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NAG Toolbox: nag_fit_glin_linf (e02gc)
Purpose
nag_fit_glin_linf (e02gc) calculates an ${l}_{\infty}$ solution to an overdetermined system of linear equations.
Syntax
[
a,
b,
relerr,
x,
resmax,
irank,
iter,
ifail] = e02gc(
n,
a,
b,
relerr, 'm',
m, 'tol',
tol)
[
a,
b,
relerr,
x,
resmax,
irank,
iter,
ifail] = nag_fit_glin_linf(
n,
a,
b,
relerr, 'm',
m, 'tol',
tol)
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
${l}_{\infty}$ solution to the overdetermined system of equations
That is to say, it calculates a vector
$x$, with
$n$ elements, which minimizes the
${l}_{\infty}$ norm of the residuals (the absolutely largest residual)
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. The solution is not unique in this case, and may not be unique even if $A$ is of full rank.
Alternatively, in applications where a complete minimization of the
${l}_{\infty}$ norm is not necessary, you may obtain an approximate solution, usually in shorter time, by giving an appropriate value to the argument
relerr.
Typically in applications to data fitting, data consisting of
$m$ points with coordinates
$\left({t}_{i},{y}_{i}\right)$ is to be approximated in the
${l}_{\infty}$ norm by a linear combination of known functions
${\varphi}_{j}\left(t\right)$,
This is equivalent to finding an
${l}_{\infty}$ solution to the overdetermined system of equations
Thus if, for each value of $i$ and $j$ the element ${a}_{ij}$ of the matrix $A$ above 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 dual formation of the
${l}_{\infty}$ problem (see
Barrodale and Phillips (1974) and
Barrodale and Phillips (1975)). The modifications are designed to improve the efficiency and stability of the simplex method for this particular application.
References
Barrodale I and Phillips C (1974) An improved algorithm for discrete Chebyshev linear approximation Proc. 4th Manitoba Conf. Numerical Mathematics 177–190 University of Manitoba, Canada
Barrodale I and Phillips C (1975) Solution of an overdetermined system of linear equations in the Chebyshev norm [F4] (Algorithm 495) ACM Trans. Math. Software 1(3) 264–270
Parameters
Compulsory Input Parameters
 1:
$\mathrm{n}$ – int64int32nag_int scalar

The number of unknowns, $n$ (the number of columns of the matrix $A$).
Constraint:
${\mathbf{n}}\ge 1$.
 2:
$\mathrm{a}\left(\mathit{lda},\mathit{sda}\right)$ – double array

lda, the first dimension of the array, must satisfy the constraint
$\mathit{lda}\ge {\mathbf{n}}+3$.
${\mathbf{a}}\left(\mathit{j},\mathit{i}\right)$ 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
$\mathit{j}=1,2,\dots ,n$, (that is, the
transpose of the matrix). The remaining elements need not be set. Preferably, the columns of the matrix
$A$ (rows of the argument
a) should be scaled before entry: see
Accuracy.
 3:
$\mathrm{b}\left({\mathbf{m}}\right)$ – double array

${\mathbf{b}}\left(\mathit{i}\right)$ must contain ${b}_{\mathit{i}}$, the $\mathit{i}$th element of the vector $b$, for $\mathit{i}=1,2,\dots ,m$.
 4:
$\mathrm{relerr}$ – double scalar

Must be set to a bound on the relative error acceptable in the maximum residual at the solution.
If
${\mathbf{relerr}}\le 0.0$, then the
${l}_{\infty}$ solution is computed, and
relerr is set to
$0.0$ on exit.
If
${\mathbf{relerr}}>0.0$, then the function obtains instead an approximate solution for which the largest residual is less than
$1.0+{\mathbf{relerr}}$ times that of the
${l}_{\infty}$ solution; on exit,
relerr contains a smaller value such that the above bound still applies. (The usual result of this option, say with
${\mathbf{relerr}}=0.1$, is a saving in the number of simplex iterations).
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:
${\mathbf{m}}\ge {\mathbf{n}}$.
 2:
$\mathrm{tol}$ – double scalar
Default:
$0.0$.
A threshold below which numbers are regarded as zero. The recommended threshold value is
$10.0\times \epsilon $, where
$\epsilon $ is the
machine precision. If
${\mathbf{tol}}\le 0.0$ on entry, the recommended value is used within the function. If premature termination occurs, a larger value for
tol may result in a valid solution.
Output Parameters
 1:
$\mathrm{a}\left(\mathit{lda},\mathit{sda}\right)$ – double array

$\mathit{sda}={\mathbf{m}}+1$.
$\mathit{lda}={\mathbf{n}}+3$.
Contains the last simplex tableau.
 2:
$\mathrm{b}\left({\mathbf{m}}\right)$ – double array

The
$\mathit{i}$th residual
${\mathit{r}}_{\mathit{i}}$ corresponding to the solution vector
$x$, for
$\mathit{i}=1,2,\dots ,m$. Note however that these residuals may contain few significant figures, especially when
resmax is within one or two orders of magnitude of
tol. Indeed if
${\mathbf{resmax}}\le {\mathbf{tol}}$, the elements
${\mathbf{b}}\left(i\right)$ may all be set to zero. It is therefore often advisable to compute the residuals directly.
 3:
$\mathrm{relerr}$ – double scalar

Is altered as described above.
 4:
$\mathrm{x}\left({\mathbf{n}}\right)$ – double array

If
${\mathbf{ifail}}={\mathbf{0}}$ or
${\mathbf{1}}$,
${\mathbf{x}}\left(\mathit{j}\right)$ contains the
$\mathit{j}$th element of the solution vector
$x$, for
$\mathit{j}=1,2,\dots ,n$. Whether this is an
${l}_{\infty}$ solution or an approximation to one, depends on the value of
relerr on entry.
 5:
$\mathrm{resmax}$ – double scalar

If
${\mathbf{ifail}}={\mathbf{0}}$ or
${\mathbf{1}}$,
resmax contains the absolute value of the largest residual(s) for the solution vector
$x$. (See
b.)
 6:
$\mathrm{irank}$ – int64int32nag_int scalar

If
${\mathbf{ifail}}={\mathbf{0}}$ or
${\mathbf{1}}$,
irank contains the computed rank of the matrix
$A$.
 7:
$\mathrm{iter}$ – int64int32nag_int scalar

If
${\mathbf{ifail}}={\mathbf{0}}$ or
${\mathbf{1}}$,
iter contains the number of iterations taken by the simplex method.
 8:
$\mathrm{ifail}$ – int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see
Error Indicators and Warnings).
Error Indicators and Warnings
Note: nag_fit_glin_linf (e02gc) may return useful information for one or more of the following detected errors or 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 (perhaps simply because the matrix $A$ is not of full rank, i.e., ${\mathbf{irank}}<{\mathbf{n}}$).
 ${\mathbf{ifail}}=2$

The calculations have terminated prematurely due to rounding errors. Experiment with larger values of
tol or try rescaling the columns of the matrix (see
Further Comments).
 ${\mathbf{ifail}}=3$

On entry,  $\mathit{lda}<{\mathbf{n}}+3$, 
or  $\mathit{sda}<{\mathbf{m}}+1$, 
or  ${\mathbf{m}}<{\mathbf{n}}$, 
or  ${\mathbf{n}}<1$. 
 ${\mathbf{ifail}}=99$
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 $n+1$ equations which have residuals of largest absolute value. 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 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
tol 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
This example approximates 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_linf (e02gc) is used to solve these in the ${l}_{\infty}$ sense.
Open in the MATLAB editor:
e02gc_example
function e02gc_example
fprintf('e02gc example results\n\n');
n = int64(3);
a = zeros(6, 6);
for i = 1:5
a(1, i) = exp((i1)/5);
a(2, i) = exp((i1)/5);
a(3, i) = 1;
end
b = [4.501; 4.36; 4.333; 4.418; 4.625];
relerr = 0;
[a, b, relerr, x, resmax, irank, iter, ifail] = ...
e02gc(n, a, b, relerr);
fprintf('Resmax = %8.4f Rank = %5d Iterations = %5d\n\n', ...
resmax, irank, iter);
disp('Solution:');
disp(x(1:irank)');
e02gc example results
Resmax = 0.0010 Rank = 3 Iterations = 4
Solution:
1.0049 2.0149 1.4822
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© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015