Re: Bug in NSolve?

*To*: mathgroup at smc.vnet.net*Subject*: [mg33974] Re: Bug in NSolve?*From*: Rodger Rosenbaum <rodgerroSPAMNOT at siteconnect.com>*Date*: Thu, 25 Apr 2002 02:59:45 -0400 (EDT)*References*: <a9oe60$d9r$1@smc.vnet.net> <a9r3eo$lrg$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

On Sat, 20 Apr 2002 06:56:24 +0000 (UTC), lalu_bhatt at yahoo.com (Bhuvanesh) wrote: >Andrzej Kozlowski <andrzej at tuins.ac.jp> wrote: > >> Unlike Solve, NSolve returns only one root when it finds multiple roots >> it considers to be sufficiently close to be identical. In fact it seems >> that it applies Union to the final result. So when you use more >> precision the roots which were considered as distinct due to possible >> error become identified and appear only once. ><snipped> >> Why do Solve and NSolve behave differently here? Well, it seems to me >> that it is probably a matter of "philosophy". Solve is basically an >> "algebraic function" and from the algebraic point of view the >> multiplicity of roots is an important property of a solution. But most >> people looking for numerical solutions would be I think inclined to >> consider multiple roots as "the same". > >But there should be two solutions, right? Solve returns: > >In[1]:= Solve[4877361379 x^2 -9754525226 x + 4877163849==0,x] > > 4877262613 - I Sqrt[2] 4877262613 + I Sqrt[2] >Out[1]= {{x -> ----------------------}, {x -> ----------------------}} > 4877361379 4877361379 > >The two solutions *are* close, as you wrote. Andrzej Kozlowski suggests that when there is a root of multiplicity greater than 1, we should not be surprised if Mathematica only reports one root. In his examples, we can determine the multiplicity by inspection; but suppose we had a polynomial of relatively high order, say fifth degree, with, say, 2 distinct roots, and 1 of multiplicity 3, e.g.: NSolve[x^5 - 15 x^4 + 74 x^3 - 142 x^2 + 117 x - 35,x,20] Mathematica gives: {{ x--> 1.000000},{ x--> 1.000000},{ x--> 5.00000000000000000},{x-->7.00000000000000000}} What am I to make of this? Why didn't Mathematica form the Union of the three identical roots and only show one? Can I be sure that the root, 5, doesn't have multiplicity greater than one? The lesser precision of the two roots, 1, implies higher multiplicity than the roots, 5 and 7, but but can I rely on such an inference? And there is that claim in the Mathematica Book: "NSolve will always give you the complete set of numerical solutions to any polynomial equation in one variable." And here is some further peculiar behavior; try: NSolve[x^3 - 3 x^2 + 3 x - 1,x,20] NSolve[x^4 - 4 x^3 + 6 x^2 - 4 x + 1,x,20] These give two roots of unity, but if you remove the third argument (the 20), then they give 3 and 4 roots of unity, respectively. Make the third argument in the cubic 25, and get three roots of unity; make the third argument 200 and get only two roots. and NSolve[x^5 - 5 x^4 + 10 x^3 - 10 x^2 + 5 x - 1,x,20] This gives 4 roots of unity; make the third argument 25 and get only 3 roots. Make the third argument 200, or 10000, and get two roots.