Re: Re: Possible bug in NSolve[equation, variable, precission]

*To*: mathgroup at smc.vnet.net*Subject*: [mg74339] Re: [mg74308] Re: Possible bug in NSolve[equation, variable, precission]*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Mon, 19 Mar 2007 02:02:07 -0500 (EST)*References*: <etg4m9$ioi$1@smc.vnet.net> <200703180545.AAA19388@smc.vnet.net>

On 18 Mar 2007, at 06:45, dimitris wrote: > Why use NSolve in your equation? Use Solve instead! If you mean only this particular equation your advice is reasonable. But for all polynomial equations it is not; for polynomial equations of large degree you will significant difference in performance between NSolve and Solve. Try for example this (random) exmple. eq = x^201 + 245*x^100 - 77*x^50 - 23*x^17 + 11*x^16 - 7*x^4 + x^3 - 2*x^2 - 25 == 0; On my 1 gigahertz PowerBook NSolve[eq,20] takes under 23 seconds but N [Solve[eq==0,x],20] takes over 72. The difference in perfomance should be even more striking for systems of polynomial equations involving polynomials of high degree. For equations and systems of small(ish) degree I think Solve is is preferable. Andrzej Kozlowski > > > In[1]:= > Clear["Global`*"] > Print[StyleForm["working version", FontColor -> Blue]] > $Version > Print[StyleForm["your polynomial", FontColor -> Blue]] > poly = 171142046150220198693105489 - 16023210221608713837587916*x - > 2020825892011586434364754*x^2 + > 190894692033395024364972*x^3 + 6039743423966949379761*x^4 - > 568929229651998950400*x^5 - 470066550477520896*x^6 + > 2821109907456*x^7 > Print[StyleForm["your second polynomial", FontColor -> Blue]] > poly2 = Expand[poly/9] > Print[StyleForm["solution of the equation poly=0", FontColor -> Blue]] > Timing[sols = Solve[poly == 0, x]] > Print[StyleForm["solution of the equation poly2=0", FontColor -> > Blue]] > Timing[sols2 = Solve[poly == 0, x]] > Print[StyleForm["numerical approximation with 20 digits precision", > FontColor -> Blue]] > (N[#1, 20] & )[x /. sols] > (N[#1, 20] & )[x /. sols2] > Print[StyleForm["numerical approximation with 100 digits precision", > FontColor -> Blue]] > (N[#1, 100] & )[x /. sols] > (N[#1, 100] & )[x /. sols2] > > > Regards > Dimitris > > > Julian Aguirre wrote: >> Dear group, >> >> Mathematica 5.2 chokes solving numerically a polynomial equation. >> >> In[1] := $Version >> Out[1]= 5.2 for Mac OS X (64 bit) (June 20, 2005) >> >> In[2]:= poly=171142046150220198693105489-16023210221608713837587916 >> x-2020825892011586434364754 x^2+190894692033395024364972 >> x^3+6039743423966949379761 x^4-568929229651998950400 >> x^5-470066550477520896 x^6+2821109907456 x^7; >> >> In[3]:= poly2=Expand[poly/9]; >> >> In[4]:= NSolve[poly==0,x] >> Out[4]= {-1211.83, -13.0015, -13.0014, 11.923, 12.0809, 12.2509, >> 167826.} >> >> (* Up to this moment, everything is O.K. But *) >> >> In[5]:= NSolve[poly==0,x,20] >> Out[5]= $Aborted (* after a loooong time *) >> >> (* However, the following works as expected*) >> >> In[6]:= x/.NSolve[poly2==0,x,20] >> Out[6]= {-1211.8267955098487289, -13.001455891126, -13.001441554521, >> 11.92303189062617, 12.08089051352363, 12.25087466630727, >> 167826.26017849924816} >> >> Let me say that I have used Mathematica to solve thousands (probably >> millions) of equations like the one above. There must be some >> magic in >> the coefficients! >> >> Julian Aguirre >> University of the Basque Country > >

**References**:**Re: Possible bug in NSolve[equation, variable, precission]***From:*"dimitris" <dimmechan@yahoo.com>