nag_zgetrf (f07arc) (PDF version)
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f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zgetrf (f07arc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zgetrf (f07arc) computes the LU factorization of a complex m by n matrix.

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_zgetrf (Nag_OrderType order, Integer m, Integer n, Complex a[], Integer pda, Integer ipiv[], NagError *fail)

3  Description

nag_zgetrf (f07arc) forms the LU factorization of a complex m by n matrix A as A=PLU, where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m>n) and U is upper triangular (upper trapezoidal if m<n). Usually A is square m=n, and both L and U are triangular. The function uses partial pivoting, with row interchanges.

4  References

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

5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     mIntegerInput
On entry: m, the number of rows of the matrix A.
Constraint: m0.
3:     nIntegerInput
On entry: n, the number of columns of the matrix A.
Constraint: n0.
4:     a[dim]ComplexInput/Output
Note: the dimension, dim, of the array a must be at least
  • max1,pda×n when order=Nag_ColMajor;
  • max1,m×pda when order=Nag_RowMajor.
The i,jth element of the matrix A is stored in
  • a[j-1×pda+i-1] when order=Nag_ColMajor;
  • a[i-1×pda+j-1] when order=Nag_RowMajor.
On entry: the m by n matrix A.
On exit: the factors L and U from the factorization A=PLU; the unit diagonal elements of L are not stored.
5:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
  • if order=Nag_ColMajor, pdamax1,m;
  • if order=Nag_RowMajor, pdamax1,n.
6:     ipiv[minm,n]IntegerOutput
On exit: the pivot indices that define the permutation matrix. At the ith step, if ipiv[i-1]>i then row i of the matrix A was interchanged with row ipiv[i-1], for i=1,2,,minm,n. ipiv[i-1]i indicates that, at the ith step, a row interchange was not required.
7:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pda=value and m=value.
Constraint: pdamax1,m.
On entry, pda=value and n=value.
Constraint: pdamax1,n.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
Uvalue,value 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 c minm,n ε P L U ,
cn is a modest linear function of n, and ε is the machine precision.

8  Further Comments

The total number of real floating point operations is approximately 83n3 if m=n (the usual case), 43n23m-n if m>n and 43m23n-m if m<n.
A call to this function with m=n may be followed by calls to the functions:
The real analogue of this function is nag_dgetrf (f07adc).

9  Example

This example computes the LU factorization of the matrix A, where
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 .

9.1  Program Text

Program Text (f07arce.c)

9.2  Program Data

Program Data (f07arce.d)

9.3  Program Results

Program Results (f07arce.r)

nag_zgetrf (f07arc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012