nag_zgetrf (f07arc) (PDF version)
f07 Chapter Contents
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 3.2.1.3 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.
Constraints:
  • 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

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
NE_INT_2
On entry, pda=value and m=value.
Constraint: pdamax1,m.
On entry, pda=value and n=value.
Constraint: pdamax1,n.
NE_INTERNAL_ERROR
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.
NE_SINGULAR
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