557 lines
30 KiB
HTML
557 lines
30 KiB
HTML
<html>
|
|
<head>
|
|
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
|
<title>The Remez Method</title>
|
|
<link rel="stylesheet" href="../math.css" type="text/css">
|
|
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
|
|
<link rel="home" href="../index.html" title="Math Toolkit 2.11.0">
|
|
<link rel="up" href="../backgrounders.html" title="Chapter 22. Backgrounders">
|
|
<link rel="prev" href="lanczos.html" title="The Lanczos Approximation">
|
|
<link rel="next" href="refs.html" title="References">
|
|
</head>
|
|
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
|
|
<table cellpadding="2" width="100%"><tr>
|
|
<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../boost.png"></td>
|
|
<td align="center"><a href="../../../../../index.html">Home</a></td>
|
|
<td align="center"><a href="../../../../../libs/libraries.htm">Libraries</a></td>
|
|
<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
|
|
<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
|
|
<td align="center"><a href="../../../../../more/index.htm">More</a></td>
|
|
</tr></table>
|
|
<hr>
|
|
<div class="spirit-nav">
|
|
<a accesskey="p" href="lanczos.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../backgrounders.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="refs.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
|
|
</div>
|
|
<div class="section">
|
|
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
|
|
<a name="math_toolkit.remez"></a><a class="link" href="remez.html" title="The Remez Method">The Remez Method</a>
|
|
</h2></div></div></div>
|
|
<p>
|
|
The <a href="http://en.wikipedia.org/wiki/Remez_algorithm" target="_top">Remez algorithm</a>
|
|
is a methodology for locating the minimax rational approximation to a function.
|
|
This short article gives a brief overview of the method, but it should not
|
|
be regarded as a thorough theoretical treatment, for that you should consult
|
|
your favorite textbook.
|
|
</p>
|
|
<p>
|
|
Imagine that you want to approximate some function <span class="emphasis"><em>f(x)</em></span>
|
|
by way of a rational function <span class="emphasis"><em>R(x)</em></span>, where <span class="emphasis"><em>R(x)</em></span>
|
|
may be either a polynomial <span class="emphasis"><em>P(x)</em></span> or a ratio of two polynomials
|
|
<span class="emphasis"><em>P(x)/Q(x)</em></span> (a rational function). Initially we'll concentrate
|
|
on the polynomial case, as it's by far the easier to deal with, later we'll
|
|
extend to the full rational function case.
|
|
</p>
|
|
<p>
|
|
We want to find the "best" rational approximation, where "best"
|
|
is defined to be the approximation that has the least deviation from <span class="emphasis"><em>f(x)</em></span>.
|
|
We can measure the deviation by way of an error function:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="serif_italic">E<sub>abs</sub>(x) = f(x) - R(x)</span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
which is expressed in terms of absolute error, but we can equally use relative
|
|
error:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="serif_italic">E<sub>rel</sub>(x) = (f(x) - R(x)) / |f(x)|</span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
And indeed in general we can scale the error function in any way we want, it
|
|
makes no difference to the maths, although the two forms above cover almost
|
|
every practical case that you're likely to encounter.
|
|
</p>
|
|
<p>
|
|
The minimax rational function <span class="emphasis"><em>R(x)</em></span> is then defined to
|
|
be the function that yields the smallest maximal value of the error function.
|
|
Chebyshev showed that there is a unique minimax solution for <span class="emphasis"><em>R(x)</em></span>
|
|
that has the following properties:
|
|
</p>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
|
<li class="listitem">
|
|
If <span class="emphasis"><em>R(x)</em></span> is a polynomial of degree <span class="emphasis"><em>N</em></span>,
|
|
then there are <span class="emphasis"><em>N+2</em></span> unknowns: the <span class="emphasis"><em>N+1</em></span>
|
|
coefficients of the polynomial, and maximal value of the error function.
|
|
</li>
|
|
<li class="listitem">
|
|
The error function has <span class="emphasis"><em>N+1</em></span> roots, and <span class="emphasis"><em>N+2</em></span>
|
|
extrema (minima and maxima).
|
|
</li>
|
|
<li class="listitem">
|
|
The extrema alternate in sign, and all have the same magnitude.
|
|
</li>
|
|
</ul></div>
|
|
<p>
|
|
That means that if we know the location of the extrema of the error function
|
|
then we can write <span class="emphasis"><em>N+2</em></span> simultaneous equations:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="serif_italic">R(x<sub>i</sub>) + (-1)<sup>i</sup>E = f(x<sub>i</sub>)</span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
where <span class="emphasis"><em>E</em></span> is the maximal error term, and <span class="emphasis"><em>x<sub>i</sub></em></span>
|
|
are the abscissa values of the <span class="emphasis"><em>N+2</em></span> extrema of the error
|
|
function. It is then trivial to solve the simultaneous equations to obtain
|
|
the polynomial coefficients and the error term.
|
|
</p>
|
|
<p>
|
|
<span class="emphasis"><em>Unfortunately we don't know where the extrema of the error function
|
|
are located!</em></span>
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.remez.h0"></a>
|
|
<span class="phrase"><a name="math_toolkit.remez.the_remez_method"></a></span><a class="link" href="remez.html#math_toolkit.remez.the_remez_method">The
|
|
Remez Method</a>
|
|
</h5>
|
|
<p>
|
|
The Remez method is an iterative technique which, given a broad range of assumptions,
|
|
will converge on the extrema of the error function, and therefore the minimax
|
|
solution.
|
|
</p>
|
|
<p>
|
|
In the following discussion we'll use a concrete example to illustrate the
|
|
Remez method: an approximation to the function e<sup>x</sup> over the range [-1, 1].
|
|
</p>
|
|
<p>
|
|
Before we can begin the Remez method, we must obtain an initial value for the
|
|
location of the extrema of the error function. We could "guess" these,
|
|
but a much closer first approximation can be obtained by first constructing
|
|
an interpolated polynomial approximation to <span class="emphasis"><em>f(x)</em></span>.
|
|
</p>
|
|
<p>
|
|
In order to obtain the <span class="emphasis"><em>N+1</em></span> coefficients of the interpolated
|
|
polynomial we need N+1 points /(x<sub>0</sub>…x<sub>N</sub>): with our interpolated form passing through
|
|
each of those points that yields <span class="emphasis"><em>N+1</em></span> simultaneous equations:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="serif_italic">f(x<sub>i</sub>) = P(x<sub>i</sub>) = c<sub>0</sub> + c<sub>1</sub>x<sub>i</sub> … + c<sub>N</sub>x<sub>i</sub><sup>N</sup></span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
Which can be solved for the coefficients <span class="emphasis"><em>c<sub>0</sub> …c<sub>N</sub></em></span> in <span class="emphasis"><em>P(x)</em></span>.
|
|
</p>
|
|
<p>
|
|
Obviously this is not a minimax solution, indeed our only guarantee is that
|
|
<span class="emphasis"><em>f(x)</em></span> and <span class="emphasis"><em>P(x)</em></span> touch at <span class="emphasis"><em>N+1</em></span>
|
|
locations, away from those points the error may be arbitrarily large. However,
|
|
we would clearly like this initial approximation to be as close to <span class="emphasis"><em>f(x)</em></span>
|
|
as possible, and it turns out that using the zeros of an orthogonal polynomial
|
|
as the initial interpolation points is a good choice. In our example we'll
|
|
use the zeros of a Chebyshev polynomial as these are particularly easy to calculate,
|
|
interpolating for a polynomial of degree 4, and measuring <span class="emphasis"><em>relative
|
|
error</em></span> we get the following error function:
|
|
</p>
|
|
<p>
|
|
<span class="inlinemediaobject"><img src="../../graphs/remez-2.png"></span>
|
|
</p>
|
|
<p>
|
|
Which has a peak relative error of 1.2x10<sup>-3</sup>.
|
|
</p>
|
|
<p>
|
|
While this is a pretty good approximation already, judging by the shape of
|
|
the error function we can clearly do better. Before starting on the Remez method
|
|
propper, we have one more step to perform: locate all the extrema of the error
|
|
function, and store these locations as our initial <span class="emphasis"><em>Chebyshev control
|
|
points</em></span>.
|
|
</p>
|
|
<div class="note"><table border="0" summary="Note">
|
|
<tr>
|
|
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../doc/src/images/note.png"></td>
|
|
<th align="left">Note</th>
|
|
</tr>
|
|
<tr><td align="left" valign="top">
|
|
<p>
|
|
In the simple case of a polynomial approximation, by interpolating through
|
|
the roots of a Chebyshev polynomial we have in fact created a <span class="emphasis"><em>Chebyshev
|
|
approximation</em></span> to the function: in terms of <span class="emphasis"><em>absolute
|
|
error</em></span> this is the best a priori choice for the interpolated form
|
|
we can achieve, and typically is very close to the minimax solution.
|
|
</p>
|
|
<p>
|
|
However, if we want to optimise for <span class="emphasis"><em>relative error</em></span>,
|
|
or if the approximation is a rational function, then the initial Chebyshev
|
|
solution can be quite far from the ideal minimax solution.
|
|
</p>
|
|
<p>
|
|
A more technical discussion of the theory involved can be found in this
|
|
<a href="http://math.fullerton.edu/mathews/n2003/ChebyshevPolyMod.html" target="_top">online
|
|
course</a>.
|
|
</p>
|
|
</td></tr>
|
|
</table></div>
|
|
<h5>
|
|
<a name="math_toolkit.remez.h1"></a>
|
|
<span class="phrase"><a name="math_toolkit.remez.remez_step_1"></a></span><a class="link" href="remez.html#math_toolkit.remez.remez_step_1">Remez
|
|
Step 1</a>
|
|
</h5>
|
|
<p>
|
|
The first step in the Remez method, given our current set of <span class="emphasis"><em>N+2</em></span>
|
|
Chebyshev control points <span class="emphasis"><em>x<sub>i</sub></em></span>, is to solve the <span class="emphasis"><em>N+2</em></span>
|
|
simultaneous equations:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="serif_italic">P(x<sub>i</sub>) + (-1)<sup>i</sup>E = f(x<sub>i</sub>)</span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
To obtain the error term <span class="emphasis"><em>E</em></span>, and the coefficients of the
|
|
polynomial <span class="emphasis"><em>P(x)</em></span>.
|
|
</p>
|
|
<p>
|
|
This gives us a new approximation to <span class="emphasis"><em>f(x)</em></span> that has the
|
|
same error <span class="emphasis"><em>E</em></span> at each of the control points, and whose
|
|
error function <span class="emphasis"><em>alternates in sign</em></span> at the control points.
|
|
This is still not necessarily the minimax solution though: since the control
|
|
points may not be at the extrema of the error function. After this first step
|
|
here's what our approximation's error function looks like:
|
|
</p>
|
|
<p>
|
|
<span class="inlinemediaobject"><img src="../../graphs/remez-3.png"></span>
|
|
</p>
|
|
<p>
|
|
Clearly this is still not the minimax solution since the control points are
|
|
not located at the extrema, but the maximum relative error has now dropped
|
|
to 5.6x10<sup>-4</sup>.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.remez.h2"></a>
|
|
<span class="phrase"><a name="math_toolkit.remez.remez_step_2"></a></span><a class="link" href="remez.html#math_toolkit.remez.remez_step_2">Remez
|
|
Step 2</a>
|
|
</h5>
|
|
<p>
|
|
The second step is to locate the extrema of the new approximation, which we
|
|
do in two stages: first, since the error function changes sign at each control
|
|
point, we must have N+1 roots of the error function located between each pair
|
|
of N+2 control points. Once these roots are found by standard root finding
|
|
techniques, we know that N extrema are bracketed between each pair of roots,
|
|
plus two more between the endpoints of the range and the first and last roots.
|
|
The N+2 extrema can then be found using standard function minimisation techniques.
|
|
</p>
|
|
<p>
|
|
We now have a choice: multi-point exchange, or single point exchange.
|
|
</p>
|
|
<p>
|
|
In single point exchange, we move the control point nearest to the largest
|
|
extrema to the absissa value of the extrema.
|
|
</p>
|
|
<p>
|
|
In multi-point exchange we swap all the current control points, for the locations
|
|
of the extrema.
|
|
</p>
|
|
<p>
|
|
In our example we perform multi-point exchange.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.remez.h3"></a>
|
|
<span class="phrase"><a name="math_toolkit.remez.iteration"></a></span><a class="link" href="remez.html#math_toolkit.remez.iteration">Iteration</a>
|
|
</h5>
|
|
<p>
|
|
The Remez method then performs steps 1 and 2 above iteratively until the control
|
|
points are located at the extrema of the error function: this is then the minimax
|
|
solution.
|
|
</p>
|
|
<p>
|
|
For our current example, two more iterations converges on a minimax solution
|
|
with a peak relative error of 5x10<sup>-4</sup> and an error function that looks like:
|
|
</p>
|
|
<p>
|
|
<span class="inlinemediaobject"><img src="../../graphs/remez-4.png"></span>
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.remez.h4"></a>
|
|
<span class="phrase"><a name="math_toolkit.remez.rational_approximations"></a></span><a class="link" href="remez.html#math_toolkit.remez.rational_approximations">Rational
|
|
Approximations</a>
|
|
</h5>
|
|
<p>
|
|
If we wish to extend the Remez method to a rational approximation of the form
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="serif_italic">f(x) = R(x) = P(x) / Q(x)</span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
where <span class="emphasis"><em>P(x)</em></span> and <span class="emphasis"><em>Q(x)</em></span> are polynomials,
|
|
then we proceed as before, except that now we have <span class="emphasis"><em>N+M+2</em></span>
|
|
unknowns if <span class="emphasis"><em>P(x)</em></span> is of order <span class="emphasis"><em>N</em></span> and
|
|
<span class="emphasis"><em>Q(x)</em></span> is of order <span class="emphasis"><em>M</em></span> This assumes that
|
|
<span class="emphasis"><em>Q(x)</em></span> is normalised so that its leading coefficient is
|
|
1, giving <span class="emphasis"><em>N+M+1</em></span> polynomial coefficients in total, plus
|
|
the error term <span class="emphasis"><em>E</em></span>.
|
|
</p>
|
|
<p>
|
|
The simultaneous equations to be solved are now:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="serif_italic">P(x<sub>i</sub>) / Q(x<sub>i</sub>) + (-1)<sup>i</sup>E = f(x<sub>i</sub>)</span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
Evaluated at the <span class="emphasis"><em>N+M+2</em></span> control points <span class="emphasis"><em>x<sub>i</sub></em></span>.
|
|
</p>
|
|
<p>
|
|
Unfortunately these equations are non-linear in the error term <span class="emphasis"><em>E</em></span>:
|
|
we can only solve them if we know <span class="emphasis"><em>E</em></span>, and yet <span class="emphasis"><em>E</em></span>
|
|
is one of the unknowns!
|
|
</p>
|
|
<p>
|
|
The method usually adopted to solve these equations is an iterative one: we
|
|
guess the value of <span class="emphasis"><em>E</em></span>, solve the equations to obtain a
|
|
new value for <span class="emphasis"><em>E</em></span> (as well as the polynomial coefficients),
|
|
then use the new value of <span class="emphasis"><em>E</em></span> as the next guess. The method
|
|
is repeated until <span class="emphasis"><em>E</em></span> converges on a stable value.
|
|
</p>
|
|
<p>
|
|
These complications extend the running time required for the development of
|
|
rational approximations quite considerably. It is often desirable to obtain
|
|
a rational rather than polynomial approximation none the less: rational approximations
|
|
will often match more difficult to approximate functions, to greater accuracy,
|
|
and with greater efficiency, than their polynomial alternatives. For example,
|
|
if we takes our previous example of an approximation to e<sup>x</sup>, we obtained 5x10<sup>-4</sup> accuracy
|
|
with an order 4 polynomial. If we move two of the unknowns into the denominator
|
|
to give a pair of order 2 polynomials, and re-minimise, then the peak relative
|
|
error drops to 8.7x10<sup>-5</sup>. That's a 5 fold increase in accuracy, for the same
|
|
number of terms overall.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.remez.h5"></a>
|
|
<span class="phrase"><a name="math_toolkit.remez.remez_practical"></a></span><a class="link" href="remez.html#math_toolkit.remez.remez_practical">Practical
|
|
Considerations</a>
|
|
</h5>
|
|
<p>
|
|
Most treatises on approximation theory stop at this point. However, from a
|
|
practical point of view, most of the work involves finding the right approximating
|
|
form, and then persuading the Remez method to converge on a solution.
|
|
</p>
|
|
<p>
|
|
So far we have used a direct approximation:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="serif_italic">f(x) = R(x)</span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
But this will converge to a useful approximation only if <span class="emphasis"><em>f(x)</em></span>
|
|
is smooth. In addition round-off errors when evaluating the rational form mean
|
|
that this will never get closer than within a few epsilon of machine precision.
|
|
Therefore this form of direct approximation is often reserved for situations
|
|
where we want efficiency, rather than accuracy.
|
|
</p>
|
|
<p>
|
|
The first step in improving the situation is generally to split <span class="emphasis"><em>f(x)</em></span>
|
|
into a dominant part that we can compute accurately by another method, and
|
|
a slowly changing remainder which can be approximated by a rational approximation.
|
|
We might be tempted to write:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="serif_italic">f(x) = g(x) + R(x)</span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
where <span class="emphasis"><em>g(x)</em></span> is the dominant part of <span class="emphasis"><em>f(x)</em></span>,
|
|
but if <span class="emphasis"><em>f(x)/g(x)</em></span> is approximately constant over the interval
|
|
of interest then:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="serif_italic">f(x) = g(x)(c + R(x))</span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
Will yield a much better solution: here <span class="emphasis"><em>c</em></span> is a constant
|
|
that is the approximate value of <span class="emphasis"><em>f(x)/g(x)</em></span> and <span class="emphasis"><em>R(x)</em></span>
|
|
is typically tiny compared to <span class="emphasis"><em>c</em></span>. In this situation if
|
|
<span class="emphasis"><em>R(x)</em></span> is optimised for absolute error, then as long as
|
|
its error is small compared to the constant <span class="emphasis"><em>c</em></span>, that error
|
|
will effectively get wiped out when <span class="emphasis"><em>R(x)</em></span> is added to
|
|
<span class="emphasis"><em>c</em></span>.
|
|
</p>
|
|
<p>
|
|
The difficult part is obviously finding the right <span class="emphasis"><em>g(x)</em></span>
|
|
to extract from your function: often the asymptotic behaviour of the function
|
|
will give a clue, so for example the function <a class="link" href="sf_erf/error_function.html" title="Error Function erf and complement erfc">erfc</a>
|
|
becomes proportional to <span class="emphasis"><em>e<sup>-x<sup>2</sup></sup>/x</em></span> as <span class="emphasis"><em>x</em></span>
|
|
becomes large. Therefore using:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="serif_italic">erfc(z) = (C + R(x)) e<sup>-x<sup>2</sup></sup>/x</span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
as the approximating form seems like an obvious thing to try, and does indeed
|
|
yield a useful approximation.
|
|
</p>
|
|
<p>
|
|
However, the difficulty then becomes one of converging the minimax solution.
|
|
Unfortunately, it is known that for some functions the Remez method can lead
|
|
to divergent behaviour, even when the initial starting approximation is quite
|
|
good. Furthermore, it is not uncommon for the solution obtained in the first
|
|
Remez step above to be a bad one: the equations to be solved are generally
|
|
"stiff", often very close to being singular, and assuming a solution
|
|
is found at all, round-off errors and a rapidly changing error function, can
|
|
lead to a situation where the error function does not in fact change sign at
|
|
each control point as required. If this occurs, it is fatal to the Remez method.
|
|
It is also possible to obtain solutions that are perfectly valid mathematically,
|
|
but which are quite useless computationally: either because there is an unavoidable
|
|
amount of roundoff error in the computation of the rational function, or because
|
|
the denominator has one or more roots over the interval of the approximation.
|
|
In the latter case while the approximation may have the correct limiting value
|
|
at the roots, the approximation is nonetheless useless.
|
|
</p>
|
|
<p>
|
|
Assuming that the approximation does not have any fatal errors, and that the
|
|
only issue is converging adequately on the minimax solution, the aim is to
|
|
get as close as possible to the minimax solution before beginning the Remez
|
|
method. Using the zeros of a Chebyshev polynomial for the initial interpolation
|
|
is a good start, but may not be ideal when dealing with relative errors and/or
|
|
rational (rather than polynomial) approximations. One approach is to skew the
|
|
initial interpolation points to one end: for example if we raise the roots
|
|
of the Chebyshev polynomial to a positive power greater than 1 then the roots
|
|
will be skewed towards the middle of the [-1,1] interval, while a positive
|
|
power less than one will skew them towards either end. More usefully, if we
|
|
initially rescale the points over [0,1] and then raise to a positive power,
|
|
we can skew them to the left or right. Returning to our example of e<sup>x</sup> over [-1,1],
|
|
the initial interpolated form was some way from the minimax solution:
|
|
</p>
|
|
<p>
|
|
<span class="inlinemediaobject"><img src="../../graphs/remez-2.png"></span>
|
|
</p>
|
|
<p>
|
|
However, if we first skew the interpolation points to the left (rescale them
|
|
to [0, 1], raise to the power 1.3, and then rescale back to [-1,1]) we reduce
|
|
the error from 1.3x10<sup>-3</sup> to 6x10<sup>-4</sup>:
|
|
</p>
|
|
<p>
|
|
<span class="inlinemediaobject"><img src="../../graphs/remez-5.png"></span>
|
|
</p>
|
|
<p>
|
|
It's clearly still not ideal, but it is only a few percent away from our desired
|
|
minimax solution (5x10<sup>-4</sup>).
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.remez.h6"></a>
|
|
<span class="phrase"><a name="math_toolkit.remez.remez_method_checklist"></a></span><a class="link" href="remez.html#math_toolkit.remez.remez_method_checklist">Remez
|
|
Method Checklist</a>
|
|
</h5>
|
|
<p>
|
|
The following lists some of the things to check if the Remez method goes wrong,
|
|
it is by no means an exhaustive list, but is provided in the hopes that it
|
|
will prove useful.
|
|
</p>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
|
<li class="listitem">
|
|
Is the function smooth enough? Can it be better separated into a rapidly
|
|
changing part, and an asymptotic part?
|
|
</li>
|
|
<li class="listitem">
|
|
Does the function being approximated have any "blips" in it?
|
|
Check for problems as the function changes computation method, or if a
|
|
root, or an infinity has been divided out. The telltale sign is if there
|
|
is a narrow region where the Remez method will not converge.
|
|
</li>
|
|
<li class="listitem">
|
|
Check you have enough accuracy in your calculations: remember that the
|
|
Remez method works on the difference between the approximation and the
|
|
function being approximated: so you must have more digits of precision
|
|
available than the precision of the approximation being constructed. So
|
|
for example at double precision, you shouldn't expect to be able to get
|
|
better than a float precision approximation.
|
|
</li>
|
|
<li class="listitem">
|
|
Try skewing the initial interpolated approximation to minimise the error
|
|
before you begin the Remez steps.
|
|
</li>
|
|
<li class="listitem">
|
|
If the approximation won't converge or is ill-conditioned from one starting
|
|
location, try starting from a different location.
|
|
</li>
|
|
<li class="listitem">
|
|
If a rational function won't converge, one can minimise a polynomial (which
|
|
presents no problems), then rotate one term from the numerator to the denominator
|
|
and minimise again. In theory one can continue moving terms one at a time
|
|
from numerator to denominator, and then re-minimising, retaining the last
|
|
set of control points at each stage.
|
|
</li>
|
|
<li class="listitem">
|
|
Try using a smaller interval. It may also be possible to optimise over
|
|
one (small) interval, rescale the control points over a larger interval,
|
|
and then re-minimise.
|
|
</li>
|
|
<li class="listitem">
|
|
Keep absissa values small: use a change of variable to keep the abscissa
|
|
over, say [0, b], for some smallish value <span class="emphasis"><em>b</em></span>.
|
|
</li>
|
|
</ul></div>
|
|
<h5>
|
|
<a name="math_toolkit.remez.h7"></a>
|
|
<span class="phrase"><a name="math_toolkit.remez.references"></a></span><a class="link" href="remez.html#math_toolkit.remez.references">References</a>
|
|
</h5>
|
|
<p>
|
|
The original references for the Remez Method and its extension to rational
|
|
functions are unfortunately in Russian:
|
|
</p>
|
|
<p>
|
|
Remez, E.Ya., <span class="emphasis"><em>Fundamentals of numerical methods for Chebyshev approximations</em></span>,
|
|
"Naukova Dumka", Kiev, 1969.
|
|
</p>
|
|
<p>
|
|
Remez, E.Ya., Gavrilyuk, V.T., <span class="emphasis"><em>Computer development of certain approaches
|
|
to the approximate construction of solutions of Chebyshev problems nonlinearly
|
|
depending on parameters</em></span>, Ukr. Mat. Zh. 12 (1960), 324-338.
|
|
</p>
|
|
<p>
|
|
Gavrilyuk, V.T., <span class="emphasis"><em>Generalization of the first polynomial algorithm
|
|
of E.Ya.Remez for the problem of constructing rational-fractional Chebyshev
|
|
approximations</em></span>, Ukr. Mat. Zh. 16 (1961), 575-585.
|
|
</p>
|
|
<p>
|
|
Some English language sources include:
|
|
</p>
|
|
<p>
|
|
Fraser, W., Hart, J.F., <span class="emphasis"><em>On the computation of rational approximations
|
|
to continuous functions</em></span>, Comm. of the ACM 5 (1962), 401-403, 414.
|
|
</p>
|
|
<p>
|
|
Ralston, A., <span class="emphasis"><em>Rational Chebyshev approximation by Remes' algorithms</em></span>,
|
|
Numer.Math. 7 (1965), no. 4, 322-330.
|
|
</p>
|
|
<p>
|
|
A. Ralston, <span class="emphasis"><em>Rational Chebyshev approximation, Mathematical Methods
|
|
for Digital Computers v. 2</em></span> (Ralston A., Wilf H., eds.), Wiley, New
|
|
York, 1967, pp. 264-284.
|
|
</p>
|
|
<p>
|
|
Hart, J.F. e.a., <span class="emphasis"><em>Computer approximations</em></span>, Wiley, New York
|
|
a.o., 1968.
|
|
</p>
|
|
<p>
|
|
Cody, W.J., Fraser, W., Hart, J.F., <span class="emphasis"><em>Rational Chebyshev approximation
|
|
using linear equations</em></span>, Numer.Math. 12 (1968), 242-251.
|
|
</p>
|
|
<p>
|
|
Cody, W.J., <span class="emphasis"><em>A survey of practical rational and polynomial approximation
|
|
of functions</em></span>, SIAM Review 12 (1970), no. 3, 400-423.
|
|
</p>
|
|
<p>
|
|
Barrar, R.B., Loeb, H.J., <span class="emphasis"><em>On the Remez algorithm for non-linear families</em></span>,
|
|
Numer.Math. 15 (1970), 382-391.
|
|
</p>
|
|
<p>
|
|
Dunham, Ch.B., <span class="emphasis"><em>Convergence of the Fraser-Hart algorithm for rational
|
|
Chebyshev approximation</em></span>, Math. Comp. 29 (1975), no. 132, 1078-1082.
|
|
</p>
|
|
<p>
|
|
G. L. Litvinov, <span class="emphasis"><em>Approximate construction of rational approximations
|
|
and the effect of error autocorrection</em></span>, Russian Journal of Mathematical
|
|
Physics, vol.1, No. 3, 1994.
|
|
</p>
|
|
</div>
|
|
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
|
<td align="left"></td>
|
|
<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
|
|
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
|
|
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
|
|
Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
|
|
Daryle Walker and Xiaogang Zhang<p>
|
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
|
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
|
</p>
|
|
</div></td>
|
|
</tr></table>
|
|
<hr>
|
|
<div class="spirit-nav">
|
|
<a accesskey="p" href="lanczos.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../backgrounders.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="refs.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
|
|
</div>
|
|
</body>
|
|
</html>
|