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

Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

Parameters

Compulsory Input Parameters

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 $n$ coefficient 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 $\max (1, nrhs_p)$ .

The $n$ by $r$ matrix of right-hand sides $B$ .

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.
The number of linear equations $n$ , 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.
The number of right-hand sides $r$ , i.e., the number of columns of the matrix $B$ .

Constraint: $nrhs_p \geq 0$ .

Output Parameters

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

If $ifail \geq 0$ , the factors $L$ and $U$ from the factorization $A = P L U$ . The unit diagonal elements of $L$ are not stored.
2: $ipiv (n)$ – int64int32nag_int array: If $ifail \geq 0$ , the pivot indices that define the permutation matrix $P$ ; at the $i$ th step row $i$ of the matrix was interchanged with row $ipiv (i)$ . $ipiv (i) = i$ indicates a row interchange was not required.
3: $b (ldb :)$ – double array: The first dimension of the array b will be $\max (1, n)$ .
The second dimension of the array b will be $\max (1, nrhs_p)$ .

If $ifail = 0$ or $n + 1$ , the $n$ by $r$ solution matrix $X$ .
4: $rcond$ – double scalar: If no constraints are violated, an estimate of the reciprocal of the condition number of the matrix $A$ , computed as $rcond = 1 / ({‖A‖}_{1} {‖A^{- 1}‖}_{1})$ .
5: $errbnd$ – double scalar: If $ifail = 0$ or $n + 1$ , an estimate of the forward error bound for a computed solution $\hat{x}$ , such that ${‖\hat{x} - x‖}_{1} / {‖x‖}_{1} \leq errbnd$ , where $\hat{x}$ is a column of the computed solution returned in the array b and $x$ is the corresponding column of the exact solution $X$ . If rcond is less than machine precision, then errbnd is returned as unity.
6: $ifail$ – 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:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W $ifail > 0 and ifail \leq n$: Diagonal element $_$ of the upper triangular factor is zero. The factorization has been completed, but the solution could not be computed.

W $ifail = n + 1$: A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.

$ifail = - 1$: Constraint: $n \geq 0$ .

$ifail = - 2$: Constraint: $nrhs_p \geq 0$ .

$ifail = - 4$: Constraint: $lda \geq \max (1, n)$ .

$ifail = - 7$: Constraint: $ldb \geq \max (1, n)$ .

$ifail = - 99$: 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.
The integer allocatable memory required is n, and the double allocatable memory required is $4 \times n$ . In this case the factorization and the solution $X$ have been computed, but rcond and errbnd have not been computed.

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. nag_linsys_real_square_solve (f04ba) uses the approximation

{‖E‖}_{1} = ε {‖A‖}_{1}

to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

Further Comments

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

A X = B

is proportional to

(\frac{2}{3} n^{3} + n^{2} r)

. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.

In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.

The complex analogue of nag_linsys_real_square_solve (f04ba) is nag_linsys_complex_square_solve (f04ca).

Example

This example solves the equations

A X = B,

where

A = (\begin{matrix} 1.80 & 2.88 & 2.05 & - 0.89 \\ 5.25 & - 2.95 & - 0.95 & - 3.80 \\ 1.58 & - 2.69 & - 2.90 & - 1.04 \\ - 1.11 & - 0.66 & - 0.59 & 0.80 \end{matrix}) and B = (\begin{matrix} 9.52 & 18.47 \\ 24.35 & 2.25 \\ 0.77 & - 13.28 \\ - 6.22 & - 6.21 \end{matrix}) .

An estimate of the condition number of

A

and an approximate error bound for the computed solutions are also printed.

Open in the MATLAB editor: f04ba_example

function f04ba_example


fprintf('f04ba example results\n\n');

% Solve Ax = b with error bound and condition number
a = [ 1.80,  2.88,  2.05, -0.89;
      5.25, -2.95, -0.95, -3.80;
      1.58, -2.69, -2.90, -1.04;
     -1.11, -0.66, -0.59,  0.80];
b = [ 9.52, 18.47;
     24.35,  2.25;
      0.77,-13.28;
     -6.22, -6.21];

[a, ipiv, x, rcond, errbnd, ifail] = f04ba(a, b);

disp('Solution');
disp(x);
disp('Estimate of condition number');
fprintf('%10.1f\n\n',1/rcond);
disp('Estimate of error bound for computed solutions');
fprintf('%10.1e\n\n',errbnd);

f04ba example results

Solution
    1.0000    3.0000
   -1.0000    2.0000
    3.0000    4.0000
   -5.0000    1.0000

Estimate of condition number
     152.2

Estimate of error bound for computed solutions
   1.7e-14

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

Chapter Contents

Chapter Introduction

NAG Toolbox