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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgtrfs (f07ch)

## Purpose

nag_lapack_dgtrfs (f07ch) computes error bounds and refines the solution to a real system of linear equations $AX=B$ or ${A}^{\mathrm{T}}X=B$, where $A$ is an $n$ by $n$ tridiagonal matrix and $X$ and $B$ are $n$ by $r$ matrices, using the $LU$ factorization returned by nag_lapack_dgttrf (f07cd) and an initial solution returned by nag_lapack_dgttrs (f07ce). Iterative refinement is used to reduce the backward error as much as possible.

## Syntax

[x, ferr, berr, info] = f07ch(trans, dl, d, du, dlf, df, duf, du2, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)
[x, ferr, berr, info] = nag_lapack_dgtrfs(trans, dl, d, du, dlf, df, duf, du2, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_dgtrfs (f07ch) should normally be preceded by calls to nag_lapack_dgttrf (f07cd) and nag_lapack_dgttrs (f07ce). nag_lapack_dgttrf (f07cd) uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix $A$ as
 $A=PLU ,$
where $P$ is a permutation matrix, $L$ is unit lower triangular with at most one nonzero subdiagonal element in each column, and $U$ is an upper triangular band matrix, with two superdiagonals. nag_lapack_dgttrs (f07ce) then utilizes the factorization to compute a solution, $\stackrel{^}{X}$, to the required equations. Letting $\stackrel{^}{x}$ denote a column of $\stackrel{^}{X}$, nag_lapack_dgtrfs (f07ch) computes a component-wise backward error, $\beta$, the smallest relative perturbation in each element of $A$ and $b$ such that $\stackrel{^}{x}$ is the exact solution of a perturbed system
 $A+E x^=b+f , with eij ≤β aij , and fj ≤β bj .$
The function also estimates a bound for the component-wise forward error in the computed solution defined by $\mathrm{max}\left|{x}_{i}-\stackrel{^}{{x}_{i}}\right|/\mathrm{max}\left|\stackrel{^}{{x}_{i}}\right|$, where $x$ is the corresponding column of the exact solution, $X$.

## References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{trans}$ – string (length ≥ 1)
Specifies the equations to be solved as follows:
${\mathbf{trans}}=\text{'N'}$
Solve $AX=B$ for $X$.
${\mathbf{trans}}=\text{'T'}$ or $\text{'C'}$
Solve ${A}^{\mathrm{T}}X=B$ for $X$.
Constraint: ${\mathbf{trans}}=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.
2:     $\mathrm{dl}\left(:\right)$ – double array
The dimension of the array dl must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
Must contain the $\left(n-1\right)$ subdiagonal elements of the matrix $A$.
3:     $\mathrm{d}\left(:\right)$ – double array
The dimension of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
Must contain the $n$ diagonal elements of the matrix $A$.
4:     $\mathrm{du}\left(:\right)$ – double array
The dimension of the array du must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
Must contain the $\left(n-1\right)$ superdiagonal elements of the matrix $A$.
5:     $\mathrm{dlf}\left(:\right)$ – double array
The dimension of the array dlf must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
Must contain the $\left(n-1\right)$ multipliers that define the matrix $L$ of the $LU$ factorization of $A$.
6:     $\mathrm{df}\left(:\right)$ – double array
The dimension of the array df must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
Must contain the $n$ diagonal elements of the upper triangular matrix $U$ from the $LU$ factorization of $A$.
7:     $\mathrm{duf}\left(:\right)$ – double array
The dimension of the array duf must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
Must contain the $\left(n-1\right)$ elements of the first superdiagonal of $U$.
8:     $\mathrm{du2}\left(:\right)$ – double array
The dimension of the array du2 must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-2\right)$
Must contain the $\left(n-2\right)$ elements of the second superdiagonal of $U$.
9:     $\mathrm{ipiv}\left(:\right)$int64int32nag_int array
The dimension of the array ipiv must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
Must contain the $n$ pivot indices that define the permutation matrix $P$. At the $i$th step, row $i$ of the matrix was interchanged with row ${\mathbf{ipiv}}\left(i\right)$, and ${\mathbf{ipiv}}\left(i\right)$ must always be either $i$ or $\left(i+1\right)$, ${\mathbf{ipiv}}\left(i\right)=i$ indicating that a row interchange was not performed.
10:   $\mathrm{b}\left(\mathit{ldb},:\right)$ – 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 $r$ matrix of right-hand sides $B$.
11:   $\mathrm{x}\left(\mathit{ldx},:\right)$ – 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 $r$ initial solution matrix $X$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the arrays b, x and the dimension of the arrays d, df, ipiv.
$n$, 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, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs_p}}\ge 0$.

### Output Parameters

1:     $\mathrm{x}\left(\mathit{ldx},:\right)$ – double array
The first dimension of the array x will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array x will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
The $n$ by $r$ refined solution matrix $X$.
2:     $\mathrm{ferr}\left({\mathbf{nrhs_p}}\right)$ – double array
Estimate of the forward error bound for each computed solution vector, such that ${‖{\stackrel{^}{x}}_{j}-{x}_{j}‖}_{\infty }/{‖{\stackrel{^}{x}}_{j}‖}_{\infty }\le {\mathbf{ferr}}\left(j\right)$, where ${\stackrel{^}{x}}_{j}$ is the $j$th column of the computed solution returned in the array x and ${x}_{j}$ is the corresponding column of the exact solution $X$. The estimate is almost always a slight overestimate of the true error.
3:     $\mathrm{berr}\left({\mathbf{nrhs_p}}\right)$ – double array
Estimate of the component-wise relative backward error of each computed solution vector ${\stackrel{^}{x}}_{j}$ (i.e., the smallest relative change in any element of $A$ or $B$ that makes ${\stackrel{^}{x}}_{j}$ an exact solution).
4:     $\mathrm{info}$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 computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $A+E x^=b ,$
where
 $E∞=OεA∞$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $x^-x ∞ x∞ ≤ κA E∞ A∞ ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{\infty }{‖A‖}_{\infty }$, the condition number of $A$ with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
Function nag_lapack_dgtcon (f07cg) can be used to estimate the condition number of $A$.

The total number of floating-point operations required to solve the equations $AX=B$ or ${A}^{\mathrm{T}}X=B$ is proportional to $nr$. At most five steps of iterative refinement are performed, but usually only one or two steps are required.
The complex analogue of this function is nag_lapack_zgtrfs (f07cv).

## Example

This example solves the equations
 $AX=B ,$
where $A$ is the tridiagonal matrix
 $A = 3.0 2.1 0.0 0.0 0.0 3.4 2.3 -1.0 0.0 0.0 0.0 3.6 -5.0 1.9 0.0 0.0 0.0 7.0 -0.9 8.0 0.0 0.0 0.0 -6.0 7.1 and B = 2.7 6.6 -0.5 10.8 2.6 -3.2 0.6 -11.2 2.7 19.1 .$
Estimates for the backward errors and forward errors are also output.
```function f07ch_example

fprintf('f07ch example results\n\n');

% Tridiagonal matrix A stored as diagonals:
du = [        2.1    -1.0      1.9     8.0];
d  = [3.0     2.3    -5.0     -0.9     7.1];
dl = [3.4     3.6     7.0     -6.0        ];
n  = numel(d);

% Factorize A.
[dlf, df, duf, du2f, ipiv, info] = ...
f07cd(dl, d, du);

% RHS
b = [ 2.7   6.6;
-0.5  10.8;
2.6  -3.2;
0.6 -11.2;
2.7  19.1];

% Solve Ax = b

trans = 'No transpose';
[x, info] = f07ce( ...
trans, dlf, df, duf, du2f, ipiv, b);

% Refine solution
[x, ferr, berr, info] = ...
f07ch( ...
trans, dl, d, du, dlf, df, duf, du2f, ipiv, b, x);

disp('Solution:');
disp(x);

fprintf('Forward  error bounds = %10.1e  %10.1e\n',ferr);
fprintf('Backward error bounds = %10.1e  %10.1e\n',berr);

```
```f07ch example results

Solution:
-4.0000    5.0000
7.0000   -4.0000
3.0000   -3.0000
-4.0000   -2.0000
-3.0000    1.0000

Forward  error bounds =    9.4e-15     1.4e-14
Backward error bounds =    7.2e-17     5.9e-17
```