Re: Roots of a Jacobi polynomial

*To*: mathgroup at smc.vnet.net*Subject*: [mg120569] Re: Roots of a Jacobi polynomial*From*: Richard Kendall-Smith <richpks at gmail.com>*Date*: Fri, 29 Jul 2011 04:43:47 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <201107261107.HAA09296@smc.vnet.net> <4E2EFAAD.60403@wolfram.com> <201107281154.HAA04077@smc.vnet.net>

Hi, I'm trying to solve a set of non-linear equations by doing the following: eqlist = {101.74 == a + ((0.055/(479248/c))*(352343 + ((479248/c)*a) - (158194*b/c))), 47205 + ((158194/c)*a) - ((61842/c)*b) == 68043.98 + (299.73*b), 460584 + (479248*(a^2)/(c^2)) - (158194*b*a/(c^2)) + (61842*(b^2)/(c^2)) == 21986 + (3576*c)} NSolve[eqlist, {a, b, c}, Reals] But it interprets c as Musical Notation. I have tried altering the variables to x y and z but then it comes up with a Visual Form error. I'm new to Mathematica but I don't see why this is happening because aren't we specifying a, b, c as variables to be solved? Thanks, Richard On Thu, Jul 28, 2011 at 5:54 AM, Jacopo Bertolotti <J.Bertolotti at utwente.nl>wrote: > On 07/26/2011 07:34 PM, Daniel Lichtblau wrote: > > On 07/26/2011 06:07 AM, Jacopo Bertolotti wrote: > >> Dear MathGroup, > >> Mathematica implements Jacobi polynomials as JacobiP[n,a,b,x] where n is > >> the order of the polynomial. As it can be checked plotting it a Jacobi > >> polynomial has n real roots in the interval [-1,1] and it goes rapidly > >> to infinity outside this interval (at least when both a and b are>-1). > >> The problem arise when you try to find the roots of such a polynomial > >> for a relatively high value of n. As an example the command > >> NSolve[JacobiP[20, -0.5, -0.5, x] == 0, x] correctly returns > >> > >> {{x -> -0.99702}, {x -> -0.972111}, {x -> -0.92418}, {x -> > >> -0.852398}, > >> {x -> -0.760555}, {x -> -0.649375}, {x -> -0.522529}, {x -> > >> -0.382672}, > >> {x -> -0.233449}, {x -> -0.0784582}, {x -> 0.0784591}, {x -> > >> 0.233445}, > >> {x -> 0.382684}, {x -> 0.522504}, {x -> 0.649423}, {x -> > >> 0.760466}, {x > >> -> 0.852539}, {x -> 0.924002}, {x -> 0.972267}, {x -> 0.996958}} > >> > >> while NSolve[JacobiP[25, -0.5, -0.5, x] == 0, x] gives > >> > >> {{x -> -1.01869}, {x -> -0.979859 - 0.0479527 I}, {x -> -0.979859 + > >> 0.0479527 I}, {x -> -0.870962 - 0.070991 I}, {x -> -0.870962 + > >> 0.070991 > >> I}, {x -> -0.71378 - 0.0505783 I}, {x -> -0.71378 + 0.0505783 I}, > >> {x -> > >> -0.571283}, {x -> -0.486396}, {x -> -0.367829}, {x -> -0.248377}, > >> {x -> > >> -0.125513}, {x -> -0.0000434329}, {x -> 0.125442}, {x -> 0.2489}, > >> {x -> > >> 0.365644}, {x -> 0.496977}, {x -> 0.555743}, {x -> 0.717741- > >> 0.0573399 > >> I}, {x -> 0.717741+ 0.0573399 I}, {x -> 0.87423- 0.0652273 I}, {x -> > >> 0.87423+ 0.0652273 I}, {x -> 0.977876- 0.0422422 I}, {x -> 0.977876+ > >> 0.0422422 I}, {x -> 1.01494}} > >> > >> i.e. both complex roots and roots outside the [-1,1] interval. > >> Substituting any of these values back into the polynomial easily show > >> that these values are not roots at all. Also notice that using the > >> command Root (e.g. Sort@Table[Root[JacobiP[25, -0.5, -0.5, x], i], {i, > >> 1, 25}]) gives different but still wrong results. > >> On a related note: NIntegrate sometimes gives wrong results when > >> integrating function of the form (1-x)^a (1+x)^b f[x] and I feel this > >> might related to the Gauss-Jacobi quadrature failing to retrieve the > >> correct roots of a Jacobi polynomial. > >> > >> Do anyone have a solution for that? > >> > >> Thank you > >> > >> Jacopo > > > > This is not a well conditioned problem. To do better you'll need to > > allow NSolve to use higher precision by not giving machine numbers in > > the input. > > > > In[102]:= solns = Sort[NSolve[JacobiP[25, -1/2, -1/2, x] == 0, x]] > > > > Out[102]= {{x -> -0.998027}, {x -> -0.982287}, {x -> -0.951057}, {x \ > > -> -0.904827}, {x -> -0.844328}, {x -> -0.770513}, {x -> -0.684547}, \ > > {x -> -0.587785}, {x -> -0.481754}, {x -> -0.368125}, {x -> \ > > -0.24869}, {x -> -0.125333}, {x -> 0.}, {x -> 0.125333}, {x -> > > 0.24869}, {x -> 0.368125}, {x -> 0.481754}, {x -> 0.587785}, {x -> > > 0.684547}, {x -> 0.770513}, {x -> 0.844328}, {x -> 0.904827}, {x -> > > 0.951057}, {x -> 0.982287}, {x -> 0.998027}} > > > > These might seem imperfect when you check residuals (see below). In > > fact they are not at all bad. We first show that by comparing to > > higher precision versions that do validate fairly well as giving > > smallish residuals. > > > > In[104]:= solns30 = > > Sort[NSolve[JacobiP[25, -1/2, -1/2, x] == 0, x, > > WorkingPrecision -> 30]] > > > > Out[104]= {{x -> -0.998026728428271561952336806863}, {x -> \ > > -0.982287250728688681085641742865}, {x -> \ > > -0.951056516295153572116439333379}, {x -> \ > > -0.904827052466019527713668647933}, {x -> \ > > -0.844327925502015078548558063967}, {x -> \ > > -0.770513242775789230803009636396}, {x -> \ > > -0.684547105928688673732283357621}, {x -> \ > > -0.587785252292473129168705954639}, {x -> \ > > -0.481753674101715274987191502872}, {x -> \ > > -0.368124552684677959156947147493}, {x -> \ > > -0.248689887164854788242283746006}, {x -> \ > > -0.125333233564304245373118759817}, {x -> 0}, {x -> > > 0.125333233564304245373118759817}, {x -> > > 0.248689887164854788242283746006}, {x -> > > 0.368124552684677959156947147493}, {x -> > > 0.481753674101715274987191502872}, {x -> > > 0.587785252292473129168705954639}, {x -> > > 0.684547105928688673732283357621}, {x -> > > 0.770513242775789230803009636396}, {x -> > > 0.844327925502015078548558063967}, {x -> > > 0.904827052466019527713668647933}, {x -> > > 0.951056516295153572116439333379}, {x -> > > 0.982287250728688681085641742865}, {x -> > > 0.998026728428271561952336806863}} > > > > This shows the two results are quite close. > > > > In[106]:= Max[Abs[(x /. solns) - (x /. solns30)]] > > Out[106]= 4.47615*10^-10 > > > > This shows the high precision solutions give reasonable residuals. > > > > In[108]:= Max[Abs[JacobiP[25, -1/2, -1/2, x] /. solns30]] > > Out[108]= 0.*10^-12 > > > > The machine precision results will not give good residuals. There are > > two possible (and related) reasons. One is that the conditioning is > > such that we are bound to lose many digits in assessing residuals (as > > seems to happen above). The other is that the form of JacobiP > > polynomials might just be inducing cancellation error. > > > > Certainly this second item is part of the problem: > > > > In[109]:= Max[Abs[Expand[JacobiP[25, -1/2, -1/2, x]] /. solns30]] > > Out[109]= 0.*10^-21 > > > > Here are the corresponding machine precision evaluations of residuals. > > > > In[112]:= Max[Abs[JacobiP[25, -1/2, -1/2, x] /. solns]] > > Out[112]= 17. > > > > In[113]:= Max[Abs[Expand[JacobiP[25, -1/2, -1/2, x]] /. solns]] > > Out[113]= 9.31323*10^-9 > > > > Daniel Lichtblau > > Wolfram Research > > > Thank you Daniel. And thank you also to everyone who wrote me > suggestions (both here and in private). The combination of not using > machine number in the input and increasing the WorkingPrecision seems to > work fine. Apparently just using one of the two tricks do not yield any > sensible result. > > Jacopo > >

**References**:**Roots of a Jacobi polynomial***From:*Jacopo Bertolotti <J.Bertolotti@utwente.nl>

**Re: Roots of a Jacobi polynomial***From:*Jacopo Bertolotti <J.Bertolotti@utwente.nl>