Re: Re: Re: Bug ??????
- To: mathgroup at smc.vnet.net
- Subject: [mg105410] Re: [mg105382] Re: [mg105341] Re: Bug ??????
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Tue, 1 Dec 2009 04:16:18 -0500 (EST)
- References: <heqf01$1m4$1@smc.vnet.net> <200911291008.FAA16050@smc.vnet.net> <200911301111.GAA13325@smc.vnet.net> <1BCD9E65-ACEB-46FD-9A03-DB8344FE86D7@mimuw.edu.pl> <op.u37k270qtgfoz2@bobbys-imac.local>
Sorry, my misunderstanding. I thought you meant that it still might be a root. ( In fact to prove that it is not a root one just a few digits would be enough (unless they turn out to be zero) since N guarantees the specified number of digits to be correct.) Andrzej On 30 Nov 2009, at 22:00, DrMajorBob wrote: > I only mean that it's hard to find (without the Roots routine) x values for which Abs@F@x is smaller. With terms like 18339659776 *x^120 included, it's not hard to get values larger than 2*10^10. The function is extremely "noisy", one might say. > > I do NOT mean that the OP's "candidate root" is a root. N[F@expr,100] proved that it isn't. > > Bobby > > On Mon, 30 Nov 2009 06:34:30 -0600, Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote: > >> What exactly do you mean? Here Mathematica has proved (I mean really *proved*) that the candidate root is not a root at all. That is, unless there is a serious bug (and I mean bug) in Mathematica's significance arithmetic. If that were so, it would be a very serious bug indeed, perhaps the worst that has ever been found. >> >> Andrzej Kozlowski >> >> On 30 Nov 2009, at 20:11, DrMajorBob wrote: >> >>> The candidate root yields a high value: >>> >>> N[F@expr, 100] >>> >>> = 1.83396597760000000000000000000000000000000000000000000000000000000000\ >>> 0000000000000000000000000000000*10^10 >>> >>> But it's not particularly high, OTOH, considering the powers and >>> coefficients involved. >>> >>> Bobby >>> >>> On Sun, 29 Nov 2009 04:08:02 -0600, Emu = <samuel.thomas.blake at gmail.com> >>> wrote: >>> >>>> On Nov 28, 12:12 am, ynb <wkfkh... at yahoo.co.jp> wrote: >>>>> F[x_]:=34880228747203264624081936 - >>>>> 464212176939061350196344960*x^2 + >>>>> 4201844995162976506469882880*x^4 - >>>>> 36736184611200699915890392480*x^6 + >>>>> 245136733977616412716801297320*x^8 - >>>>> 1144143594851571569661248433072*x^10 + >>>>> 3682862525053500791559515638600*x^12 - >>>>> 8693355704402316431096075720520*x^14 + >>>>> 16394872503384952006491292949865*x^16 - >>>>> 26387316917169915527289585290460*x^18 + >>>>> 37452280566060594746358503070858*x^20 - >>>>> 47740404486181766316209780642820*x^22 + >>>>> 55423947476122401752437921213065*x^24 - >>>>> 58870208625780045323379674540820*x^26 + >>>>> 58030587837504412314635631719520*x^28 - >>>>> 54472073947308977321830018366176*x^30 + >>>>> 49239457796351067392552601696240*x^32 - >>>>> 43012853616400258712689244528460*x^34 + >>>>> 36323948931672906173046609029970*x^36 - >>>>> 29377569489403484765569859203920*x^38 + >>>>> 22788548915181561726713932258680*x^40 - >>>>> 16857194550514400031853658104200*x^42 + >>>>> 11584615647879044636617246631070*x^44 - >>>>> 7411292928519764848064641481820*x^46 + >>>>> 4455112744096674126517658718330*x^48 - >>>>> 2438996599504313974964504461440*x^50 + >>>>> 1194689292448727425260627641460*x^52 - >>>>> 524949326441431396920558140380*x^54 + >>>>> 201021537824162724562860099525*x^56 - >>>>> 61015761298172117757282456180*x^58 + >>>>> 8304189679978507974953617206*x^60 + >>>>> 2576525048464159376125949700*x^62 - >>>>> 2090208393662742383940297195*x^64 + >>>>> 1986814425386740056472178280*x^66 - >>>>> 689825144661940289046969960*x^68 - >>>>> 74165160041784503310561360*x^70 - >>>>> 43639409581797171854387880*x^72 - >>>>> 306779359014073038922080*x^74 + >>>>> 29021239224919123514667120*x^76 + >>>>> 3148715202822489687194520*x^78 - >>>>> 1180110005143725763548459*x^80 - >>>>> 1198749024197941338242580*x^82 - >>>>> 491140297003511546045670*x^84 + >>>>> 69048887622760819121580*x^86 + >>>>> 69823737459557420754765*x^88 + >>>>> 14776899216873553079620*x^90 - >>>>> 1463855286795400794960*x^92 - >>>>> 2352108554547064743120*x^94 - >>>>> 381175702618028601675*x^96 + >>>>> 126522213276402173400*x^98 + >>>>> 35845283140073787252*x^100 - >>>>> 2394735843271729380*x^102 - >>>>> 1421523086424723225*x^104 - >>>>> 37328586803289300*x^106 + >>>>> 29410426690606450*x^108 + >>>>> 2647220666999700*x^110 - >>>>> 300290705882655*x^112 - 51254703758400* >>>>> x^114 + 500254901760*x^116 + >>>>> 403671859200*x^118 + 18339659776 + 18339659776 *x^120 >>>>> >>>>> (* Bug ?; F[Sqrt[Sqrt[2] + 3^(1/3)] + 1/Sqrt[3^(1/3) + 5^(1/5)]] >>>>> //N >>>>> =3.828176627860558*^38<---Bug ? *) >>>>> >>>>> (* =0? *) >>>> >>>> It appears that the expression Sqrt[Sqrt[2] + 3^(1/3)] + 1/Sqrt[3^ >>>> (1/3) + 5^(1/5)] is very close to one of the roots of F[x], but it = is >>>> not a zero of F[x]. >>>> >>>> >>>> In[117]:= Select[Solve[F[x] == 0, x], (x /. N[#]) \[Element] = Reals &] >>>> [[-1, 1, -1]]; (* Returns a large Root object. *) >>>> >>>> In[118]:= N[Sqrt[Sqrt[2] + 3^(1/3)] + 1/Sqrt[3^(1/3) + 5^(1/5)] - = %, >>>> 100] >>>> Out[118]= >>>> = 2.267459811963931497406941878036067357307998685435567904057927238778317779= 198056926405182471544211907*10^-44 >>>> >>>> Sam >>>> >>>> >>> >>> >>> -- >>> DrMajorBob at yahoo.com >>> >> > > > -- > DrMajorBob at yahoo.com