The function may be called by the names: e02gcc, nag_fit_glin_linf or nag_linf_fit.
3Description
Given a matrix $A$ with $m$ rows and $n$ columns $(m\ge n)$ and a vector $b$ with $m$ elements, the function calculates an ${l}_{\infty}$ solution to the overdetermined system of equations
$$Ax=b\text{.}$$
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)
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 $({t}_{i},{y}_{i})$ is to be approximated in the ${l}_{\infty}$ norm by a linear combination of known functions ${\varphi}_{j}\left(t\right)$,
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.
4References
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. Software1(3) 264–270
5Arguments
1: $\mathbf{order}$ – Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag\_RowMajor}$. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
${\mathbf{order}}=\mathrm{Nag\_RowMajor}$ or $\mathrm{Nag\_ColMajor}$.
2: $\mathbf{m}$ – IntegerInput
On entry: the number of equations, $m$ (the number of rows of the matrix $A$).
Constraint:
${\mathbf{m}}\ge {\mathbf{n}}$.
3: $\mathbf{n}$ – IntegerInput
On entry: the number of unknowns, $n$ (the number of columns of the matrix $A$).
Note: the dimension, dim, of the array
a
must be at least
$\left({\mathbf{n}}+3\right)\times \left({\mathbf{m}}+1\right)$.
where ${\mathbf{A}}(j,i)$ appears in this document, it refers to the array element
${\mathbf{a}}\left[(i-1)\times \left({\mathbf{n}}+3\right)+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag\_ColMajor}$;
${\mathbf{a}}\left[(j-1)\times \left({\mathbf{m}}+1\right)+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag\_RowMajor}$.
On entry: ${\mathbf{A}}(\mathit{j},\mathit{i})$ 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 Section 7.
On entry: ${\mathbf{b}}\left[\mathit{i}-1\right]$ must contain ${b}_{\mathit{i}}$, the $\mathit{i}$th element of the vector $b$, for $\mathit{i}=1,2,\dots ,m$.
On exit: 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 ${\mathbf{b}}\left[i-1\right]$ may all be set to zero. It is, therefore, often advisable to compute the residuals directly.
6: $\mathbf{tol}$ – doubleInput
On entry: 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.
Suggested value:
$0.0$.
7: $\mathbf{relerr}$ – double *Input/Output
On entry: must be set to a bound on the relative error acceptable in the maximum residual at the solution.
If ${\mathbf{relerr}}\le 0.0$, the ${l}_{\infty}$ solution is computed, and relerr is set to $0.0$ on exit.
If ${\mathbf{relerr}}>0.0$, 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).
On exit: if an optimal but not necessarily unique solution is found,
${\mathbf{x}}\left[\mathit{j}-1\right]$ 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.
9: $\mathbf{resmax}$ – double *Output
On exit: if an optimal but not necessarily unique solution is found, resmax contains the absolute value of the largest residual(s) for the solution vector $x$. (See b.)
10: $\mathbf{rank}$ – Integer *Output
On exit: if an optimal but not necessarily unique solution is found, rank contains the computed rank of the matrix $A$.
11: $\mathbf{iter}$ – Integer *Output
On exit: if an optimal but not necessarily unique solution is found, iter contains the number of iterations taken by the simplex method.
12: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 1$.
NE_INT_2
On entry, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{m}}\ge {\mathbf{n}}$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_NON_UNIQUE
An optimal solution has been obtained, but may not be unique.
NE_TERMINATION_FAILURE
Premature termination due to rounding errors. Try using larger value of tol: ${\mathbf{tol}}=\u27e8\mathit{\text{value}}\u27e9$.
7Accuracy
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.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
e02gcc is not threaded in any implementation.
9Further 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 $A$.
10Example
This example approximates 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 ${y}_{i}$ at $5$ points ${t}_{i}$ we may form the overdetermined set of equations for $K$, $L$ and $M$