on Mon Jul 02 2007, "John Maddock" <john-AT-johnmaddock.co.uk> wrote:
> The main index page you refer to there scrolls so slowly on my > system as to be unusable - the sample page > http://randysimons.com/overige/multicolumn/ is much better in this > respect, but the amount of scrolling up and down to read the columns > is IMO intolerable. How would this cope with a page like this: > http://freespace.virgin.net/boost.regex/toolkit/html/math_toolkit/backgrounders/remez.html > that which as well as being much longer than the average reference > page has more than it's fair share of media objects? One would columnize entire sections, or tuples of sections, before moving on to a separate area. So for example (your page uses a subheading that matches the main heading, which is a bit weird. I also used elipses liberally because you'd be able to fit lots more horizontally in a column than I can here with a fixed-width font. +------------------------------+------------------------------+------------------------------+ |** The Remez Method ** | We want to find the "best" | In the following discussion | | |rational approximation, where |we'll use a concrete example | |The Remez algorithm is a |"best" is defined to be the |to illustrate the Remez | |methodology for locating the |approximation that has the |method: an approximation to | |minimax rational approximation|least deviation from f(x). We |the function ex over the range| |to a function. This short |can measure the deviation by |[-1, 1]. | |article gives a brief overview|way of an error function: | | |of the method, but it should | |Before we can begin the Remez | |not be regarded as a thorough |Eabs(x) = f(x) - R(x) |method, we must obtain an | |theoretical treatment, for | |initial value for the location| |that you should consult your |which is expressed in terms of|of the extrema of the error | |favorite textbook. |absolute error, but we can |function. We could "guess" | | |equally use relative error: |these, but a much closer first| |Imagine that you want to |... |approximation can be obtained | |approximate some function f(x)| |by first constructing an | |by way of a rational function | Unfortunately we don't know |interpolated polynomial | |R(x), where R(x) may be either|where the extrema of the error|approximation to f(x). | |a polynomial P(x) or a ratio |function are located! | | |of two polynomials P(x)/Q(x) | |In order to obtain the N+1 | |(a rational | ** The Remez Method ** |coefficients of the | |function). Initially we'll | |interpolated polynomial we | |concentrate on the polynomial |The Remez method is an |need N+1 points (x0...xN): | |case, as it's by far the |iterative technique which, |with our interpolated form | |easier to deal with, later |given a broad range of |passing through each of those | |we'll extend to the full |assumptions, will converge on |points that yields N+1 | |rational function case. |the extrema of the error |simultaneous equations: | | |function, and therefore the | | | |minimax solution. | | +------------------------------+------------------------------+------------------------------+ | f(xi) = P(xi) = c0+ c1xi... +| | |cNxiN | | | | | |Which can be solved for the | | |coefficients c0...cNin P(x). | | | | | |Obviously this is not a | | |minimax solution, indeed our | Initial Interpolated Approximation | |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 | Which has a peak relative | ** Remez Step 1 *** | |zeros of an orthogonal |error of 1.2x10-3. | ... | |polynomial as the initial | | | |interpolation points is a good|While this is a pretty good | | |choice. In our example we'll |approximation already, judging| | |use the zeros of a Chebyshev |by the shape of the error | | |polynomial as these are |function we can clearly do | | |particularly easy to |better. Before starting on the| | |calculate, interpolating for a|Remez method propper, we have | | |polynomial of degree 4, and |one more step to perform: | | |measuring relative error we |locate all the extrema of the | | |get the following error |error function, and store | | |function: |these locations as our initial| | | |Chebyshev control points. | | | | ... | | +------------------------------+------------------------------+------------------------------+ -- Dave Abrahams Boost Consulting http://www.boost-consulting.com The Astoria Seminar ==> http://www.astoriaseminar.com ------------------------------------------------------------------------- This SF.net email is sponsored by DB2 Express Download DB2 Express C - the FREE version of DB2 express and take control of your XML. No limits. Just data. 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