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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="document_to_test_formatting.remez"></a><a class="link" href="remez.html" title="Sample Article (The Remez Method)"> Sample Article (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 f(x) by way of a rational
function R(x), where R(x) may be either a polynomial P(x) or a ratio of two
polynomials P(x)/Q(x) (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 f(x).
We can measure the deviation by way of an error function:
</p>
<p>
E<sub>abs</sub>(x) = f(x) - R(x)
</p>
<p>
which is expressed in terms of absolute error, but we can equally use relative
error:
</p>
<p>
E<sub>rel</sub>(x) = (f(x) - R(x)) / |f(x)|
</p>
<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 R(x) 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 R(x) that has the following properties:
</p>
<div class="itemizedlist"><ul class="itemizedlist" type="disc">
<li class="listitem">
If R(x) is a polynomial of degree N, then there are N+2 unknowns: the N+1
coefficients of the polynomial, and maximal value of the error function.
</li>
<li class="listitem">
The error function has N+1 roots, and N+2 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 N+2 simultaneous equations:
</p>
<p>
R(x<sub>i</sub>) + (-1)<sup>i</sup>E = f(x<sub>i</sub>)
</p>
<p>
where E is the maximal error term, and x<sub>i</sub> are the abscissa values of the N+2
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>
<a name="document_to_test_formatting.remez.the_remez_method"></a><h5>
<a name="id771060"></a>
<a class="link" href="remez.html#document_to_test_formatting.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 f(x).
</p>
<p>
In order to obtain the N+1 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 N+1 simultaneous equations:
</p>
<p>
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>
</p>
<p>
Which can be solved for the coefficients c<sub>0</sub>...c<sub>N</sub> in P(x).
</p>
<p>
Obviously this is not a minimax solution, indeed our only guarantee is that
f(x) and P(x) touch at N+1 locations, away from those points the error may
be arbitrarily large. However, we would clearly like this initial approximation
to be as close to f(x) 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="../images/remez-2.png" alt="remez-2"></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>
<a name="document_to_test_formatting.remez.remez_step_1"></a><h5>
<a name="id771248"></a>
<a class="link" href="remez.html#document_to_test_formatting.remez.remez_step_1">Remez Step 1</a>
</h5>
<p>
The first step in the Remez method, given our current set of N+2 Chebyshev
control points x<sub>i</sub>, is to solve the N+2 simultaneous equations:
</p>
<p>
P(x<sub>i</sub>) + (-1)<sup>i</sup>E = f(x<sub>i</sub>)
</p>
<p>
To obtain the error term E, and the coefficients of the polynomial P(x).
</p>
<p>
This gives us a new approximation to f(x) 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="../images/remez-3.png" alt="remez-3"></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>
<a name="document_to_test_formatting.remez.remez_step_2"></a><h5>
<a name="id771342"></a>
<a class="link" href="remez.html#document_to_test_formatting.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>
<a name="document_to_test_formatting.remez.iteration"></a><h5>
<a name="id771387"></a>
<a class="link" href="remez.html#document_to_test_formatting.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="../images/remez-4.png" alt="remez-4"></span>
</p>
<a name="document_to_test_formatting.remez.rational_approximations"></a><h5>
<a name="id771441"></a>
<a class="link" href="remez.html#document_to_test_formatting.remez.rational_approximations">Rational
Approximations</a>
</h5>
<p>
If we wish to extend the Remez method to a rational approximation of the form
</p>
<p>
f(x) = R(x) = P(x) / Q(x)
</p>
<p>
where P(x) and Q(x) are polynomials, then we proceed as before, except that
now we have N+M+2 unknowns if P(x) is of order N and Q(x) is of order M. This
assumes that Q(x) is normalised so that it's leading coefficient is 1, giving
N+M+1 polynomial coefficients in total, plus the error term E.
</p>
<p>
The simultaneous equations to be solved are now:
</p>
<p>
P(x<sub>i</sub>) / Q(x<sub>i</sub>) + (-1)<sup>i</sup>E = f(x<sub>i</sub>)
</p>
<p>
Evaluated at the N+M+2 control points x<sub>i</sub>.
</p>
<p>
Unfortunately these equations are non-linear in the error term E: we can only
solve them if we know E, and yet E is one of the unknowns!
</p>
<p>
The method usually adopted to solve these equations is an iterative one: we
guess the value of E, solve the equations to obtain a new value for E (as well
as the polynomial coefficients), then use the new value of E as the next guess.
The method is repeated until E 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>
<a name="document_to_test_formatting.remez.practical_considerations"></a><h5>
<a name="id771550"></a>
<a class="link" href="remez.html#document_to_test_formatting.remez.practical_considerations">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>
<p>
f(x) = R(x)
</p>
<p>
But this will converge to a useful approximation only if f(x) 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 f(x) 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>
<p>
f(x) = g(x) + R(x)
</p>
<p>
where g(x) is the dominant part of f(x), but if f(x)/g(x) is approximately
constant over the interval of interest then:
</p>
<p>
f(x) = g(x)(c + R(x))
</p>
<p>
Will yield a much better solution: here <span class="emphasis"><em>c</em></span> is a constant
that is the approximate value of f(x)/g(x) and R(x) is typically tiny compared
to <span class="emphasis"><em>c</em></span>. In this situation if R(x) 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 R(x) is added to <span class="emphasis"><em>c</em></span>.
</p>
<p>
The difficult part is obviously finding the right g(x) to extract from your
function: often the asymptotic behaviour of the function will give a clue,
so for example the function __erfc becomes proportional to e<sup>-x<sup>2</sup></sup>/x as x becomes
large. Therefore using:
</p>
<p>
erfc(z) = (C + R(x)) e<sup>-x<sup>2</sup></sup>/x
</p>
<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="../images/remez-2.png" alt="remez-2"></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="../images/remez-5.png" alt="remez-5"></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>
<a name="document_to_test_formatting.remez.remez_method_checklist"></a><h5>
<a name="id771737"></a>
<a class="link" href="remez.html#document_to_test_formatting.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" 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>
<a name="document_to_test_formatting.remez.references"></a><h5>
<a name="id771857"></a>
<a class="link" href="remez.html#document_to_test_formatting.remez.references">References</a>
</h5>
<p>
The original references for the Remez Method and it's 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>
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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>)
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