NewtonZero never comes down

*To*: mathgroup at smc.vnet.net*Subject*: [mg51416] NewtonZero never comes down*From*: Roger Bagula <tftn at earthlink.net>*Date*: Sat, 16 Oct 2004 04:21:07 -0400 (EDT)*Reply-to*: tftn at earthlink.net*Sender*: owner-wri-mathgroup at wolfram.com

I've tried several ways to find better values for zeta zeros fast and easy... FindRoot[] just doesn't give much accuracy. NewtonZero[] tends to just spin it's wheels on my machine.... Here's what I came up with that gets about 20 places and pretty fast relatively. It's much better than FindRoot[] anyway! The zeta zeros all seem to be transcendental irrational numbers and are harder to calculate than most numbers in modern number theory. (* finding Zeta Zero roots*) (* using approximation from analyitic continuation on 0<=z<=1 strip where zeta dosesn't converge*) Clear[RSzeta,t,g3,f,g,a] RSzeta[t_]=RiemannSiegelZ[t]/Exp[I*RiemannSiegelTheta[t]] Digits=10000 t=(2 Sqrt[2])*25*Pi^2*Sqrt[n]/(6*Sqrt[Digits])-16 (* gives the zeta zeros given in Jahnke-Emde Tables of Functions, Dover*) g3=Delete[Union[Table[If[N[Abs[RSzeta[t]]]<0.01,N[t,20],0],{n,1,Digits}]],1] Dimensions[g3] ListPlot[g3,PlotJoined->True] (* definition of Newton function*) f[x_]:=N[x-Zeta[x]/Zeta'[x],20] (* ten iterations toward the zero*) g[x_]:=NestList[f,1/2+I*x,10][[10]] (* values starting at the continuation values approximated*) a=Table[g[g3[[n]]],{n,1,Dimensions[g3][[1]]}] ListPlot[Im[a],PlotJoined->True] Respectfully, Roger L. Bagula tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : alternative email: rlbtftn at netscape.net URL : http://home.earthlink.net/~tftn