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  1[section:remez Sample Article (The Remez Method)]
  2
  3The [@http://en.wikipedia.org/wiki/Remez_algorithm Remez algorithm]
  4is a methodology for locating the minimax rational approximation
  5to a function.  This short article gives a brief overview of the method, but
  6it should not be regarded as a thorough theoretical treatment, for that you
  7should consult your favorite textbook.
  8
  9Imagine that you want to approximate some function f(x) by way of a rational
 10function R(x), where R(x) may be either a polynomial P(x) or a ratio of two
 11polynomials P(x)/Q(x) (a rational function).  Initially we'll concentrate on the 
 12polynomial case, as it's by far the easier to deal with, later we'll extend 
 13to the full rational function case.  
 14
 15We want to find the "best" rational approximation, where
 16"best" is defined to be the approximation that has the least deviation
 17from f(x).  We can measure the deviation by way of an error function:
 18
 19E[sub abs](x) = f(x) - R(x)
 20
 21which is expressed in terms of absolute error, but we can equally use
 22relative error:
 23
 24E[sub rel](x) = (f(x) - R(x)) / |f(x)|
 25
 26And indeed in general we can scale the error function in any way we want, it
 27makes no difference to the maths, although the two forms above cover almost
 28every practical case that you're likely to encounter.
 29
 30The minimax rational function R(x) is then defined to be the function that
 31yields the smallest maximal value of the error function.  Chebyshev showed
 32that there is a unique minimax solution for R(x) that has the following
 33properties:
 34
 35* If R(x) is a polynomial of degree N, then there are N+2 unknowns:
 36the N+1 coefficients of the polynomial, and maximal value of the error
 37function.
 38* The error function has N+1 roots, and N+2 extrema (minima and maxima).
 39* The extrema alternate in sign, and all have the same magnitude.
 40
 41That means that if we know the location of the extrema of the error function
 42then we can write N+2 simultaneous equations:
 43
 44R(x[sub i]) + (-1)[super i]E = f(x[sub i])
 45
 46where E is the maximal error term, and x[sub i] are the abscissa values of the
 47N+2 extrema of the error function.  It is then trivial to solve the simultaneous
 48equations to obtain the polynomial coefficients and the error term.
 49
 50['Unfortunately we don't know where the extrema of the error function are located!]
 51
 52[h4 The Remez Method]
 53
 54The Remez method is an iterative technique which, given a broad range of
 55assumptions, will converge on the extrema of the error function, and therefore
 56the minimax solution.
 57
 58In the following discussion we'll use a concrete example to illustrate
 59the Remez method: an approximation to the function e[super x][space] over
 60the range \[-1, 1\].
 61
 62Before we can begin the Remez method, we must obtain an initial value
 63for the location of the extrema of the error function.  We could "guess"
 64these, but a much closer first approximation can be obtained by first  
 65constructing an interpolated polynomial approximation to f(x).
 66
 67In order to obtain the N+1 coefficients of the interpolated polynomial
 68we need N+1 points (x[sub 0]...x[sub N]): with our interpolated form 
 69passing through each of those points
 70that yields N+1 simultaneous equations:
 71
 72f(x[sub i]) = P(x[sub i]) = c[sub 0] + c[sub 1]x[sub i] ... + c[sub N]x[sub i][super N]
 73
 74Which can be solved for the coefficients c[sub 0]...c[sub N] in P(x).
 75
 76Obviously this is not a minimax solution, indeed our only guarantee is that f(x) and 
 77P(x) touch at N+1 locations, away from those points the error may be arbitrarily
 78large.  However, we would clearly like this initial approximation to be as close to
 79f(x) as possible, and it turns out that using the zeros of an orthogonal polynomial
 80as the initial interpolation points is a good choice.  In our example we'll use the 
 81zeros of a Chebyshev polynomial as these are particularly easy to calculate, 
 82interpolating for a polynomial of degree 4, and measuring /relative error/
 83we get the following error function:
 84
 85[$images/remez-2.png]
 86
 87Which has a peak relative error of 1.2x10[super -3].
 88
 89While this is a pretty good approximation already, judging by the 
 90shape of the error function we can clearly do better.  Before starting
 91on the Remez method propper, we have one more step to perform: locate
 92all the extrema of the error function, and store
 93these locations as our initial ['Chebyshev control points].
 94
 95[note
 96In the simple case of a polynomial approximation, by interpolating through
 97the roots of a Chebyshev polynomial we have in fact created a ['Chebyshev
 98approximation] to the function: in terms of /absolute error/
 99this is the best a priori choice for the interpolated form we can
100achieve, and typically is very close to the minimax solution.
101
102However, if we want to optimise for /relative error/, or if the approximation
103is a rational function, then the initial Chebyshev solution can be quite far
104from the ideal minimax solution.  
105
106A more technical discussion of the theory involved can be found in this
107[@http://math.fullerton.edu/mathews/n2003/ChebyshevPolyMod.html online course].]
108
109[h4 Remez Step 1]
110
111The first step in the Remez method, given our current set of
112N+2 Chebyshev control points x[sub i], is to solve the N+2 simultaneous
113equations:
114
115P(x[sub i]) + (-1)[super i]E = f(x[sub i])
116
117To obtain the error term E, and the coefficients of the polynomial P(x).
118
119This gives us a new approximation to f(x) that has the same error /E/ at
120each of the control points, and whose error function ['alternates in sign]
121at the control points.  This is still not necessarily the minimax 
122solution though: since the control points may not be at the extrema of the error
123function.  After this first step here's what our approximation's error
124function looks like:
125
126[$images/remez-3.png]
127
128Clearly this is still not the minimax solution since the control points
129are not located at the extrema, but the maximum relative error has now
130dropped to 5.6x10[super -4].
131
132[h4 Remez Step 2]
133
134The second step is to locate the extrema of the new approximation, which we do 
135in two stages:  first, since the error function changes sign at each
136control point, we must have N+1 roots of the error function located between
137each pair of N+2 control points.  Once these roots are found by standard root finding 
138techniques, we know that N extrema are bracketed between each pair of
139roots, plus two more between the endpoints of the range and the first and last roots.
140The N+2 extrema can then be found using standard function minimisation techniques.
141
142We now have a choice: multi-point exchange, or single point exchange.
143
144In single point exchange, we move the control point nearest to the largest extrema to
145the absissa value of the extrema.
146
147In multi-point exchange we swap all the current control points, for the locations
148of the extrema.
149
150In our example we perform multi-point exchange.
151
152[h4 Iteration]
153
154The Remez method then performs steps 1 and 2 above iteratively until the control
155points are located at the extrema of the error function: this is then
156the minimax solution.
157
158For our current example, two more iterations converges on a minimax
159solution with a peak relative error of
1605x10[super -4] and an error function that looks like:
161
162[$images/remez-4.png]
163
164[h4 Rational Approximations]
165
166If we wish to extend the Remez method to a rational approximation of the form
167
168f(x) = R(x) = P(x) / Q(x)
169
170where P(x) and Q(x) are polynomials, then we proceed as before, except that now
171we have N+M+2 unknowns if P(x) is of order N and Q(x) is of order M.  This assumes
172that Q(x) is normalised so that it's leading coefficient is 1, giving
173N+M+1 polynomial coefficients in total, plus the error term E.
174
175The simultaneous equations to be solved are now:
176
177P(x[sub i]) / Q(x[sub i]) + (-1)[super i]E = f(x[sub i])
178
179Evaluated at the N+M+2 control points x[sub i].
180
181Unfortunately these equations are non-linear in the error term E: we can only
182solve them if we know E, and yet E is one of the unknowns!
183
184The method usually adopted to solve these equations is an iterative one: we guess the
185value of E, solve the equations to obtain a new value for E (as well as the polynomial
186coefficients), then use the new value of E as the next guess.  The method is
187repeated until E converges on a stable value.
188
189These complications extend the running time required for the development
190of rational approximations quite considerably. It is often desirable
191to obtain a rational rather than polynomial approximation none the less:
192rational approximations will often match more difficult to approximate
193functions, to greater accuracy, and with greater efficiency, than their
194polynomial alternatives.  For example, if we takes our previous example
195of an approximation to e[super x], we obtained 5x10[super -4] accuracy
196with an order 4 polynomial.  If we move two of the unknowns into the denominator
197to give a pair of order 2 polynomials, and re-minimise, then the peak relative error drops
198to 8.7x10[super -5].  That's a 5 fold increase in accuracy, for the same number 
199of terms overall.
200
201[h4 Practical Considerations]
202
203Most treatises on approximation theory stop at this point.  However, from
204a practical point of view, most of the work involves finding the right
205approximating form, and then persuading the Remez method to converge
206on a solution.
207
208So far we have used a direct approximation:
209
210f(x) = R(x)
211
212But this will converge to a useful approximation only if f(x) is smooth.  In
213addition round-off errors when evaluating the rational form mean that this
214will never get closer than within a few epsilon of machine precision.  
215Therefore this form of direct approximation is often reserved for situations
216where we want efficiency, rather than accuracy.
217
218The first step in improving the situation is generally to split f(x) into
219a dominant part that we can compute accurately by another method, and a 
220slowly changing remainder which can be approximated by a rational approximation.
221We might be tempted to write:
222
223f(x) = g(x) + R(x)
224
225where g(x) is the dominant part of f(x), but if f(x)\/g(x) is approximately
226constant over the interval of interest then:
227
228f(x) = g(x)(c + R(x))
229
230Will yield a much better solution: here /c/ is a constant that is the approximate
231value of f(x)\/g(x) and R(x) is typically tiny compared to /c/.  In this situation
232if R(x) is optimised for absolute error, then as long as its error is small compared
233to the constant /c/, that error will effectively get wiped out when R(x) is added to
234/c/.
235
236The difficult part is obviously finding the right g(x) to extract from your
237function: often the asymptotic behaviour of the function will give a clue, so
238for example the function __erfc becomes proportional to 
239e[super -x[super 2]]\/x as x becomes large.  Therefore using:
240
241erfc(z) = (C + R(x)) e[super -x[super 2]]/x
242
243as the approximating form seems like an obvious thing to try, and does indeed
244yield a useful approximation.
245
246However, the difficulty then becomes one of converging the minimax solution.
247Unfortunately, it is known that for some functions the Remez method can lead
248to divergent behaviour, even when the initial starting approximation is quite good.
249Furthermore, it is not uncommon for the solution obtained in the first Remez step
250above to be a bad one: the equations to be solved are generally "stiff", often
251very close to being singular, and assuming a solution is found at all, round-off
252errors and a rapidly changing error function, can lead to a situation where the
253error function does not in fact change sign at each control point as required.
254If this occurs, it is fatal to the Remez method.  It is also possible to
255obtain solutions that are perfectly valid mathematically, but which are
256quite useless computationally: either because there is an unavoidable amount
257of roundoff error in the computation of the rational function, or because
258the denominator has one or more roots over the interval of the approximation.
259In the latter case while the approximation may have the correct limiting value at
260the roots, the approximation is nonetheless useless.
261
262Assuming that the approximation does not have any fatal errors, and that the only
263issue is converging adequately on the minimax solution, the aim is to
264get as close as possible to the minimax solution before beginning the Remez method.
265Using the zeros of a Chebyshev polynomial for the initial interpolation is a 
266good start, but may not be ideal when dealing with relative errors and\/or
267rational (rather than polynomial) approximations.  One approach is to skew
268the initial interpolation points to one end: for example if we raise the
269roots of the Chebyshev polynomial to a positive power greater than 1 
270then the roots will be skewed towards the middle of the \[-1,1\] interval, 
271while a positive power less than one
272will skew them towards either end.  More usefully, if we initially rescale the
273points over \[0,1\] and then raise to a positive power, we can skew them to the left 
274or right.  Returning to our example of e[super x][space] over \[-1,1\], the initial
275interpolated form was some way from the minimax solution:
276
277[$images/remez-2.png]
278
279However, if we first skew the interpolation points to the left (rescale them
280to \[0, 1\], raise to the power 1.3, and then rescale back to \[-1,1\]) we
281reduce the error from 1.3x10[super -3][space]to 6x10[super -4]:
282
283[$images/remez-5.png]
284
285It's clearly still not ideal, but it is only a few percent away from
286our desired minimax solution (5x10[super -4]).
287
288[h4 Remez Method Checklist]
289
290The following lists some of the things to check if the Remez method goes wrong, 
291it is by no means an exhaustive list, but is provided in the hopes that it will
292prove useful.
293
294* Is the function smooth enough?  Can it be better separated into
295a rapidly changing part, and an asymptotic part?
296* Does the function being approximated have any "blips" in it?  Check
297for problems as the function changes computation method, or
298if a root, or an infinity has been divided out.  The telltale
299sign is if there is a narrow region where the Remez method will
300not converge.
301* Check you have enough accuracy in your calculations: remember that
302the Remez method works on the difference between the approximation
303and the function being approximated: so you must have more digits of
304precision available than the precision of the approximation
305being constructed.  So for example at double precision, you
306shouldn't expect to be able to get better than a float precision
307approximation.
308* Try skewing the initial interpolated approximation to minimise the
309error before you begin the Remez steps.
310* If the approximation won't converge or is ill-conditioned from one starting
311location, try starting from a different location.
312* If a rational function won't converge, one can minimise a polynomial
313(which presents no problems), then rotate one term from the numerator to
314the denominator and minimise again.  In theory one can continue moving
315terms one at a time from numerator to denominator, and then re-minimising, 
316retaining the last set of control points at each stage.
317* Try using a smaller interval.  It may also be possible to optimise over
318one (small) interval, rescale the control points over a larger interval,
319and then re-minimise.
320* Keep absissa values small: use a change of variable to keep the abscissa
321over, say \[0, b\], for some smallish value /b/.
322
323[h4 References]
324
325The original references for the Remez Method and it's extension
326to rational functions are unfortunately in Russian:
327
328Remez, E.Ya., ['Fundamentals of numerical methods for Chebyshev approximations], 
329"Naukova Dumka", Kiev, 1969.
330
331Remez, E.Ya., Gavrilyuk, V.T., ['Computer development of certain approaches 
332to the approximate construction of solutions of Chebyshev problems 
333nonlinearly depending on parameters], Ukr. Mat. Zh. 12 (1960), 324-338.
334
335Gavrilyuk, V.T., ['Generalization of the first polynomial algorithm of 
336E.Ya.Remez for the problem of constructing rational-fractional 
337Chebyshev approximations], Ukr. Mat. Zh. 16 (1961), 575-585.
338
339Some English language sources include:
340
341Fraser, W., Hart, J.F., ['On the computation of rational approximations 
342to continuous functions], Comm. of the ACM 5 (1962), 401-403, 414.
343
344Ralston, A., ['Rational Chebyshev approximation by Remes' algorithms], 
345Numer.Math. 7 (1965), no. 4, 322-330.
346
347A. Ralston, ['Rational Chebyshev approximation, Mathematical 
348Methods for Digital Computers v. 2] (Ralston A., Wilf H., eds.), 
349Wiley, New York, 1967, pp. 264-284.
350
351Hart, J.F. e.a., ['Computer approximations], Wiley, New York a.o., 1968.
352
353Cody, W.J., Fraser, W., Hart, J.F., ['Rational Chebyshev approximation 
354using linear equations], Numer.Math. 12 (1968), 242-251.
355
356Cody, W.J., ['A survey of practical rational and polynomial 
357approximation of functions], SIAM Review 12 (1970), no. 3, 400-423.
358
359Barrar, R.B., Loeb, H.J., ['On the Remez algorithm for non-linear 
360families], Numer.Math. 15 (1970), 382-391.
361
362Dunham, Ch.B., ['Convergence of the Fraser-Hart algorithm for rational 
363Chebyshev approximation], Math. Comp. 29 (1975), no. 132, 1078-1082.
364
365G. L. Litvinov, ['Approximate construction of rational
366approximations and the effect of error autocorrection],
367Russian Journal of Mathematical Physics, vol.1, No. 3, 1994.
368
369[endsect][/section:remez The Remez Method]
370
371
372