1: $m$ – Integer Input: On entry: $m$ , the number of rows of the matrix $A$ .

Constraint: $m \geq 0$ .
2: $n$ – Integer Input: On entry: $n$ , the number of columns of the matrix $A$ .

Constraint: $n \geq 0$ .
3: $a (lda *)$ – Complex (Kind=nag_wp) array Input/Output: Note: the second dimension of the array a must be at least $\max (1, n)$ .

On entry: the $m$ by $n$ matrix $A$ .

On exit: the factors $L$ and $U$ from the factorization $A = P L U$ ; the unit diagonal elements of $L$ are not stored.
4: $lda$ – Integer Input: On entry: the first dimension of the array a as declared in the (sub)program from which f07arf is called.

Constraint: $lda \geq \max (1, m)$ .
5: $ipiv (\min (m, n))$ – Integer array Output: On exit: 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.
6: $info$ – Integer Output: On exit: $info = 0$ unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$info > 0$: Element $〈value〉$ 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.

7 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.

8 Parallelism and Performance

f07arf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f07arf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 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 routine with

m = n

may be followed by calls to the routines:

f07asf to solve $A X = B$ , $A^{T} X = B$ or $A^{H} X = B$ ;
f07auf to estimate the condition number of $A$ ;
f07awf to compute the inverse of $A$ .

The real analogue of this routine is f07adf.

10 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}) .

Interfaces: FL CL AD

NAG FL Interface Introduction

F07 (Lapacklin) Chapter Contents

F07 (Lapacklin) Chapter Introduction

f07ar: FL CL AD

NAG FL Interfacef07arf (zgetrf)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
f07arf (zgetrf)