NAG Library Function Document

nag_real_lin_eqn (f04arc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

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

L U

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,

A x = b

, the function first computes an

L U

factorization of

A

with partial pivoting,

P A = L U

, 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

L y = P b

and

U x = 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: n – IntegerInput: On entry: $n$ , the order of the matrix A.
Constraint: $n \geq 1$ .
2: a[ $n \times tda$ ] – doubleInput/Output: Note: the $(i, j)$ th element of the matrix $A$ is stored in $a [(i - 1) \times tda + j - 1]$ .

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: tda – IntegerInput: On entry: the stride separating matrix column elements in the array a.

Constraint: $tda \geq n$ .
4: b[n] – const doubleInput: On entry: the right-hand side vector $b$ .
5: x[n] – doubleOutput: On exit: the solution vector $x$ .
6: fail – NagError *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 $tda \geq n$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_INT_ARG_LT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 1$ .
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 Parallelism and Performance

Not applicable.

9 Further Comments

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

n^{3}

10 Example

To solve the set of linear equations

A x = b

where

A = (\begin{matrix} 33 & 16 & 72 \\ - 24 & - 10 & - 57 \\ - 8 & - 4 & - 17 \end{matrix}) and B = (\begin{matrix} - 359 \\ 281 \\ 85 \end{matrix}) .

NAG Library Function Documentnag_real_lin_eqn (f04arc)