Re: Change in NSolve algorithm

*To*: mathgroup at smc.vnet.net*Subject*: [mg98537] Re: Change in NSolve algorithm*From*: Szabolcs <szhorvat at gmail.com>*Date*: Sun, 12 Apr 2009 03:46:22 -0400 (EDT)*References*: <grpif7$lnr$1@smc.vnet.net>

On Apr 11, 10:56 am, mark mcclure <mcmcc... at unca.edu> wrote: > Has anyone noticed that the algorithm used by NSolve has > changed in V7? I think the change is mostly for the better, > but not exclusively. To see the change, enter the following > in both V7 (7.0.0 or 7.0.1) and any version < 7.0: > Abs[sols = z /. NSolve[z^200 == 1, z]] > > In V7, Mathematica correctly generates a list of 1.0s. In > versions prior to V7, numerical error creeps in so you get > numbers ranging from 0.004 to 2.55. V7 is noticeably faster, > too. To visualize the error in V6, you can enter > ListPlot[{Re[#], Im[#]} & /@ sols, > AspectRatio -> Automatic] > > Of course, you should get a circle. > > On the other hand, V7 fares slightly worse with Wilkinson's > polynomial: > sols = x/.NSolve[Expand[Product[x-i, {i,1,20}] == 0,x]]; > Norm[Table[sols[[i]] - i, {i, 1, 20}]] > > I get 0.087351 from V6 and 0.26578 in V7. Also, V6 returns > all real solutions, while V7 returns some complex. > > Of course, the change is platform independent. I haven't > played with it extensively, but I think this is probably a > net big improvement. > Thank you, Mark, for this little demonstration! For the people reading this post and wondering about whether Mathematica can find more accurate solutions to those equations: using the WorkingPrecision option with a non-machine-precision setting will usually improve the results (but the computation will take longer).