Re: Beware of NSolve - nastier example

*To*: mathgroup at smc.vnet.net*Subject*: [mg50220] Re: [mg50207] Beware of NSolve - nastier example*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Sat, 21 Aug 2004 03:04:23 -0400 (EDT)*References*: <200408200858.EAA12533@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Carlos Felippa wrote: > This is a nasty one: > > f=5/432-11/(27*Sqrt[70]*Sqrt[19-1890*x])+x/(2*Sqrt[38/35-(108+1/10000000)*x]); > > Print[N[Solve[f==0,x]]]; (* OK, 3 roots #2,4,5 of a sextic poly *) > Print[NSolve[f,x,16]]; (* 6 roots, 3 wrong *) > Print[NSolve[f,x,21]]; (* 4 roots, 2 correct, 1 double wrong, > 1 single wrong *) > Print[NSolve[f,x,24]]; (* 3 roots, 1 wrong *) > Print[NSolve[f,x,28]]; (* 3 roots, 1 wrong *) > Print[NSolve[f,x,32]]; (* 2 correct roots, missed 1 *) > Print[NSolve[f,x,64]]; (* 2 correct roots, missed 1 *) > Print[NSolve[f,x,128]]; (* 3 correct roots *) > > All 3 roots are single and well conditioned, no problem getting them > in single precision with NR. > > Note: please reply to this newsgroup; DONT send mail to me directly > (there is a white list, if not in list your msg >null) The claim that all roots are well conditioned is not correct. One of them is quite problematic. Look at all six solutions, including parasites, to the Solve (do this via VerifySolutions->False). You will see that there is very nearly, but not exactly, a double root. This means NSolve will, at modest precision, be forced to decide the thing is a double root, average the values, and get somenthing that is not in fact a correct solution. The verification process will then discard this. To make matters worse, the two agree to about 15 digits, and moreover agree with a root of something in a denominator to about 11 digits. So far as I can tell any numeric polynomial solver is going to be in for a difficult time with this, because near such a denominator residuals can fluctuate wildly. On the flip side, the heuristics that decide when a root actually satisfies the equation can be a bit lax. Hence parasites can wind up in the returned solution set. One way to influence this is to rescale the equation e.g. multiplying by some large number. I'll look into tightening some of the decisions as to what constitutes a sufficiently small residual. The near-double-root issue goes with the territory and is in no way incorrect behavior. What is NR, by the way? Daniel Lichtblau Wolfram Research

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