F07ANF (ZGESV) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F07ANF (ZGESV)

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

F07ANF (ZGESV) computes the solution to a complex system of linear equations
AX=B ,  
where A is an n by n matrix and X and B are n by r matrices.

2  Specification

SUBROUTINE F07ANF ( N, NRHS, A, LDA, IPIV, B, LDB, INFO)
INTEGER  N, NRHS, LDA, IPIV(N), LDB, INFO
COMPLEX (KIND=nag_wp)  A(LDA,*), B(LDB,*)
The routine may be called by its LAPACK name zgesv.

3  Description

F07ANF (ZGESV) uses the LU decomposition with partial pivoting and row interchanges to factor A as
A=PLU ,  
where P is a permutation matrix, L is unit lower triangular, and U is upper triangular. The factored form of A is then used to solve the system of equations AX=B.

4  References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5  Arguments

1:     N – INTEGERInput
On entry: n, the number of linear equations, i.e., the order of the matrix A.
Constraint: N0.
2:     NRHS – INTEGERInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: NRHS0.
3:     ALDA* – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least max1,N.
On entry: the n by n coefficient 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 F07ANF (ZGESV) is called.
Constraint: LDAmax1,N.
5:     IPIVN – INTEGER arrayOutput
On exit: if no constraints are violated, the pivot indices that define the permutation matrix P; at the ith step row i of the matrix was interchanged with row IPIVi. IPIVi=i indicates a row interchange was not required.
6:     BLDB* – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array B must be at least max1,NRHS.
On entry: the n by r right-hand side matrix B.
On exit: if INFO=0, the n by r solution matrix X.
7:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07ANF (ZGESV) is called.
Constraint: LDBmax1,N.
8:     INFO – INTEGEROutput
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, so the solution could not be computed.

7  Accuracy

The computed solution for a single right-hand side, x^ , satisfies the equation of the form
A+E x^=b ,  
where
E1 = Oε A1  
and ε  is the machine precision. An approximate error bound for the computed solution is given by
x^ - x 1 x 1 κA E 1 A 1  
where κA = A-1 1 A 1 , the condition number of A  with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
Following the use of F07ANF (ZGESV), F07AUF (ZGECON) can be used to estimate the condition number of A  and F07AVF (ZGERFS) can be used to obtain approximate error bounds. Alternatives to F07ANF (ZGESV), which return condition and error estimates directly are F04CAF and F07APF (ZGESVX).

8  Parallelism and Performance

F07ANF (ZGESV) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F07ANF (ZGESV) 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 floating-point operations is approximately 83 n3 + 8n2 r , where r  is the number of right-hand sides.
The real analogue of this routine is F07AAF (DGESV).

10  Example

This example solves the equations
Ax = b ,  
where A is the general matrix
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   and   b = 26.26+51.78i 6.43-08.68i -5.75+25.31i 1.16+02.57i .  
Details of the LU factorization of A are also output.

10.1  Program Text

Program Text (f07anfe.f90)

10.2  Program Data

Program Data (f07anfe.d)

10.3  Program Results

Program Results (f07anfe.r)


F07ANF (ZGESV) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016