NAG Library Function Document
nag_linf_fit (e02gcc)
1 Purpose
nag_linf_fit (e02gcc) calculates an ${l}_{\infty}$ solution to an overdetermined system of linear equations.
2 Specification
#include <nag.h> 
#include <nage02.h> 
void 
nag_linf_fit (Nag_OrderType order,
Integer m,
Integer n,
double a[],
double b[],
double tol,
double *relerr,
double x[],
double *resmax,
Integer *rank,
Integer *iter,
NagError *fail) 

3 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.
4 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
5 Arguments
 1:
$\mathbf{order}$ – Nag_OrderTypeInput

On entry: the
order argument specifies the twodimensional storage scheme being used, i.e., rowmajor ordering or columnmajor ordering. C language defined storage is specified by
${\mathbf{order}}=\mathrm{Nag\_RowMajor}$. See
Section 3.2.1.3 in the Essential Introduction 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 $A$).
Constraint:
${\mathbf{n}}\ge 1$.
 4:
$\mathbf{a}\left[\mathit{dim}\right]$ – doubleInput/Output

Note: the dimension,
dim, of the array
a
must be at least
$\left({\mathbf{n}}+3\right)\times \left({\mathbf{m}}+1\right)$.
Where
${\mathbf{A}}\left(j,i\right)$ appears in this document, it refers to the array element
 ${\mathbf{a}}\left[\left(i1\right)\times \left({\mathbf{n}}+3\right)+j1\right]$ when ${\mathbf{order}}=\mathrm{Nag\_ColMajor}$;
 ${\mathbf{a}}\left[\left(j1\right)\times \left({\mathbf{m}}+1\right)+i1\right]$ when ${\mathbf{order}}=\mathrm{Nag\_RowMajor}$.
On entry:
${\mathbf{A}}\left(\mathit{j},\mathit{i}\right)$ 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
Section 7.
On exit: contains the last simplex tableau.
 5:
$\mathbf{b}\left[{\mathbf{m}}\right]$ – doubleInput/Output

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
${\mathbf{b}}\left[i1\right]$ 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
${\mathbf{relerr}}\le 0.0$, 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).
On exit: is altered as described above.
 8:
$\mathbf{x}\left[{\mathbf{n}}\right]$ – doubleOutput

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 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.2.1.2 in the Essential Introduction for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_INT

On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 1$.
 NE_INT_2

On entry, ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
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.
An unexpected error has been triggered by this function. Please contact
NAG.
See
Section 3.6.6 in the Essential Introduction for further information.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 3.6.5 in the Essential Introduction 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:
${\mathbf{tol}}=\u2329\mathit{\text{value}}\u232a$.
7 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.
8 Parallelism and Performance
Not applicable.
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
$A$.
10 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_linf_fit (e02gcc) is used to solve these in the ${l}_{\infty}$ sense.
10.1 Program Text
Program Text (e02gcce.c)
10.2 Program Data
Program Data (e02gcce.d)
10.3 Program Results
Program Results (e02gcce.r)