hide long namesshow long names

Integer type: int32 int64 nag_int show int32 show int32 show int64 show int64 show nag_int show nag_int

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_linsys_real_gen_lsqsol (f04am)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_linsys_real_gen_lsqsol (f04am) calculates the accurate least squares solution of a set of

m

linear equations in

n

unknowns,

m \geq n

and rank

= n

, with multiple right-hand sides,

A X = B

, using a

Q R

factorization and iterative refinement.

Syntax

[x, qr, alpha, ipiv, ifail] = f04am(a, b, eps, 'm', m, 'n', n, 'ir', ir)

[x, qr, alpha, ipiv, ifail] = nag_linsys_real_gen_lsqsol(a, b, eps, 'm', m, 'n', n, 'ir', ir)

Note: the interface to this routine has changed since earlier releases of the toolbox:

At Mark 22:

m was made optional

Description

To compute the least squares solution to a set of

m

linear equations in

n

unknowns

(m \geq n)

A X = B

, nag_linsys_real_gen_lsqsol (f04am) first computes a

Q R

factorization of

A

with column pivoting,

A P = Q R

, where

R

is upper triangular,

Q

is an

m

m

orthogonal matrix, and

P

is a permutation matrix.

Q^{T}

is applied to the

m

r

right-hand side matrix

B

to give

C = Q^{T} B

, and the

n

r

solution matrix

X

is calculated, to a first approximation, by back-substitution in

R X = C

. The residual matrix

S = B - A X

is calculated using additional precision, and a correction

D

X

is computed as the least squares solution to

A D = S

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: $a (lda, n)$ – double array: lda, the first dimension of the array, must satisfy the constraint $lda \geq m$ .
The $m$ by $n$ matrix $A$ .
2: $b (ldb, ir)$ – double array: ldb, the first dimension of the array, must satisfy the constraint $ldb \geq m$ .
The $m$ by $r$ right-hand side matrix $B$ .
3: $eps$ – double scalar: Must be set to the value of the machine precision.

Optional Input Parameters

1: $m$ – int64int32nag_int scalar: Default: the first dimension of the arrays a, b. (An error is raised if these dimensions are not equal.)
$m$ , the number of rows of the matrix $A$ , i.e., the number of equations.

Constraint: $m \geq 1$ .
2: $n$ – int64int32nag_int scalar: Default: the second dimension of the array a.
$n$ , the number of columns of the matrix $A$ , i.e., the number of unknowns.

Constraint: $0 \leq n \leq m$ .
3: $ir$ – int64int32nag_int scalar: Default: the second dimension of the array b.
$r$ , the number of right-hand sides.

Output Parameters

1: $x (ldx, ir)$ – double array: The $n$ by $r$ solution matrix $X$ .
2: $qr (ldqr, n)$ – double array: Details of the $Q R$ factorization.
3: $alpha (n)$ – double array: The diagonal elements of the upper triangular matrix $R$ .
4: $ipiv (n)$ – int64int32nag_int array: Details of the column interchanges.
5: $ifail$ – 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 rank of $A$ is less than $n$ ; the problem does not have a unique solution.

$ifail = 2$: The iterative refinement fails to converge, i.e., the matrix $A$ is too ill-conditioned.

$ifail = - 99$: 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

Although the correction process is continued until the solution has converged to full machine accuracy, all the figures in the final solution may not be correct since the correction

D

X

is itself the solution to a linear least squares problem. For a detailed error analysis see page 116 of Wilkinson and Reinsch (1971).

Further Comments

The time taken by nag_linsys_real_gen_lsqsol (f04am) is approximately proportional to

n^{2} (3 m - n)

, provided

r

is small compared with

n

Example

This example calculates the accurate least squares solution of the equations

\begin{array}{l} 1.1 x_{1} + 0.9 x_{2} = 2.2 \\ 1.2 x_{1} + 1.0 x_{2} = 2.3 \\ 1.0 x_{1} + 1.0 x_{2} = 2.1 \end{array}

Open in the MATLAB editor: f04am_example

function f04am_example


fprintf('f04am example results\n\n');

% Least-squares solution to Ax = b, m>n
a = [1.1, 0.9;
     1.2, 1;
     1,   1];
b = [2.2;
     2.3;
     2.1];

epsilon = x02aj;
[x, qr, alpha, ipiv, ifail] = f04am( ...
                                     a, b, epsilon);

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

f04am example results

Solution
    1.3010
    0.7935

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox