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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_fit_glin_l1sol (e02ga)

▸▿ 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_fit_glin_l1sol (e02ga) calculates an

l_{1}

solution to an over-determined system of linear equations.

Syntax

[a, b, x, resid, irank, iter, ifail] = e02ga(a, b, 'm', m, 'nplus2', nplus2, 'toler', toler)

[a, b, x, resid, irank, iter, ifail] = nag_fit_glin_l1sol(a, b, 'm', m, 'nplus2', nplus2, 'toler', toler)

Description

Given a matrix

A

with

m

rows and

n

columns

(m \geq n)

and a vector

b

with

m

elements, the function calculates an

l_{1}

solution to the over-determined system of equations

A x = b .

That is to say, it calculates a vector

x

, with

n

elements, which minimizes the

l_{1}

norm (the sum of the absolute values) of the residuals

r (x) = \sum_{i = 1}^{m} |r_{i}|,

where the residuals

r_{i}

are given by

r_{i} = b_{i} - \sum_{j = 1}^{n} a_{i j} x_{j}, i = 1, 2, \dots, m .

Here

a_{i j}

is the element in row

i

and column

j

A

b_{i}

is the

i

th element of

b

and

x_{j}

the

j

th element of

x

. The matrix

A

need not be of full rank.

Typically in applications to data fitting, data consisting of

m

points with coordinates

(t_{i}, y_{i})

are to be approximated in the

l_{1}

norm by a linear combination of known functions

ϕ_{j} (t)

α_{1} ϕ_{1} (t) + α_{2} ϕ_{2} (t) + \dots + α_{n} ϕ_{n} (t) .

This is equivalent to fitting an

l_{1}

solution to the over-determined system of equations

\sum_{j = 1}^{n} ϕ_{j} (t_{i}) α_{j} = y_{i}, i = 1, 2, \dots, m .

Thus if, for each value of

i

and

j

, the element

a_{i j}

of the matrix

A

in the previous paragraph is set equal to the value of

ϕ_{j} (t_{i})

and

b_{i}

is set equal to

y_{i}

, the solution vector

x

will contain the required values of the

α_{j}

. Note that the independent variable

t

above can, instead, be a vector of several independent variables (this includes the case where each

ϕ_{i}

is a function of a different variable, or set of variables).

The algorithm is a modification of the simplex method of linear programming applied to the primal formulation of the

l_{1}

problem (see Barrodale and Roberts (1973) and Barrodale and Roberts (1974)). The modification allows several neighbouring simplex vertices to be passed through in a single iteration, providing a substantial improvement in efficiency.

References

Barrodale I and Roberts F D K (1973) An improved algorithm for discrete

l_{1}

linear approximation SIAM J. Numer. Anal. 10 839–848

Barrodale I and Roberts F D K (1974) Solution of an overdetermined system of equations in the

l_{1}

-norm Comm. ACM 17(6) 319–320

Parameters

Compulsory Input Parameters

1: $a (lda, nplus2)$ – double array: lda, the first dimension of the array, must satisfy the constraint $lda \geq m + 2$ .
$a (i, j)$ must contain $a_{i j}$ , the element in the $i$ th row and $j$ th column of the matrix $A$ , for $i = 1, 2, \dots, m$ and $j = 1, 2, \dots, n$ . The remaining elements need not be set.
2: $b (m)$ – double array: $b (i)$ must contain $b_{i}$ , the $i$ th element of the vector $b$ , for $i = 1, 2, \dots, m$ .

Optional Input Parameters

1: $m$ – int64int32nag_int scalar: Default: the dimension of the array b.
The number of equations, $m$ (the number of rows of the matrix $A$ ).

Constraint: $m \geq n \geq 1$ .
2: $nplus2$ – int64int32nag_int scalar: Default: the second dimension of the array a.
$n + 2$ , where $n$ is the number of unknowns (the number of columns of the matrix $A$ ).

Constraint: $3 \leq nplus2 \leq m + 2$ .
3: $toler$ – double scalar: Default: $0.0$ .
A non-negative value. In general toler specifies a threshold below which numbers are regarded as zero. The recommended threshold value is $ε^{2 / 3}$ where $ε$ is the machine precision. The recommended value can be computed within the function by setting toler to zero. If premature termination occurs a larger value for toler may result in a valid solution.

Output Parameters

1: $a (lda, nplus2)$ – double array: $lda = m + 2$ .
Contains the last simplex tableau generated by the simplex method.
2: $b (m)$ – double array: The $i$ th residual $r_{i}$ corresponding to the solution vector $x$ , for $i = 1, 2, \dots, m$ .
3: $x (nplus2)$ – double array: $x (j)$ contains the $j$ th element of the solution vector $x$ , for $j = 1, 2, \dots, n$ . The elements $x (n + 1)$ and $x (n + 2)$ are unused.
4: $resid$ – double scalar: The sum of the absolute values of the residuals for the solution vector $x$ .
5: $irank$ – int64int32nag_int scalar: The computed rank of the matrix $A$ .
6: $iter$ – int64int32nag_int scalar: The number of iterations taken by the simplex method.
7: $ifail$ – 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:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W $ifail = 1$: An optimal solution has been obtained but this may not be unique.

$ifail = 2$: The calculations have terminated prematurely due to rounding errors. Experiment with larger values of toler or try scaling the columns of the matrix (see Further Comments).

$ifail = 3$

On entry,	$nplus2 < 3$ ,
or	$nplus2 > m + 2$ ,
or	$lda < m + 2$ .

$ifail = - 99$: 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

Experience suggests that the computational accuracy of the solution

x

is comparable with the accuracy that could be obtained by applying Gaussian elimination with partial pivoting to the

n

equations satisfied by this algorithm (i.e., those equations with zero residuals). The accuracy therefore varies with the conditioning of the problem, but has been found generally very satisfactory in practice.

Further Comments

The effects of

m

and

n

on the time and on the number of iterations in the Simplex Method vary from problem to problem, but typically the number of iterations is a small multiple of

n

and the total time taken is approximately proportional to

m n^{2}

It is recommended that, before the function is entered, the columns of the matrix

A

are scaled so that the largest element in each column is of the order of unity. This should improve the conditioning of the matrix, and also enable the argument toler to perform its correct function. The solution

x

obtained will then, of course, relate to the scaled form of the matrix. Thus if the scaling is such that, for each

j = 1, 2, \dots, n

, the elements of the

j

th column are multiplied by the constant

k_{j}

, the element

x_{j}

of the solution vector

x

must be multiplied by

k_{j}

if it is desired to recover the solution corresponding to the original matrix

A

Example

Suppose we wish to approximate a set of data by a curve of the form

y = K e^{t} + L e^{- t} + M

where

K

L

and

M

are unknown. Given values

y_{i}

5

points

t_{i}

we may form the over-determined set of equations for

K

L

and

M

e^{x_{i}} K + e^{- x_{i}} L + M = y_{i}, i = 1, 2, \dots, 5 .

nag_fit_glin_l1sol (e02ga) is used to solve these in the

l_{1}

sense.

Open in the MATLAB editor: e02ga_example

function e02ga_example


fprintf('e02ga example results\n\n');

a = zeros(7, 5);
for i = 1:5
  a(i, 1) = exp((i-1)/5);
  a(i, 2) = exp(-(i-1)/5);
  a(i, 3) = 1;
end
b = [4.501;   4.36;   4.333;   4.418;   4.625];

[a, b, x, resid, irank, iter, ifail] = ...
  e02ga(a, b);

fprintf('Resid = %8.4f  Rank = %5d  Iterations = %5d\n\n',resid, irank, iter);
disp('Solution:');
disp(x(1:irank)');

e02ga example results

Resid =   0.0028  Rank =     3  Iterations =     5

Solution:
    1.0014    2.0035    1.4960

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Chapter Contents

Chapter Introduction

NAG Toolbox