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

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

1: $a (lda :)$ – complex 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: $b (ldb :)$ – complex 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 $r$ right-hand side matrix $B$ .

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array b.
$n$ , 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: $a (lda :)$ – complex array: The first dimension of the array a will be $\max (1, n)$ .
The second dimension of the array a will be $\max (1, n)$ .

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 no constraints are violated, 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: $b (ldb :)$ – complex 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 $info = 0$ , the $n$ by $r$ solution matrix $X$ .
4: $info$ – 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. The factorization has been completed, but the factor $U$ is exactly singular, so the solution could not be computed.

Accuracy

The computed solution for a single right-hand side,

\hat{x}

, satisfies the 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.

Following the use of nag_lapack_zgesv (f07an), nag_lapack_zgecon (f07au) can be used to estimate the condition number of

A

and nag_lapack_zgerfs (f07av) can be used to obtain approximate error bounds. Alternatives to nag_lapack_zgesv (f07an), which return condition and error estimates directly are nag_linsys_complex_square_solve (f04ca) and nag_lapack_zgesvx (f07ap).

Further Comments

The total number of floating-point operations is approximately

\frac{8}{3} n^{3} + 8 n^{2} r

, where

r

is the number of right-hand sides.

The real analogue of this function is nag_lapack_dgesv (f07aa).

Example

This example solves the equations

A x = b,

where

A

is the general matrix

A = (\begin{array}{r} - 1.34 + 2.55 i & 0.28 + 3.17 i & - 6.39 - 2.20 i & 0.72 - 0.92 i \\ - 0.17 - 1.41 i & 3.31 - 0.15 i & - 0.15 + 1.34 i & 1.29 + 1.38 i \\ - 3.29 - 2.39 i & - 1.91 + 4.42 i & - 0.14 - 1.35 i & 1.72 + 1.35 i \\ 2.41 + 0.39 i & - 0.56 + 1.47 i & - 0.83 - 0.69 i & - 1.96 + 0.67 i \end{array}) and b = (\begin{array}{r} 26.26 + 51.78 i \\ 6.43 - 8.68 i \\ - 5.75 + 25.31 i \\ 1.16 + 2.57 i \end{array}) .

Details of the

L U

factorization of

A

are also output.

Open in the MATLAB editor: f07an_example

function f07an_example


fprintf('f07an example results\n\n');

% Linear system Ax = B
a = [ -1.34 + 2.55i,  0.28 + 3.17i,  -6.39 - 2.2i,   0.72 - 0.92i;
      -0.17 - 1.41i,  3.31 - 0.15i,  -0.15 + 1.34i,  1.29 + 1.38i;
      -3.29 - 2.39i, -1.91 + 4.42i,  -0.14 - 1.35i,  1.72 + 1.35i;
       2.41 + 0.39i, -0.56 + 1.47i,  -0.83 - 0.69i, -1.96 + 0.67i];
b = [ 26.26 + 51.78i;
       6.43 -  8.68i;
      -5.75 + 25.31i;
       1.16 +  2.57i];

[LU, ipiv, x, info] = f07an(a, b);

disp('Solution');
disp(x');
disp('Details of factorization');
disp(LU);
disp('Pivot indices');
disp(double(ipiv'));

f07an example results

Solution
   1.0000 - 1.0000i   2.0000 + 3.0000i  -4.0000 + 5.0000i   0.0000 - 6.0000i

Details of factorization
  -3.2900 - 2.3900i  -1.9100 + 4.4200i  -0.1400 - 1.3500i   1.7200 + 1.3500i
   0.2376 + 0.2560i   4.8952 - 0.7114i  -0.4623 + 1.6966i   1.2269 + 0.6190i
  -0.1020 - 0.7010i  -0.6691 + 0.3689i  -5.1414 - 1.1300i   0.9983 + 0.3850i
  -0.5359 + 0.2707i  -0.2040 + 0.8601i   0.0082 + 0.1211i   0.1482 - 0.1252i

Pivot indices
     3     2     3     4

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

Chapter Contents

Chapter Introduction

NAG Toolbox