1: $a (lda :)$ – double 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: $a (lda :)$ – double 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 floating-point operations is approximately

\frac{2}{3} n^{3}

m = n

(the usual case),

\frac{1}{3} n^{2} (3 m - n)

m > n

and

\frac{1}{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_dgetrs (f07ae) to solve $A X = B$ or $A^{T} X = B$ ;
nag_lapack_dgecon (f07ag) to estimate the condition number of $A$ ;
nag_lapack_dgetri (f07aj) to compute the inverse of $A$ .

The complex analogue of this function is nag_lapack_zgetrf (f07ar).

Example

This example computes the

L U

factorization of the matrix

A

, 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}) .

Open in the MATLAB editor: f07ad_example

function f07ad_example


fprintf('f07ad example results\n\n');

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];

% LU Factorize
[LU, ipiv, info] = f07ad(a);

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

f07ad example results

Details of factorization
    5.2500   -2.9500   -0.9500   -3.8000
    0.3429    3.8914    2.3757    0.4129
    0.3010   -0.4631   -1.5139    0.2948
   -0.2114   -0.3299    0.0047    0.1314

Pivot indices
     2     2     3     4

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

Chapter Contents

Chapter Introduction

NAG Toolbox