RE: How Many Users Are Happy With FindRoot?

*To*: mathgroup at smc.vnet.net*Subject*: [mg49181] RE: How Many Users Are Happy With FindRoot?*From*: "David Park" <djmp at earthlink.net>*Date*: Tue, 6 Jul 2004 03:34:10 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Tom, Your routine looks nice enough, but still most users aren't going to think of it. One of my beliefs is that Mathematica is not so much a toolbox for doing mathematics as a meta toolbox for making the tools to do mathematics. But I make an exception for FindRoot. I think it should be more robust and have simpler usage out of the box for real roots of real functions. This is a bread and butter function. The completion of Andrzej's routine is << "NumericalMath`IntervalRoots`" f[x_] = Tan[x] - (x/2)^2*Tanh[x]; intervals = IntervalBisection[f[x], x, Interval[{-3*N[Pi], 3*N[Pi]}], 0.1, MaxRecursion -> 10]; MapThread[FindRoot[f[x], {x, Plus @@ #/2}] &, {List @@ intervals}] {{x -> -7.78813}, {x -> -4.51891}, {x -> 0.}, {x -> 4.51891}, {x -> 7.78813}} But I was unaware of or had forgotten about IntervalRoots. Again, how many people are going to think of it BEFORE spending a lot of time with FindRoot, which they might expect to easily solve a problem like this? It's true that there is no such thing as perfect root finder even for real functions. I once wrote a very robust routine for solving the mass action chemical equilibrium equation. It will quickly solve for all concentrations to nearly the full precision of the floating point processor over nearly the full number range of the processor. But it uses a special transformation of the function and only works for that type of equation. I tried Ted's routine on these equations and found it worked extremely well. I was able to break it but it wasn't easy. That's why I think a significantly better FindRoot could be provided to Mathematica users. David Park djmp at earthlink.net http://home.earthlink.net/~djmp/ From: Thomas Burton [mailto:tburton at brahea.com] To: mathgroup at smc.vnet.net [[ This message was both posted and mailed: see the 'To' and 'Newsgroups' headers for details. ]] Golly! I'm pleased as punch with FindRoot :-) Seriously, the following short, simple-minded, and obvious extension has so far bailed me out of the kind of problem you describe: rootSearch[f_,{x_,xMin_,xMax_,dx_,eps_}]:= Union[Table[FindRoot[f,{x,x0}],{x0,xMin,xMax,dx}], SameTest -> (Abs[#1[[1,-1]]-#2[[1,-1]]]<eps&)] I plot the function and then choose limits xMin and xMax and increment dx appropriately. I get more frustrated when an upgrade to Mathematica breaks existing code. I had a client frozen to version 4.x because I didn't have time to sort out why version 5 broke code I wrote for them. Among the issues were new failures in (my usage of) FindRoot (not for closed-form expressions). Which is partly why I am wary of Mathgroup extensions. If the evolution of core Mathematica functions breaks my code, what about MathSource? Finding real roots of closed-form expressions is, I should think, a common application of Mathematica that deserves a method more sophisticated than Newton and secant. One one hand, to upgrade Mr Ersek's RootSearch from MathSource to Mathematica's core entails a lot of work: vetting of methods, extensive quality assurance, commitment to perpetual maintenance, etc. On the other hand, Mr. Ersek has already done some of the work. Also it's easy to check the results of a root finder. I always check FindRoot. RootSearch would be no different. Perhaps the case could be made that Wolfram should upgrade Ersek's RootSearch to a standard add-on. Tom Burton On Sun, 4 Jul 2004 23:09:59 -1000, David Park wrote (in article <ccb5t7$enc$1 at smc.vnet.net>): > Dear MathGroup, > > I am continually being frustrated with the usage and performance of > FindRoot.... > There is a solution. There is a package in MathSource, > Enhancements`RootSearch`, by Ted Ersek. It is far better at handling roots > of real expressions.

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