 -  4.25i,  4.15 + 0.8i,  0    + 0i;
-1.89 +  1.15i,   0.04 -  3.69i, -0.02 + 0.46i, 0.33 - 0.26i];
b = [-14.78 - 32.36i, -18.02 + 28.46i;
2.98 -  2.14i,  14.22 + 15.42i;
-20.96 + 17.06i,   5.62 + 35.89i;
9.54 +  9.91i, -16.46 -  1.73i];

% Solve
uplo  = 'L';
trans = 'N';
diag  = 'N';
[x, info] = f07ts( ...
uplo, trans, diag, a, b);

% Get error bounds
[ferr, berr, info] = f07tv( ...
uplo, trans, diag, a, b, x);

disp('Solution(s)');
disp(x);

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

```
```f07tv example results

Solution(s)
-5.0000 - 2.0000i   1.0000 + 5.0000i
-3.0000 - 1.0000i  -2.0000 - 2.0000i
2.0000 + 1.0000i   3.0000 + 4.0000i
4.0000 + 3.0000i   4.0000 - 3.0000i

Backward errors (machine-dependent)
6.2e-17    3.5e-17
Estimated forward error bounds (machine-dependent)
2.9e-14    3.2e-14
```