Re: Re: Beware of NSolve - nastier example

*To*: mathgroup at smc.vnet.net*Subject*: [mg50432] Re: [mg50414] Re: Beware of NSolve - nastier example*From*: DrBob <drbob at bigfoot.com>*Date*: Fri, 3 Sep 2004 03:35:32 -0400 (EDT)*References*: <200408200858.EAA12533@smc.vnet.net> <cg6srb$odf$1@smc.vnet.net> <200408280837.EAA19074@smc.vnet.net> <200408311028.GAA18653@smc.vnet.net> <opsdluzdvziz9bcq@monster.cox-internet.com> <ch3olv$33$1@smc.vnet.net> <200409020834.EAA02117@smc.vnet.net>*Reply-to*: drbob at bigfoot.com*Sender*: owner-wri-mathgroup at wolfram.com

I was iterating to 50 digit answers with starting points from 1/3 to 4 times the correct root. You're getting a machine precision answer with a start that's already correct to 13 places. The difference between the converged answer and your starting value, in fact, was -3.189115638235762*^-14 It's not surprising you could get convergence starting there. If you say the convergence interval is +/-10^-1, OK, but that's awfully close for an initial guess!! However, I was very far off in my post. I should have had g[x_]:=x - f[x]/f'[x] This is exactly what you were doing, and what I thought I was doing, but wasn't! Now, the derivative at the machine precision and 50-digit solutions are: g'@single g'@fifty 8.942337077226623*^-7 0``39.84918837339986 And that's very nice! But at other nearby points: g'/@Range[.005,.015,.001]; Norm/@% {0.611662,0.832022,1.05746,1.32013,1.6874,2.63267,2.89968,2.80032,2.70822,2.\ 62331,2.54518} If that norm is greater than one anywhere on the interval between the desired root and the starting point, then g isn't a contraction, and convergence of the basic NR is very iffy. Bobby On Thu, 2 Sep 2004 04:34:54 -0400 (EDT), Carlos Felippa <carlos at colorado.edu> wrote: > DrBob <drbob at bigfoot.com> wrote in message news:<ch3olv$33$1 at smc.vnet.net>... >> Oops, in my last paragraph I claimed convergence for a wider range of starting values than I actually get. (I had a typo when doing some of the tests.) 0.003`n to 0.04`n looks fine. At least it converges for SOME input, which is more than the raw Newton iteration can do. >> >> Of course, I used Solve to get the h function--the same Solve routine that solved the original problem already! > > No problem whatsoever for conventional NR in 16-digits > (double precision) if you are sufficiently near a root. Just run this > NR iteration, which is at the level of the homeworks I assign to > engineering sophomores: > > ClearAll[f,x]; > f=(5/432-11/(27*Sqrt[70]*Sqrt[19-1890*x])+x/(2*Sqrt[38/35-(108+1/10000000)*x])); > fprime=D[f,x]; > Print[N[Solve[f==0,x]]//InputForm]; (* gives 3 correct roots *) > > SetPrecision[{xn,xnext,r,f,fprime,fp},16]; > xn=0.0100529100415; (* the slippery root *) > n=10; (* actually n=3 is enough to get limit accuracy *) > For [i=1,i<=n,i++, r=N[f/.x->xn]; fp=N[fprime/.x->xn]; > dx=-r/fp; xnext=xn+dx; > Print[{i,r,fp,dx,xn,xnext}//InputForm]; xn=xnext]; > > The derivative f' is large, O(10^13) whereas the residual goes down to > O(10^(-4)). The interval where NR converges to that root is tiny, of > O(10^(-10), but finite. If one tries single precision (~ 6-7 digits) > NR fails, as noted by Daniel Lichtblau, since the convergence > interval falls in the noise. > > What is a bit surprising to me is that NSolve applied directly to f, > (not as N[Solve][..]]) needs 128-digit working precision to resolve > that root. > > > -- DrBob at bigfoot.com www.eclecticdreams.net

**References**:**Re: Beware of NSolve - nastier example***From:*carlos@colorado.edu (Carlos Felippa)