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From: johnmaddock_at_[hidden]
Date: 2007-07-02 05:14:42

Author: johnmaddock
Date: 2007-07-02 05:14:41 EDT (Mon, 02 Jul 2007)
New Revision: 7334

Sectionalised the docs a bit better, and added a sample longer article.

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+[section:remez Sample Article (The Remez Method)]
+The [@ Remez algorithm]
+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.
+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.
+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:
+E[sub abs](x) = f(x) - R(x)
+which is expressed in terms of absolute error, but we can equally use
+relative error:
+E[sub rel](x) = (f(x) - R(x)) / |f(x)|
+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.
+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
+* 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
+* The error function has N+1 roots, and N+2 extrema (minima and maxima).
+* The extrema alternate in sign, and all have the same magnitude.
+That means that if we know the location of the extrema of the error function
+then we can write N+2 simultaneous equations:
+R(x[sub i]) + (-1)[super i]E = f(x[sub i])
+where E is the maximal error term, and x[sub i] 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.
+['Unfortunately we don't know where the extrema of the error function are located!]
+[h4 The Remez Method]
+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.
+In the following discussion we'll use a concrete example to illustrate
+the Remez method: an approximation to the function e[super x][space] over
+the range \[-1, 1\].
+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).
+In order to obtain the N+1 coefficients of the interpolated polynomial
+we need N+1 points (x[sub 0]...x[sub N]): with our interpolated form
+passing through each of those points
+that yields N+1 simultaneous equations:
+f(x[sub i]) = P(x[sub i]) = c[sub 0] + c[sub 1]x[sub i] ... + c[sub N]x[sub i][super N]
+Which can be solved for the coefficients c[sub 0]...c[sub N] in P(x).
+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 /relative error/
+we get the following error function:
+Which has a peak relative error of 1.2x10[super -3].
+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 ['Chebyshev control points].
+In the simple case of a polynomial approximation, by interpolating through
+the roots of a Chebyshev polynomial we have in fact created a ['Chebyshev
+approximation] to the function: in terms of /absolute error/
+this is the best a priori choice for the interpolated form we can
+achieve, and typically is very close to the minimax solution.
+However, if we want to optimise for /relative error/, or if the approximation
+is a rational function, then the initial Chebyshev solution can be quite far
+from the ideal minimax solution.
+A more technical discussion of the theory involved can be found in this
+[@ online course].]
+[h4 Remez Step 1]
+The first step in the Remez method, given our current set of
+N+2 Chebyshev control points x[sub i], is to solve the N+2 simultaneous
+P(x[sub i]) + (-1)[super i]E = f(x[sub i])
+To obtain the error term E, and the coefficients of the polynomial P(x).
+This gives us a new approximation to f(x) that has the same error /E/ at
+each of the control points, and whose error function ['alternates in sign]
+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:
+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[super -4].
+[h4 Remez Step 2]
+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.
+We now have a choice: multi-point exchange, or single point exchange.
+In single point exchange, we move the control point nearest to the largest extrema to
+the absissa value of the extrema.
+In multi-point exchange we swap all the current control points, for the locations
+of the extrema.
+In our example we perform multi-point exchange.
+[h4 Iteration]
+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.
+For our current example, two more iterations converges on a minimax
+solution with a peak relative error of
+5x10[super -4] and an error function that looks like:
+[h4 Rational Approximations]
+If we wish to extend the Remez method to a rational approximation of the form
+f(x) = R(x) = P(x) / Q(x)
+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.
+The simultaneous equations to be solved are now:
+P(x[sub i]) / Q(x[sub i]) + (-1)[super i]E = f(x[sub i])
+Evaluated at the N+M+2 control points x[sub i].
+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!
+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.
+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[super x], we obtained 5x10[super -4] 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[super -5]. That's a 5 fold increase in accuracy, for the same number
+of terms overall.
+[h4 Practical Considerations]
+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.
+So far we have used a direct approximation:
+f(x) = R(x)
+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.
+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:
+f(x) = g(x) + R(x)
+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:
+f(x) = g(x)(c + R(x))
+Will yield a much better solution: here /c/ is a constant that is the approximate
+value of f(x)\/g(x) and R(x) is typically tiny compared to /c/. In this situation
+if R(x) is optimised for absolute error, then as long as its error is small compared
+to the constant /c/, that error will effectively get wiped out when R(x) is added to
+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[super -x[super 2]]\/x as x becomes large. Therefore using:
+erfc(z) = (C + R(x)) e[super -x[super 2]]/x
+as the approximating form seems like an obvious thing to try, and does indeed
+yield a useful approximation.
+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.
+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[super x][space] over \[-1,1\], the initial
+interpolated form was some way from the minimax solution:
+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[super -3][space]to 6x10[super -4]:
+It's clearly still not ideal, but it is only a few percent away from
+our desired minimax solution (5x10[super -4]).
+[h4 Remez Method Checklist]
+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.
+* Is the function smooth enough? Can it be better separated into
+a rapidly changing part, and an asymptotic part?
+* 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.
+* 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
+* Try skewing the initial interpolated approximation to minimise the
+error before you begin the Remez steps.
+* If the approximation won't converge or is ill-conditioned from one starting
+location, try starting from a different location.
+* 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.
+* 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.
+* Keep absissa values small: use a change of variable to keep the abscissa
+over, say \[0, b\], for some smallish value /b/.
+[h4 References]
+The original references for the Remez Method and it's extension
+to rational functions are unfortunately in Russian:
+Remez, E.Ya., ['Fundamentals of numerical methods for Chebyshev approximations],
+"Naukova Dumka", Kiev, 1969.
+Remez, E.Ya., Gavrilyuk, V.T., ['Computer development of certain approaches
+to the approximate construction of solutions of Chebyshev problems
+nonlinearly depending on parameters], Ukr. Mat. Zh. 12 (1960), 324-338.
+Gavrilyuk, V.T., ['Generalization of the first polynomial algorithm of
+E.Ya.Remez for the problem of constructing rational-fractional
+Chebyshev approximations], Ukr. Mat. Zh. 16 (1961), 575-585.
+Some English language sources include:
+Fraser, W., Hart, J.F., ['On the computation of rational approximations
+to continuous functions], Comm. of the ACM 5 (1962), 401-403, 414.
+Ralston, A., ['Rational Chebyshev approximation by Remes' algorithms],
+Numer.Math. 7 (1965), no. 4, 322-330.
+A. Ralston, ['Rational Chebyshev approximation, Mathematical
+Methods for Digital Computers v. 2] (Ralston A., Wilf H., eds.),
+Wiley, New York, 1967, pp. 264-284.
+Hart, J.F. e.a., ['Computer approximations], Wiley, New York a.o., 1968.
+Cody, W.J., Fraser, W., Hart, J.F., ['Rational Chebyshev approximation
+using linear equations], Numer.Math. 12 (1968), 242-251.
+Cody, W.J., ['A survey of practical rational and polynomial
+approximation of functions], SIAM Review 12 (1970), no. 3, 400-423.
+Barrar, R.B., Loeb, H.J., ['On the Remez algorithm for non-linear
+families], Numer.Math. 15 (1970), 382-391.
+Dunham, Ch.B., ['Convergence of the Fraser-Hart algorithm for rational
+Chebyshev approximation], Math. Comp. 29 (1975), no. 132, 1078-1082.
+G. L. Litvinov, ['Approximate construction of rational
+approximations and the effect of error autocorrection],
+Russian Journal of Mathematical Physics, vol.1, No. 3, 1994.
+[endsect][/section:remez The Remez Method]

Modified: sandbox/doc_test/doc/test.qbk
--- sandbox/doc_test/doc/test.qbk (original)
+++ sandbox/doc_test/doc/test.qbk 2007-07-02 05:14:41 EDT (Mon, 02 Jul 2007)
@@ -42,17 +42,21 @@
 [section Basic Formatting Tests]
-[h4 Font Styles]
+[section Font Styles]
 Here we go with some inline formatting:
 ['italic], [*bold], [_underline], [^teletype], [-strikethrough],
 we can combine styles as well: ['[*bold italic]], [_[^teletype with underline]].
-[h4 Replaceable Text]
+[section Replaceable Text]
 Text that is intended to be user-replaceable is [~rendered like this].
-[h4 Quotations]
+[section Quotations]
 Here we go: ["A question that sometimes drives me hazy: am I or are the others crazy?]--Einstein
@@ -63,23 +67,31 @@
 ["Here's the rule for bargains: ["Do other men, for they would do you.] That's
 the true business precept.]
-[h4 Inline Code]
+[section Inline Code]
 This text has inlined code `int main() { return 0; }` in it.
 The code should be syntax highlighted.
-[h4 Links]
+[section Links]
 Try this: [@ this is [*boost's] website....] it should
 be visible as a link.
-[h4 Footnotes]
+[section Footnotes]
 Here's one [footnote A sample footnote].
 And here's another [footnote Another sample footnote].
-[h4 Lists]
+[section Lists]
 A numbered list:
@@ -116,7 +128,9 @@
             * 2.b.2.a
             * 2.b.2.b
-[h4 Code Blocks]
+[section Code Blocks]
 These should be syntax highlighted:
@@ -380,7 +394,9 @@
    } // namespace boost
-[h4 Preformatted text]
+[section Preformatted text]
 Here's some sample program output:
@@ -412,13 +428,17 @@
 Standard deviation 1 is greater than standard deviation 2 REJECTED'''
-[h4 Blockquote]
+[section Blockquote]
 Lets indent the next paragraph:
 [:Here we go!!!]
-[h4 Admonishments]
+[section Admonishments]
 There are four admonishments supported by Docbook XML:
@@ -435,7 +455,9 @@
 They can contain more than one paragraph.
-[h4 Headings]
+[section Headings]
 Now try rendering some heading styles:
@@ -460,7 +482,9 @@
 They can contain more than one paragraph.
-[h4 Tables]
+[section Tables]
 Here's a big table with code and other tricky things:
@@ -500,7 +524,9 @@
 [[`estimate_beta`][`ibeta_invb(alpha, x, probability)`]]
-[h4 Variable Lists]
+[section Variable Lists]
 [variablelist A Variable List
     [[term 1] [The definition of term 1]]
@@ -508,7 +534,9 @@
     [[term 3] [The definition of term 3]]
-[h4 Callouts]
+[section Callouts]
 [import stub.cpp]
@@ -522,6 +550,8 @@
 [section Images]
 These are tricky enough that they warrent their own section.
@@ -540,3 +570,4 @@
 [include test_HTML4_symbols.qbk]
+[include remez.qbk]
\ No newline at end of file

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