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<title>The Remez Method</title>
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<div class="section">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.remez"></a><a class="link" href="remez.html" title="The Remez Method">The Remez Method</a>
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</h2></div></div></div>
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<p>
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The <a href="http://en.wikipedia.org/wiki/Remez_algorithm" target="_top">Remez algorithm</a>
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is a methodology for locating the minimax rational approximation to a function.
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This short article gives a brief overview of the method, but it should not
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be regarded as a thorough theoretical treatment, for that you should consult
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your favorite textbook.
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</p>
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<p>
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Imagine that you want to approximate some function f(x) by way of a rational
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function R(x), where R(x) may be either a polynomial P(x) or a ratio of two
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polynomials P(x)/Q(x) (a rational function). Initially we'll concentrate on
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the polynomial case, as it's by far the easier to deal with, later we'll extend
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to the full rational function case.
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</p>
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<p>
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We want to find the "best" rational approximation, where "best"
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is defined to be the approximation that has the least deviation from f(x).
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We can measure the deviation by way of an error function:
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</p>
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<p>
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E<sub>abs</sub>(x) = f(x) - R(x)
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</p>
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<p>
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which is expressed in terms of absolute error, but we can equally use relative
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error:
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</p>
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<p>
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E<sub>rel</sub>(x) = (f(x) - R(x)) / |f(x)|
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</p>
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<p>
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And indeed in general we can scale the error function in any way we want, it
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makes no difference to the maths, although the two forms above cover almost
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every practical case that you're likely to encounter.
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</p>
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<p>
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The minimax rational function R(x) is then defined to be the function that
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yields the smallest maximal value of the error function. Chebyshev showed that
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there is a unique minimax solution for R(x) that has the following properties:
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</p>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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If R(x) is a polynomial of degree N, then there are N+2 unknowns: the N+1
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coefficients of the polynomial, and maximal value of the error function.
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</li>
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<li class="listitem">
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The error function has N+1 roots, and N+2 extrema (minima and maxima).
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</li>
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<li class="listitem">
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The extrema alternate in sign, and all have the same magnitude.
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</li>
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</ul></div>
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<p>
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That means that if we know the location of the extrema of the error function
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then we can write N+2 simultaneous equations:
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</p>
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<p>
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R(x<sub>i</sub>) + (-1)<sup>i</sup>E = f(x<sub>i</sub>)
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</p>
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<p>
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where E is the maximal error term, and x<sub>i</sub> are the abscissa values of the N+2
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extrema of the error function. It is then trivial to solve the simultaneous
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equations to obtain the polynomial coefficients and the error term.
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</p>
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<p>
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<span class="emphasis"><em>Unfortunately we don't know where the extrema of the error function
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are located!</em></span>
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</p>
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<h5>
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<a name="math_toolkit.remez.h0"></a>
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<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
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Remez Method</a>
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</h5>
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<p>
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The Remez method is an iterative technique which, given a broad range of assumptions,
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will converge on the extrema of the error function, and therefore the minimax
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solution.
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</p>
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<p>
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In the following discussion we'll use a concrete example to illustrate the
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Remez method: an approximation to the function e<sup>x</sup>   over the range [-1, 1].
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</p>
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<p>
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Before we can begin the Remez method, we must obtain an initial value for the
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location of the extrema of the error function. We could "guess" these,
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but a much closer first approximation can be obtained by first constructing
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an interpolated polynomial approximation to f(x).
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</p>
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<p>
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In order to obtain the N+1 coefficients of the interpolated polynomial we need
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N+1 points (x<sub>0</sub>...x<sub>N</sub>): with our interpolated form passing through each of those
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points that yields N+1 simultaneous equations:
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</p>
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<p>
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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>
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</p>
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<p>
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Which can be solved for the coefficients c<sub>0</sub>...c<sub>N</sub> in P(x).
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</p>
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<p>
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Obviously this is not a minimax solution, indeed our only guarantee is that
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f(x) and P(x) touch at N+1 locations, away from those points the error may
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be arbitrarily large. However, we would clearly like this initial approximation
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to be as close to f(x) as possible, and it turns out that using the zeros of
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an orthogonal polynomial as the initial interpolation points is a good choice.
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In our example we'll use the zeros of a Chebyshev polynomial as these are particularly
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easy to calculate, interpolating for a polynomial of degree 4, and measuring
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<span class="emphasis"><em>relative error</em></span> we get the following error function:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../graphs/remez-2.png"></span>
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</p>
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<p>
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Which has a peak relative error of 1.2x10<sup>-3</sup>.
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</p>
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<p>
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While this is a pretty good approximation already, judging by the shape of
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the error function we can clearly do better. Before starting on the Remez method
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propper, we have one more step to perform: locate all the extrema of the error
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function, and store these locations as our initial <span class="emphasis"><em>Chebyshev control
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points</em></span>.
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</p>
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<div class="note"><table border="0" summary="Note">
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<tr>
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<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../doc/src/images/note.png"></td>
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<th align="left">Note</th>
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</tr>
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<tr><td align="left" valign="top">
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<p>
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In the simple case of a polynomial approximation, by interpolating through
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the roots of a Chebyshev polynomial we have in fact created a <span class="emphasis"><em>Chebyshev
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approximation</em></span> to the function: in terms of <span class="emphasis"><em>absolute
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error</em></span> this is the best a priori choice for the interpolated form
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we can achieve, and typically is very close to the minimax solution.
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</p>
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<p>
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However, if we want to optimise for <span class="emphasis"><em>relative error</em></span>,
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or if the approximation is a rational function, then the initial Chebyshev
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solution can be quite far from the ideal minimax solution.
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</p>
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<p>
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A more technical discussion of the theory involved can be found in this
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<a href="http://math.fullerton.edu/mathews/n2003/ChebyshevPolyMod.html" target="_top">online
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course</a>.
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</p>
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</td></tr>
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</table></div>
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<h5>
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<a name="math_toolkit.remez.h1"></a>
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<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
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Step 1</a>
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</h5>
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<p>
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The first step in the Remez method, given our current set of N+2 Chebyshev
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control points x<sub>i</sub>, is to solve the N+2 simultaneous equations:
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</p>
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<p>
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P(x<sub>i</sub>) + (-1)<sup>i</sup>E = f(x<sub>i</sub>)
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</p>
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<p>
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To obtain the error term E, and the coefficients of the polynomial P(x).
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</p>
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<p>
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This gives us a new approximation to f(x) that has the same error <span class="emphasis"><em>E</em></span>
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at each of the control points, and whose error function <span class="emphasis"><em>alternates
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in sign</em></span> at the control points. This is still not necessarily the
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minimax solution though: since the control points may not be at the extrema
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of the error function. After this first step here's what our approximation's
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error function looks like:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../graphs/remez-3.png"></span>
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</p>
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<p>
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Clearly this is still not the minimax solution since the control points are
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not located at the extrema, but the maximum relative error has now dropped
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to 5.6x10<sup>-4</sup>.
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</p>
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<h5>
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<a name="math_toolkit.remez.h2"></a>
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<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
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Step 2</a>
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</h5>
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<p>
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The second step is to locate the extrema of the new approximation, which we
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do in two stages: first, since the error function changes sign at each control
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point, we must have N+1 roots of the error function located between each pair
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of N+2 control points. Once these roots are found by standard root finding
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techniques, we know that N extrema are bracketed between each pair of roots,
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plus two more between the endpoints of the range and the first and last roots.
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The N+2 extrema can then be found using standard function minimisation techniques.
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</p>
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<p>
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We now have a choice: multi-point exchange, or single point exchange.
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</p>
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<p>
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In single point exchange, we move the control point nearest to the largest
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extrema to the absissa value of the extrema.
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</p>
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<p>
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In multi-point exchange we swap all the current control points, for the locations
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of the extrema.
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</p>
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<p>
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In our example we perform multi-point exchange.
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</p>
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<h5>
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<a name="math_toolkit.remez.h3"></a>
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<span class="phrase"><a name="math_toolkit.remez.iteration"></a></span><a class="link" href="remez.html#math_toolkit.remez.iteration">Iteration</a>
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</h5>
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<p>
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The Remez method then performs steps 1 and 2 above iteratively until the control
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points are located at the extrema of the error function: this is then the minimax
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solution.
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</p>
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<p>
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For our current example, two more iterations converges on a minimax solution
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with a peak relative error of 5x10<sup>-4</sup> and an error function that looks like:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../graphs/remez-4.png"></span>
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</p>
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<h5>
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<a name="math_toolkit.remez.h4"></a>
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<span class="phrase"><a name="math_toolkit.remez.rational_approximations"></a></span><a class="link" href="remez.html#math_toolkit.remez.rational_approximations">Rational
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Approximations</a>
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</h5>
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<p>
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If we wish to extend the Remez method to a rational approximation of the form
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</p>
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<p>
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f(x) = R(x) = P(x) / Q(x)
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</p>
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<p>
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where P(x) and Q(x) are polynomials, then we proceed as before, except that
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now we have N+M+2 unknowns if P(x) is of order N and Q(x) is of order M. This
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assumes that Q(x) is normalised so that its leading coefficient is 1, giving
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N+M+1 polynomial coefficients in total, plus the error term E.
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</p>
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<p>
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The simultaneous equations to be solved are now:
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</p>
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<p>
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P(x<sub>i</sub>) / Q(x<sub>i</sub>) + (-1)<sup>i</sup>E = f(x<sub>i</sub>)
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</p>
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<p>
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Evaluated at the N+M+2 control points x<sub>i</sub>.
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</p>
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<p>
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Unfortunately these equations are non-linear in the error term E: we can only
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solve them if we know E, and yet E is one of the unknowns!
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</p>
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<p>
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The method usually adopted to solve these equations is an iterative one: we
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guess the value of E, solve the equations to obtain a new value for E (as well
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as the polynomial coefficients), then use the new value of E as the next guess.
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The method is repeated until E converges on a stable value.
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</p>
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<p>
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These complications extend the running time required for the development of
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rational approximations quite considerably. It is often desirable to obtain
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a rational rather than polynomial approximation none the less: rational approximations
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will often match more difficult to approximate functions, to greater accuracy,
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and with greater efficiency, than their polynomial alternatives. For example,
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if we takes our previous example of an approximation to e<sup>x</sup>, we obtained 5x10<sup>-4</sup> accuracy
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with an order 4 polynomial. If we move two of the unknowns into the denominator
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to give a pair of order 2 polynomials, and re-minimise, then the peak relative
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error drops to 8.7x10<sup>-5</sup>. That's a 5 fold increase in accuracy, for the same
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number of terms overall.
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</p>
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<h5>
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<a name="math_toolkit.remez.h5"></a>
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<span class="phrase"><a name="math_toolkit.remez.practical_considerations"></a></span><a class="link" href="remez.html#math_toolkit.remez.practical_considerations">Practical
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Considerations</a>
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</h5>
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<p>
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Most treatises on approximation theory stop at this point. However, from a
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practical point of view, most of the work involves finding the right approximating
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form, and then persuading the Remez method to converge on a solution.
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</p>
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<p>
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So far we have used a direct approximation:
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</p>
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<p>
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f(x) = R(x)
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</p>
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<p>
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But this will converge to a useful approximation only if f(x) is smooth. In
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addition round-off errors when evaluating the rational form mean that this
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will never get closer than within a few epsilon of machine precision. Therefore
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this form of direct approximation is often reserved for situations where we
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want efficiency, rather than accuracy.
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</p>
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<p>
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The first step in improving the situation is generally to split f(x) into a
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dominant part that we can compute accurately by another method, and a slowly
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changing remainder which can be approximated by a rational approximation. We
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might be tempted to write:
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</p>
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<p>
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f(x) = g(x) + R(x)
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</p>
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<p>
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where g(x) is the dominant part of f(x), but if f(x)/g(x) is approximately
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constant over the interval of interest then:
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</p>
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<p>
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f(x) = g(x)(c + R(x))
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</p>
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<p>
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Will yield a much better solution: here <span class="emphasis"><em>c</em></span> is a constant
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that is the approximate value of f(x)/g(x) and R(x) is typically tiny compared
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to <span class="emphasis"><em>c</em></span>. In this situation if R(x) is optimised for absolute
|
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error, then as long as its error is small compared to the constant <span class="emphasis"><em>c</em></span>,
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that error will effectively get wiped out when R(x) is added to <span class="emphasis"><em>c</em></span>.
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</p>
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<p>
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The difficult part is obviously finding the right g(x) to extract from your
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function: often the asymptotic behaviour of the function will give a clue,
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so for example the function <a class="link" href="sf_erf/error_function.html" title="Error Functions">erfc</a>
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becomes proportional to e<sup>-x<sup>2</sup></sup>/x as x becomes large. Therefore using:
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</p>
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<p>
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erfc(z) = (C + R(x)) e<sup>-x<sup>2</sup></sup>/x
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</p>
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<p>
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as the approximating form seems like an obvious thing to try, and does indeed
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yield a useful approximation.
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</p>
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<p>
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However, the difficulty then becomes one of converging the minimax solution.
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Unfortunately, it is known that for some functions the Remez method can lead
|
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to divergent behaviour, even when the initial starting approximation is quite
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good. Furthermore, it is not uncommon for the solution obtained in the first
|
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Remez step above to be a bad one: the equations to be solved are generally
|
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"stiff", often very close to being singular, and assuming a solution
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is found at all, round-off errors and a rapidly changing error function, can
|
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lead to a situation where the error function does not in fact change sign at
|
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each control point as required. If this occurs, it is fatal to the Remez method.
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It is also possible to obtain solutions that are perfectly valid mathematically,
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but which are quite useless computationally: either because there is an unavoidable
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amount of roundoff error in the computation of the rational function, or because
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the denominator has one or more roots over the interval of the approximation.
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In the latter case while the approximation may have the correct limiting value
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at the roots, the approximation is nonetheless useless.
|
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</p>
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<p>
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Assuming that the approximation does not have any fatal errors, and that the
|
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only issue is converging adequately on the minimax solution, the aim is to
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get as close as possible to the minimax solution before beginning the Remez
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method. Using the zeros of a Chebyshev polynomial for the initial interpolation
|
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is a good start, but may not be ideal when dealing with relative errors and/or
|
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rational (rather than polynomial) approximations. One approach is to skew the
|
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initial interpolation points to one end: for example if we raise the roots
|
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of the Chebyshev polynomial to a positive power greater than 1 then the roots
|
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will be skewed towards the middle of the [-1,1] interval, while a positive
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power less than one will skew them towards either end. More usefully, if we
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initially rescale the points over [0,1] and then raise to a positive power,
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we can skew them to the left or right. Returning to our example of e<sup>x</sup>   over [-1,1],
|
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the initial interpolated form was some way from the minimax solution:
|
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</p>
|
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<p>
|
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<span class="inlinemediaobject"><img src="../../graphs/remez-2.png"></span>
|
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</p>
|
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<p>
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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 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>
|
|
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
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<td align="left"></td>
|
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<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
|
|
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
|
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
|
|
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
|
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
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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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</p>
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</div></td>
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</tr></table>
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