NAG FL Interface
f07aaf (dgesv)

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1 Purpose

f07aaf computes the solution to a real system of linear equations
AX=B ,  
where A is an n×n matrix and X and B are n×r matrices.

2 Specification

Fortran Interface
Subroutine f07aaf ( n, nrhs, a, lda, ipiv, b, ldb, info)
Integer, Intent (In) :: n, nrhs, lda, ldb
Integer, Intent (Out) :: ipiv(n), info
Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*)
C Header Interface
#include <nag.h>
void  f07aaf_ (const Integer *n, const Integer *nrhs, double a[], const Integer *lda, Integer ipiv[], double b[], const Integer *ldb, Integer *info)
The routine may be called by the names f07aaf, nagf_lapacklin_dgesv or its LAPACK name dgesv.

3 Description

f07aaf 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 https://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 Integer Input
On entry: n, the number of linear equations, i.e., the order of the matrix A.
Constraint: n0.
2: nrhs Integer Input
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
3: a(lda,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least max(1,n).
On entry: the n×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 Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f07aaf is called.
Constraint: ldamax(1,n).
5: ipiv(n) Integer array Output
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 ipiv(i). ipiv(i)=i indicates a row interchange was not required.
6: b(ldb,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array b must be at least max(1,nrhs).
On entry: the n×r right-hand side matrix B.
On exit: if info=0, the n×r solution matrix X.
7: ldb Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f07aaf is called.
Constraint: ldbmax(1,n).
8: 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, 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 = O(ε) A1  
and ε is the machine precision. An approximate error bound for the computed solution is given by
x^-x1 x1 κ(A) E1 A1  
where κ(A) = A-11 A1 , 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 f07aaf, f07agf can be used to estimate the condition number of A and f07ahf can be used to obtain approximate error bounds. Alternatives to f07aaf, which return condition and error estimates directly are f04baf and f07abf.

8 Parallelism and Performance

f07aaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07aaf 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 23 n3 + 2n2 r , where r is the number of right-hand sides.
The complex analogue of this routine is f07anf.

10 Example

This example solves the equations
Ax = b ,  
where A is the general matrix
A = ( 1.80 2.88 2.05 -0.89 5.25 -2.95 -0.95 -3.80 1.58 -2.69 -2.90 -1.04 -1.11 -0.66 -0.59 0.80 )   and   b = ( 9.52 24.35 0.77 -6.22 ) .  
Details of the LU factorization of A are also output.

10.1 Program Text

Program Text (f07aafe.f90)

10.2 Program Data

Program Data (f07aafe.d)

10.3 Program Results

Program Results (f07aafe.r)