F07ARF (ZGETRF) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

F07ARF (ZGETRF) computes the LU factorization of a complex m by n matrix.

2  Specification

COMPLEX (KIND=nag_wp)  A(LDA,*)
The routine may be called by its LAPACK name zgetrf.

3  Description

F07ARF (ZGETRF) 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 routine 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  Parameters

1:     M – INTEGERInput
On entry: m, the number of rows of the matrix A.
Constraint: M0.
2:     N – INTEGERInput
On entry: n, the number of columns of the matrix A.
Constraint: N0.
3:     A(LDA,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least max1,N.
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.
4:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F07ARF (ZGETRF) is called.
Constraint: LDAmax1,M.
5:     IPIV(minM,N) – INTEGER arrayOutput
On exit: the pivot indices that define the permutation matrix. At the ith step, if IPIVi>i then row i of the matrix A was interchanged with row IPIVi, for i=1,2,,minm,n. IPIVii indicates that, at the ith step, a row interchange was not required.
6:     INFO – INTEGEROutput
On exit: INFO=0 unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

Errors or warnings detected by the routine:
If INFO=-i, the ith parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.
If INFO=i, Ui,i 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 routine with m=n may be followed by calls to the routines:
The real analogue of this routine is F07ADF (DGETRF).

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 (f07arfe.f90)

9.2  Program Data

Program Data (f07arfe.d)

9.3  Program Results

Program Results (f07arfe.r)

F07ARF (ZGETRF) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

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