1: $a (lda :)$ – double array: The first dimension of the array a must be at least $\max (1, n)$ .
The second dimension of the array a must be at least $\max (1, n)$ .

The $n$ by $n$ matrix $A$ .
2: $b (ldb :)$ – 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 $\max (1, m)$ .

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

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the arrays a, b and the second dimension of the array a.
$n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .
2: $m$ – int64int32nag_int scalar: Default: the second dimension of the array b.
$m$ , the number of right-hand sides.

Constraint: $m \geq 0$ .

Output Parameters

1: $c (ldc :)$ – double array: The first dimension of the array c will be $\max (1, n)$ .
The second dimension of the array c will be $\max (1, m)$ .

The $n$ by $m$ solution matrix $X$ .
2: $aa (ldaa :)$ – double array: The first dimension of the array aa will be $\max (1, n)$ .
The second dimension of the array aa will be $\max (1, n)$ .

The triangular factors $L$ and $U$ , except that the unit diagonal elements of $U$ are not stored.
3: $bb (ldbb :)$ – double array: The first dimension of the array bb will be $\max (1, n)$ .
The second dimension of the array bb will be $\max (1, m)$ .

The final $n$ by $m$ residual matrix $R = B - A X$ .
4: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: The matrix $A$ is singular, possibly due to rounding errors.

$ifail = 2$: Iterative refinement fails to improve the solution, i.e., the matrix $A$ is too ill-conditioned.

$ifail = 3$

On entry,	$n < 0$ ,
or	$m < 0$ ,
or	$lda < \max (1, n)$ ,
or	$ldb < \max (1, n)$ ,
or	$ldc < \max (1, n)$ ,
or	$ldaa < \max (1, n)$ ,
or	$ldbb < \max (1, n)$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The computed solutions should be correct to full machine accuracy. For a detailed error analysis see page 107 of Wilkinson and Reinsch (1971).

Further Comments

The time taken by nag_linsys_real_square_solve_ref (f04ae) is approximately proportional to

n^{3}

If there is only one right-hand side, it is simpler to use nag_linsys_real_square_solve_1rhs (f04at).

Example

This example solves the set of linear equations

A X = B

where

A = (\begin{array}{r} 33 & 16 & 72 \\ - 24 & - 10 & - 57 \\ - 8 & - 4 & - 17 \end{array}) and B = (\begin{array}{r} - 359 \\ 281 \\ 85 \end{array}) .

Open in the MATLAB editor: f04ae_example

function f04ae_example


fprintf('f04ae example results\n\n');

% Solve Ax = b for general matrix A
a = [  33,  16,  72;
      -24, -10, -57;
       -8,  -4, -17];
b = [-359; 281;  85];

[x, LU, resid, ifail] = f04ae(a, b);

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

f04ae example results

Solution
     1
    -2
    -5

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

Chapter Contents

Chapter Introduction

NAG Toolbox