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: $dl (:)$ – double array: The dimension of the array dl must be at least $\max (1, n - 1)$

Must contain the $(n - 1)$ subdiagonal elements of the matrix $A$ .
2: $d (:)$ – double array: The dimension of the array d must be at least $\max (1, n)$

Must contain the $n$ diagonal elements of the matrix $A$ .
3: $du (:)$ – double array: The dimension of the array du must be at least $\max (1, n - 1)$

Must contain the $(n - 1)$ superdiagonal elements of the matrix $A$ .
4: $b (ldb :)$ – double array: The first dimension of the array b must be at least $\max (1, n)$ .
The second dimension of the array b must be at least $\max (1, nrhs_p)$ .

Note: to solve the equations $A x = b$ , where $b$ is a single right-hand side, b may be supplied as a one-dimensional array with length $ldb = \max (1, n)$ .

The $n$ by $r$ right-hand side matrix $B$ .

Optional Input Parameters

1: $n$ – 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: $n \geq 0$ .
2: $nrhs_p$ – 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: $nrhs_p \geq 0$ .

Output Parameters

1: $dl (:)$ – double array: The dimension of the array dl will be $\max (1, n - 1)$

If no constraints are violated, dl stores the ( $n - 2$ ) elements of the second superdiagonal of the upper triangular matrix $U$ from the $L U$ factorization of $A$ , in $dl (1), dl (2), \dots, dl (n - 2)$ .
2: $d (:)$ – double array: The dimension of the array d will be $\max (1, n)$

If no constraints are violated, d stores the $n$ diagonal elements of the upper triangular matrix $U$ from the $L U$ factorization of $A$ .
3: $du (:)$ – double array: The dimension of the array du will be $\max (1, n - 1)$

If no constraints are violated, du stores the $(n - 1)$ elements of the first superdiagonal of $U$ .
4: $b (ldb :)$ – double array: The first dimension of the array b will be $\max (1, n)$ .
The second dimension of the array b will be $\max (1, nrhs_p)$ .

Note: to solve the equations $A x = b$ , where $b$ is a single right-hand side, b may be supplied as a one-dimensional array with length $ldb = \max (1, n)$ .

If $info = 0$ , the $n$ by $r$ solution matrix $X$ .
5: $info$ – int64int32nag_int scalar: $info = 0$ 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.

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

W $info > 0$: Element $_$ of the diagonal is exactly zero, and the solution has not been computed. The factorization has not been completed unless $n =_$ .

Accuracy

The computed solution for a single right-hand side,

\hat{x}

, satisfies an equation of the form

(A + E) \hat{x} = b,

where

{‖E‖}_{1} = O (ε) {‖A‖}_{1}

and

ε

is the machine precision. An approximate error bound for the computed solution is given by

\frac{{‖\hat{x} - x‖}_{1}}{{‖x‖}_{1}} \leq κ (A) \frac{{‖E‖}_{1}}{{‖A‖}_{1}},

where

κ (A) = {‖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).

Further Comments

The total number of floating-point operations required to solve the equations

A X = B

is proportional to

n r

The complex analogue of this function is nag_lapack_zgtsv (f07cn).

Example

This example solves the equations

A x = b,

where

A

is the tridiagonal matrix

A = (\begin{array}{r} 3.0 & 2.1 & 0 & 0 & 0 \\ 3.4 & 2.3 & - 1.0 & 0 & 0 \\ 0 & 3.6 & - 5.0 & 1.9 & 0 \\ 0 & 0 & 7.0 & - 0.9 & 8.0 \\ 0 & 0 & 0 & - 6.0 & 7.1 \end{array}) and b = (\begin{matrix} 2.7 \\ - 0.5 \\ 2.6 \\ 0.6 \\ 2.7 \end{matrix}) .

Open in the MATLAB editor: f07ca_example

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

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox