nag_real_lin_eqn (f04arc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_real_lin_eqn (f04arc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_real_lin_eqn (f04arc) calculates the approximate solution of a set of real linear equations with a single right-hand side, using an LU  factorization with partial pivoting.

2  Specification

#include <nag.h>
#include <nagf04.h>
void  nag_real_lin_eqn (Integer n, double a[], Integer tda, const double b[], double x[], NagError *fail)

3  Description

Given a set of linear equations, Ax = b , the function first computes an LU  factorization of A  with partial pivoting, PA = LU , where P  is a permutation matrix, L  is lower triangular and U  is unit upper triangular. The approximate solution x  is found by forward and backward substitution in Ly = Pb  and Ux = y , where b  is the right-hand side.

4  References

Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

5  Arguments

1:     nIntegerInput
On entry: n , the order of the matrix A.
Constraint: n1 .
2:     a[n×tda]doubleInput/Output
Note: the i,jth element of the matrix A is stored in a[i-1×tda+j-1].
On entry: the n  by n  matrix A .
On exit: A  is overwritten by the lower triangular matrix L  and the off-diagonal elements of the upper triangular matrix U . The unit diagonal elements of U  are not stored.
3:     tdaIntegerInput
On entry: the stride separating matrix column elements in the array a.
Constraint: tdan .
4:     b[n]const doubleInput
On entry: the right-hand side vector b .
5:     x[n]doubleOutput
On exit: the solution vector x .
6:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_2_INT_ARG_LT
On entry, tda=value  while n=value . These arguments must satisfy tdan .
NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_INT_ARG_LT
On entry, n=value.
Constraint: n1.
NE_SINGULAR
The matrix A  is singular, possibly due to rounding errors.

7  Accuracy

The accuracy of the computed solution depends on the conditioning of the original matrix. For a detailed error analysis see page 107 of Wilkinson and Reinsch (1971).

8  Further Comments

The time taken by nag_real_lin_eqn (f04arc) is approximately proportional to n 3 .

9  Example

To solve the set of linear equations Ax = b  where
A = 33 16 72 -24 -10 -57 -8 -4 -17   and   B = -359 281 85 .

9.1  Program Text

Program Text (f04arce.c)

9.2  Program Data

Program Data (f04arce.d)

9.3  Program Results

Program Results (f04arce.r)


nag_real_lin_eqn (f04arc) (PDF version)
f04 Chapter Contents
f04 Chapter Introduction
NAG C Library Manual

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