       Re: How Many Users Are Happy With FindRoot?

• To: mathgroup at smc.vnet.net
• Subject: [mg49170] Re: [mg49150] How Many Users Are Happy With FindRoot?
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Tue, 6 Jul 2004 03:33:35 -0400 (EDT)
• References: <200407050854.EAA14774@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```On 5 Jul 2004, at 17:54, David Park wrote:

> *This message was transferred with a trial version of CommuniGate(tm)
> Pro*
> Dear MathGroup,
>
> I am continually being frustrated with the usage and performance of
> FindRoot. Consider the problem of finding the roots lying between -3
> Pi and 3 Pi of the following function.
>
> f[x_] = Tan[x] - (x/2)^2*Tanh[x];
>
> If we plot the function we can see why FindRoot runs into difficulty.
>
> Plot[f[x], {x, -3*Pi, 3*Pi}, Frame -> True,
>    Axes -> {True, False}, PlotRange -> {-15, 15}];
>
> If we pick starting values that are close enough we obtain the actual
> roots.
>
> FindRoot[f[x], {x, {-7.5, -4, 2, 4, 7.5}}]
> {x -> {-7.78813, -4.51891, 0., 4.51891, 7.78813}}
>
> But I object to having to pick the initial values that closely. This
> is not convenient and should not be necessary. If we pick the starting
> values to just be in the middle of each branch, we only obtain one
> correct root out of five.
>
> FindRoot[f[x], {x, {-2*Pi, -Pi, 0, Pi, 2*Pi}}]
> {x -> {-4.51891, 0., 0., 0., 4.51891}}
>
> With the algorithm that Mathematica uses there is no reason we should
> expect any better. The root searching is jumping from branch to
> branch. So let's confine it to a single branch. We obtain an error
> message and an incorrect answer.
>
> FindRoot[f[x], {x, -2*Pi, -5*(Pi/2), -3*(Pi/2)}]
> {x -> -5.36881}
>
> It appears that in being ambitious enought to handle complex roots and
> multiple equations FindRoot has skimped on finding real roots of real
> equations. Mathematica should use more adaptive methods, perhaps using
> bisection or other methods to locate roots in a fixed region. One
> should not even have to give a starting point. These are probably the
> most common applications of FindRoot and we should expect better
> performance.
>
> 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.
>
> Needs["Enhancements`RootSearch`"]
>
> RootSearch[f[x] == 0, {x, -3*Pi, 3*Pi}]
> {{x -> -7.78813}, {x -> -4.51891}, {x -> 0.}, {x -> 4.51891}, {x ->
> 7.78813}}
>
> This shouldn't be buried away in MathSource. It should be part of
> Mathematica proper, on an equal par with FindRoot. It should be the
> user's first choice for finding real roots of real equations. One
> shouldn't spend time fumbling with FindRoot and then searching outside
> Mathematica proper for a solution.
>
> David Park
> djmp at earthlink.net
>
>
I agree that Ted's package is nice, but Mathematica already included
means of solving this sort of problem, e.g.

<< "NumericalMath`IntervalRoots`"

f[x_] = Tan[x] - (x/2)^2*Tanh[x];

IntervalBisection[f[x], x, Interval[{-3*N[Pi], 3*N[Pi]}],
0.1, MaxRecursion -> 10]

Interval[{-7.8048942487621495, -7.7312631709436355},
{-4.565126824747676, -4.491495746929162},
{-0.07363107781851091, 0.07363107781851091},
{4.491495746929162, 4.565126824747676},
{7.7312631709436355, 7.8048942487621495}]

Andrzej Kozlowski
Chiba, Japan
http://www.mimuw.edu.pl/~akoz/

```

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