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NAG Toolbox: nag_lapack_dgtsv (f07ca)

Purpose

nag_lapack_dgtsv (f07ca) computes the solution to a real system of linear equations
 $AX=B ,$
where $A$ is an $n$ by $n$ tridiagonal matrix and $X$ and $B$ are $n$ by $r$ matrices.

Syntax

[dl, d, du, b, info] = f07ca(dl, d, du, b, 'n', n, 'nrhs_p', nrhs_p)
[dl, d, du, b, info] = nag_lapack_dgtsv(dl, d, du, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dgtsv (f07ca) uses Gaussian elimination with partial pivoting and row interchanges to solve the equations $AX=B$. The matrix $A$ is factorized as $A=PLU$, where $P$ is a permutation matrix, $L$ is unit lower triangular with at most one nonzero subdiagonal element per column, and $U$ is an upper triangular band matrix, with two superdiagonals.
Note that the equations ${A}^{\mathrm{T}}X=B$ may be solved by interchanging the order of the arguments du and dl.

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:     $\mathrm{dl}\left(:\right)$ – 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 $A$.
2:     $\mathrm{d}\left(:\right)$ – 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 $A$.
3:     $\mathrm{du}\left(:\right)$ – 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 $A$.
4:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
Note: to solve the equations $Ax=b$, where $b$ is a single right-hand side, b may be supplied as a one-dimensional array with length $\mathit{ldb}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $n$ by $r$ right-hand side matrix $B$.

Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the array b and the dimension of the array d.
$n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathrm{nrhs_p}$int64int32nag_int scalar
Default: the second dimension of the array b.
$r$, 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:     $\mathrm{dl}\left(:\right)$ – double array
The dimension of the array dl will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
If no constraints are violated, dl stores the ($n-2$) elements of the second superdiagonal of the upper triangular matrix $U$ from the $LU$ factorization of $A$, in ${\mathbf{dl}}\left(1\right),{\mathbf{dl}}\left(2\right),\dots ,{\mathbf{dl}}\left(n-2\right)$.
2:     $\mathrm{d}\left(:\right)$ – double array
The dimension of the array d will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
If no constraints are violated, d stores the $n$ diagonal elements of the upper triangular matrix $U$ from the $LU$ factorization of $A$.
3:     $\mathrm{du}\left(:\right)$ – double array
The dimension of the array du will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
If no constraints are violated, du stores the $\left(n-1\right)$ elements of the first superdiagonal of $U$.
4:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double array
The first dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
Note: to solve the equations $Ax=b$, where $b$ is a single right-hand side, b may be supplied as a one-dimensional array with length $\mathit{ldb}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{info}}={\mathbf{0}}$, the $n$ by $r$ solution matrix $X$.
5:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_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.
W  ${\mathbf{info}}>0$
Element $_$ of the diagonal is exactly zero, and the solution has not been computed. The factorization has not been completed unless ${\mathbf{n}}=_$.

Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $A+E x^ = b ,$
where
 $E1 = Oε A1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $x^-x 1 x1 ≤ κA E1 A1 ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, 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.
Alternatives to nag_lapack_dgtsv (f07ca), which return condition and error estimates are nag_linsys_real_tridiag_solve (f04bc) and nag_lapack_dgtsvx (f07cb).

The total number of floating-point operations required to solve the equations $AX=B$ is proportional to $nr$.
The complex analogue of this function is nag_lapack_zgtsv (f07cn).

Example

This example solves the equations
 $Ax=b ,$
where $A$ is the tridiagonal matrix
 $A = 3.0 2.1 0.0 0.0 0.0 3.4 2.3 -1.0 0.0 0.0 0.0 3.6 -5.0 1.9 0.0 0.0 0.0 7.0 -0.9 8.0 0.0 0.0 0.0 -6.0 7.1 and b = 2.7 -0.5 2.6 0.6 2.7 .$
```function f07ca_example

fprintf('f07ca 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        ];

% RHS B
b  = [2.7;   -0.5;    2.6;     0.6;    2.7];

% Solve Ax = B
[dlf, df, duf, x, info] = f07ca( ...
dl, d, du, b);

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

```
```f07ca example results

Solution
-4.0000    7.0000    3.0000   -4.0000   -3.0000

```

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