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NAG Toolbox: nag_fit_glin_l1sol (e02ga)
Purpose
nag_fit_glin_l1sol (e02ga) calculates an ${l}_{1}$ solution to an overdetermined 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
$\left(m\ge n\right)$ and a vector
$b$ with
$m$ elements, the function calculates an
${l}_{1}$ solution to the overdetermined system of equations
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
where the residuals
${r}_{i}$ are given by
Here
${a}_{ij}$ is the element in row
$i$ and column
$j$ of
$A$,
${b}_{i}$ 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
$\left({t}_{i},{y}_{i}\right)$ are to be approximated in the
${l}_{1}$ norm by a linear combination of known functions
${\varphi}_{j}\left(t\right)$,
This is equivalent to fitting an
${l}_{1}$ solution to the overdetermined system of equations
Thus if, for each value of
$i$ and
$j$, the element
${a}_{ij}$ of the matrix
$A$ in the previous paragraph is set equal to the value of
${\varphi}_{j}\left({t}_{i}\right)$ and
${b}_{i}$ is set equal to
${y}_{i}$, the solution vector
$x$ will contain the required values of the
${\alpha}_{j}$. Note that the independent variable
$t$ above can, instead, be a vector of several independent variables (this includes the case where each
${\varphi}_{i}$ 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:
$\mathrm{a}\left(\mathit{lda},{\mathbf{nplus2}}\right)$ – double array

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

${\mathbf{b}}\left(\mathit{i}\right)$ must contain ${b}_{\mathit{i}}$, the $\mathit{i}$th element of the vector $b$, for $\mathit{i}=1,2,\dots ,m$.
Optional Input Parameters
 1:
$\mathrm{m}$ – int64int32nag_int scalar

Default:
the dimension of the array
b.
The number of equations, $m$ (the number of rows of the matrix $A$).
Constraint:
${\mathbf{m}}\ge n\ge 1$.
 2:
$\mathrm{nplus2}$ – 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:
$3\le {\mathbf{nplus2}}\le {\mathbf{m}}+2$.
 3:
$\mathrm{toler}$ – double scalar
Default:
$0.0$.
A nonnegative value. In general
toler specifies a threshold below which numbers are regarded as zero. The recommended threshold value is
${\epsilon}^{2/3}$ where
$\epsilon $ 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:
$\mathrm{a}\left(\mathit{lda},{\mathbf{nplus2}}\right)$ – double array

$\mathit{lda}={\mathbf{m}}+2$.
Contains the last simplex tableau generated by the simplex method.
 2:
$\mathrm{b}\left({\mathbf{m}}\right)$ – double array

The
$\mathit{i}$th residual ${r}_{\mathit{i}}$ corresponding to the solution vector $x$, for $\mathit{i}=1,2,\dots ,m$.
 3:
$\mathrm{x}\left({\mathbf{nplus2}}\right)$ – double array

${\mathbf{x}}\left(\mathit{j}\right)$ contains the $\mathit{j}$th element of the solution vector $x$, for $\mathit{j}=1,2,\dots ,n$. The elements ${\mathbf{x}}\left(n+1\right)$ and ${\mathbf{x}}\left(n+2\right)$ are unused.
 4:
$\mathrm{resid}$ – double scalar

The sum of the absolute values of the residuals for the solution vector $x$.
 5:
$\mathrm{irank}$ – int64int32nag_int scalar

The computed rank of the matrix $A$.
 6:
$\mathrm{iter}$ – int64int32nag_int scalar

The number of iterations taken by the simplex method.
 7:
$\mathrm{ifail}$ – int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ 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 ${\mathbf{ifail}}=1$

An optimal solution has been obtained but this may not be unique.
 ${\mathbf{ifail}}=2$

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).
 ${\mathbf{ifail}}=3$

On entry,  ${\mathbf{nplus2}}<3$, 
or  ${\mathbf{nplus2}}>{\mathbf{m}}+2$, 
or  $\mathit{lda}<{\mathbf{m}}+2$. 
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
 ${\mathbf{ifail}}=999$
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
where
$K$,
$L$ and
$M$ are unknown. Given values
${y}_{i}$ at
$5$ points
${t}_{i}$ we may form the overdetermined set of equations for
$K$,
$L$ and
$M$
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((i1)/5);
a(i, 2) = exp((i1)/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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© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015