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NAG Toolbox: nag_linsys_real_square_solve_ref (f04ae)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_linsys_real_square_solve_ref (f04ae) calculates the accurate solution of a set of real linear equations with multiple right-hand sides using an LU factorization with partial pivoting, and iterative refinement.

Syntax

[c, aa, bb, ifail] = f04ae(a, b, 'n', n, 'm', m)
[c, aa, bb, ifail] = nag_linsys_real_square_solve_ref(a, b, 'n', n, 'm', m)

Description

Given a set of real 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. An approximation to X is found by forward and backward substitution. The residual matrix R=B-AX is then calculated using additional precision, and a correction D to X is found by solving LUD=PR. X is replaced by X+D and this iterative refinement of the solution is repeated until full machine accuracy has been obtained.

References

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

Parameters

Compulsory Input Parameters

1:     alda: – double array
The first dimension of the array a must be at least max1,n.
The second dimension of the array a must be at least max1,n.
The n by n matrix A.
2:     bldb: – double array
The first dimension of the array b must be at least max1,n.
The second dimension of the array b must be at least max1,m.
The n by m right-hand side matrix B.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the arrays a, b and the second dimension of the array a.
n, the order of the matrix A.
Constraint: n0.
2:     m int64int32nag_int scalar
Default: the second dimension of the array b.
m, the number of right-hand sides.
Constraint: m0.

Output Parameters

1:     cldc: – double array
The first dimension of the array c will be max1,n.
The second dimension of the array c will be max1,m.
The n by m solution matrix X.
2:     aaldaa: – double array
The first dimension of the array aa will be max1,n.
The second dimension of the array aa will be max1,n.
The triangular factors L and U, except that the unit diagonal elements of U are not stored.
3:     bbldbb: – double array
The first dimension of the array bb will be max1,n.
The second dimension of the array bb will be max1,m.
The final n by m residual matrix R=B-AX.
4:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
   ifail=1
The matrix A is singular, possibly due to rounding errors.
   ifail=2
Iterative refinement fails to improve the solution, i.e., the matrix A is too ill-conditioned.
   ifail=3
On entry,n<0,
orm<0,
orlda<max1,n,
orldb<max1,n,
orldc<max1,n,
orldaa<max1,n,
orldbb<max1,n.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

Accuracy

The computed solutions should be correct to full machine accuracy. For a detailed error analysis see page 107 of Wilkinson and Reinsch (1971).

Further Comments

The time taken by nag_linsys_real_square_solve_ref (f04ae) is approximately proportional to n3.
If there is only one right-hand side, it is simpler to use nag_linsys_real_square_solve_1rhs (f04at).

Example

This example solves the set of linear equations AX=B where
A= 33 16 72 -24 -10 -57 -8 -4 -17   and  B= -359 281 85 .  
function f04ae_example


fprintf('f04ae example results\n\n');

% Solve Ax = b for general matrix A
a = [  33,  16,  72;
      -24, -10, -57;
       -8,  -4, -17];
b = [-359; 281;  85];

[x, LU, resid, ifail] = f04ae(a, b);

disp('Solution');
disp(x);


f04ae example results

Solution
     1
    -2
    -5


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Chapter Introduction
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