Yes, I found it, but it won't let me re-post. Tobias
On Thursday, January 14, 2016 at 4:18:55 PM UTC, Aaron Meurer wrote: > > Here is the mpmath mailing list > https://groups.google.com/forum/#!forum/mpmath. > > Aaron Meurer > > On Thu, Jan 14, 2016 at 11:17 AM, Tobias Hartung > <[email protected] <javascript:>> wrote: > > Thank you and sorry about the wrong group post. Do you happen to know, > how I > > can crosspost it? > > > > Tobias > > > > On Thursday, January 14, 2016 at 3:35:34 PM UTC, Aaron Meurer wrote: > >> > >> You may want to crosspost this to the mpmath list. > >> > >> Aaron Meurer > >> > >> On Thu, Jan 14, 2016 at 10:14 AM, Tobias Hartung > >> <[email protected]> wrote: > >> > Hi everyone, > >> > > >> > I am trying to integrate polynomials on an n-sphere and implemented > (see > >> > attached) the algorithm published in Alan Genz "Fully Symmetric > >> > Interpolatory Rules for Multiple Integrals over Hyper-Spherical > >> > Surfaces" > >> > Journal of Computational and Applied Mathematics 157: 187-195, 2003. > On > >> > double precision, everything is fine; however, I lose precision in > some > >> > integrals using more than double prec. > >> > > >> > I am using randomized samples on the sphere which (by construction) > >> > should > >> > integrate the polynomials (that I tested) exactly. However, > integrating > >> > all > >> > polynomials up to degree 6 on a 5-dimensional sphere with points that > >> > should > >> > integrate all polynomials up to degree 7 exactly, I obtain the > following > >> > test results : > >> > > >> > mp.prec = 1024 (aka mp.dps = 307) > >> > > >> > polynomial | degree of polynomial | log_10 ( difference of two > >> > randomized integrations ) | log_10 ( relative error of weights ) | > >> > analytic value of integral > >> > > >> > [0, 0, 0, 0, 0, 0] 0 -inf -306.962262 > >> > 31.0062767 > >> > [1, 0, 0, 0, 0, 0] 1 -308.111506 -306.962262 0.0 > >> > [2, 0, 0, 0, 0, 0] 2 -14.519354 -306.962262 > >> > 5.16771278 > >> > [1, 1, 0, 0, 0, 0] 2 -14.8787358 -306.962262 0.0 > >> > [3, 0, 0, 0, 0, 0] 3 -308.583548 -306.962262 0.0 > >> > [2, 1, 0, 0, 0, 0] 3 -309.157806 -306.962262 0.0 > >> > [1, 1, 1, 0, 0, 0] 3 -309.759866 -306.962262 0.0 > >> > [4, 0, 0, 0, 0, 0] 4 -14.6442927 -306.962262 > >> > 1.93789229 > >> > [3, 1, 0, 0, 0, 0] 4 -15.3047045 -306.962262 0.0 > >> > [2, 2, 0, 0, 0, 0] 4 -15.0673319 -306.962262 > >> > 0.645964098 > >> > [2, 1, 1, 0, 0, 0] 4 -16.0594155 -306.962262 0.0 > >> > [1, 1, 1, 1, 0, 0] 4 -31.8146065 -306.962262 0.0 > >> > [5, 0, 0, 0, 0, 0] 5 -309.37189 -306.962262 0.0 > >> > [4, 1, 0, 0, 0, 0] 5 -309.481766 -306.962262 0.0 > >> > [3, 2, 0, 0, 0, 0] 5 -309.739262 -306.962262 0.0 > >> > [3, 1, 1, 0, 0, 0] 5 -310.095075 -306.962262 0.0 > >> > [2, 2, 1, 0, 0, 0] 5 -310.474598 -306.962262 0.0 > >> > [2, 1, 1, 1, 0, 0] 5 -310.714154 -306.962262 0.0 > >> > [1, 1, 1, 1, 1, 0] 5 -311.46643 -306.962262 0.0 > >> > [6, 0, 0, 0, 0, 0] 6 -14.7692314 -306.962262 > >> > 0.968946146 > >> > [5, 1, 0, 0, 0, 0] 6 -15.6057345 -306.962262 0.0 > >> > [4, 2, 0, 0, 0, 0] 6 -15.4314091 -306.962262 > >> > 0.193789229 > >> > [3, 3, 0, 0, 0, 0] 6 -15.8275833 -306.962262 0.0 > >> > [4, 1, 1, 0, 0, 0] 6 -16.5822942 -306.962262 0.0 > >> > [3, 2, 1, 0, 0, 0] 6 -16.4753651 -306.962262 0.0 > >> > [2, 2, 2, 0, 0, 0] 6 -16.0997045 -306.962262 > >> > 0.0645964098 > >> > [3, 1, 1, 1, 0, 0] 6 -32.3374853 -306.962262 0.0 > >> > [2, 2, 1, 1, 0, 0] 6 -17.3552924 -306.962262 0.0 > >> > [2, 1, 1, 1, 1, 0] 6 -33.4100353 -306.962262 0.0 > >> > [1, 1, 1, 1, 1, 1] 6 -49.1751847 -306.962262 0.0 > >> > > >> > It should be noted that the polynomial are written in the following > >> > form: a > >> > list p of length n represents the polynomial > >> > > >> > z_0^{p[0]} z_1^{p[1]} z_2^{p[2]} ... z_{n-1}^{p[n-1]}. > >> > > >> > If we look at the second to last column, we can see that the weights > add > >> > are > >> > exact with respect to mp.prec=1024 but do not add up to 1.0 exactly. > >> > Hence, > >> > the loss in precision is unlikely to be caused by the weights being > only > >> > on > >> > double prec (and even if it is, then I don't understand why because > they > >> > are > >> > calculated with mp.prec=1024, as well). > >> > > >> > The really interesting column is the middle one. Here, I took two > >> > randomized > >> > set of integration points (both should integrate all polynomials > exactly > >> > up > >> > to machine error) and printed the decadic logarithm of the absolute > >> > value of > >> > their difference. In other words, we are expecting the middle column > to > >> > be > >> > populated with numbers in the vicinity of -307 (just like the fourth > >> > column). However, we only get these values for odd degree polynomials > >> > (and > >> > the constant 1 but that is just the sum of weights, i.e., the > >> > randomization > >> > doesn't do anything). For even degree polynomials, we have > significantly > >> > reduced precision. > >> > > >> > Does anyone have an idea what might be the reason for such behavior? > >> > > >> > Thank you very much, > >> > Tobias > >> > > >> > -- > >> > You received this message because you are subscribed to the Google > >> > Groups > >> > "sympy" group. > >> > To unsubscribe from this group and stop receiving emails from it, > send > >> > an > >> > email to [email protected]. > >> > To post to this group, send email to [email protected]. > >> > Visit this group at https://groups.google.com/group/sympy. > >> > To view this discussion on the web visit > >> > > >> > > https://groups.google.com/d/msgid/sympy/cf76f223-96a0-4789-8530-db224622f6bb%40googlegroups.com. > > > >> > For more options, visit https://groups.google.com/d/optout. > > > > -- > > You received this message because you are subscribed to the Google > Groups > > "sympy" group. > > To unsubscribe from this group and stop receiving emails from it, send > an > > email to [email protected] <javascript:>. > > To post to this group, send email to [email protected] > <javascript:>. > > Visit this group at https://groups.google.com/group/sympy. > > To view this discussion on the web visit > > > https://groups.google.com/d/msgid/sympy/015dd6ad-d2e8-495c-82f7-1c157b0c7ab0%40googlegroups.com. > > > > > > For more options, visit https://groups.google.com/d/optout. > -- You received this message because you are subscribed to the Google Groups "sympy" group. 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