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] <javascript:>> 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] <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/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]. 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