RE: Re: ALL roots of non-polynomial equation
- To: mathgroup at smc.vnet.net
- Subject: [mg35996] RE: [mg35942] Re: [mg35926] ALL roots of non-polynomial equation
- From: "DrBob" <majort at cox-internet.com>
- Date: Sun, 11 Aug 2002 05:13:59 -0400 (EDT)
- Reply-to: <drbob at bigfoot.com>
- Sender: owner-wri-mathgroup at wolfram.com
Good luck with a function like 1 + Sin[x], which has infinitely many roots but never changes sign! Or Sin[1/x], which has infinitely many roots converging on 0, but no limit for the function at 0. Or... still worse... 1 + Sin[1/x], which has BOTH problems. Bobby -----Original Message----- From: Andrzej Kozlowski [mailto:andrzej at platon.c.u-tokyo.ac.jp] To: mathgroup at smc.vnet.net Subject: [mg35996] [mg35942] Re: [mg35926] ALL roots of non-polynomial equation In your example, yes. Here is one way (adapted from a slightly different problem in Stan Wagon's "Mathematica in Action") We make use of Mathematica's ability to plot graphs: In[1]:= g = Plot[Sin[x], {x, 0.1, 10.1*Pi}, DisplayFunction -> Identity]; We make a list of all the coordinates of the points represented on the graph. In[2]:= points = Cases[g, Line[x_] -> x, Infinity][[1]]; We make a list of the signs of the y values: In[3]:= signs = Sign /@ Transpose[Cases[g, Line[x_] -> x, Infinity][[ 1]]][[2]]; We find the points where the sign changes: In[4]:= positions = Position[Rest[signs]*Rest[RotateRight[signs]], -1] Out[4]= {{27}, {51}, {74}, {101}, {126}, {149}, {177}, {200}, {226}, {252}} We make a list of starting points for FindRoot: In[5]:= starts = First[Transpose[Extract[points, positions]]] Out[5]= {2.7825096162536145, 6.080185995733974, 8.787418231655966, 12.198138489619575, 15.464841498197309, 18.61672099859868, 21.92859710988888, 24.46767425065356, 27.840417480532142, 31.139545383515845} We find the roots: In[6]:= (FindRoot[Sin[x] == 0, {x, #1}, WorkingPrecision -> 20] & ) /@ starts Out[6]= {{x -> 3.141592653589793238462643383255068`20}, {x -> 6.283185307179586476925286766538051`20}, {x -> 9.424777960769379715387930149825109`20}, {x -> 12.566370614359172953850573533079026`20}, {x -> 15.707963267948966192313216916378673`20}, {x -> 18.849555921538759430775860299681079`20}, {x -> 21.991148575128552669238503682979946`20}, {x -> 25.132741228718345907701147066183302`20}, {x -> 28.274333882308139146163790449476032`20}, {x -> 31.415926535897932384626433832775678`20}} This question has been asked frequently so you can find various approaches, including this one, in the archives. Of course there is no guarantee. For very complex functions you may well miss some roots. The situation can become a lot more complicated if your equation has multiple roots. Andrzej On Thursday, August 8, 2002, at 07:06 PM, Mihajlo Vanevic wrote: > > Can Mathematica find (localize) ALL roots of non-polynomial equation > > eq[x]==0 > > on a given segment x \in [a,b], a,b=Real?? > > (for example Sin[x]==0, for 0.1<x<10.1 Pi ) > > > > > > >