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

The $m$ by $n$ matrix $A$ .

Optional Input Parameters

1: $m$ – int64int32nag_int scalar: Default: the first dimension of the array a.
$m$ , the number of rows of the matrix $A$ .

Constraint: $m \geq 0$ .
2: $n$ – int64int32nag_int scalar: Default: the second dimension of the array a.
$n$ , the number of columns of the matrix $A$ .

Constraint: $n \geq 0$ .

Output Parameters

1: $a (lda :)$ – complex array: The first dimension of the array a will be $\max (1, m)$ .
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: $ipiv (\min (m, n))$ – int64int32nag_int array: The pivot indices that define the permutation matrix. At the $i$ th step, if $ipiv (i) > i$ then row $i$ of the matrix $A$ was interchanged with row $ipiv (i)$ , for $i = 1, 2, \dots, \min (m, n)$ . $ipiv (i) \leq i$ indicates that, at the $i$ th step, a row interchange was not required.
3: $info$ – 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, and division by zero will occur if it is used to solve a system of equations.

Accuracy

The computed factors

L

and

U

are the exact factors of a perturbed matrix

A + E

, where

|E| \leq c (\min (m, n)) ε P |L| |U|,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision.

Further Comments

The total number of real floating-point operations is approximately

\frac{8}{3} n^{3}

m = n

(the usual case),

\frac{4}{3} n^{2} (3 m - n)

m > n

and

\frac{4}{3} m^{2} (3 n - m)

m < n

A call to this function with

m = n

may be followed by calls to the functions:

nag_lapack_zgetrs (f07as) to solve $A X = B$ , $A^{T} X = B$ or $A^{H} X = B$ ;
nag_lapack_zgecon (f07au) to estimate the condition number of $A$ ;
nag_lapack_zgetri (f07aw) to compute the inverse of $A$ .

The real analogue of this function is nag_lapack_dgetrf (f07ad).

Example

This example computes the

L U

factorization of the matrix

A

, where

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}) .

Open in the MATLAB editor: f07ar_example

function f07ar_example


fprintf('f07ar example results\n\n');

a = [-1.34 + 2.55i,  0.28 + 3.17i, -6.39 - 2.20i,  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];

[LU, ipiv, info] = f07ar(a);

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

f07ar example results

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