Thanks so much for all the efforts for making sage output consistent results for the numerical approximation. However, the main problem I had was the fact that Sage returns different answers when I preparse the string of the same sage expression as in the examples of the original post.
On Wednesday, 11 March 2015 17:18:12 UTC-4, William wrote: > > On Wed, Mar 11, 2015 at 2:07 PM, Dima Pasechnik <[email protected] > <javascript:>> wrote: > > > > > > On Wednesday, 11 March 2015 16:44:32 UTC, William wrote: > >> > >> On Wed, Mar 11, 2015 at 9:35 AM, Dima Pasechnik <[email protected]> > wrote: > >> > On 2015-03-11, Nils Bruin <[email protected]> wrote: > >> >> On Wednesday, March 11, 2015 at 2:46:25 AM UTC-7, Dima Pasechnik > wrote: > >> >>> > >> >>> I tried this integral directly in Maxima, and taking bfloat of it > >> >>> outputs nonsense. > >> >>> > >> >> > >> >> I have noticed before that bfloats aren't infectious enough: > operations > >> >> on > >> >> bfloats can easily result in a normal "double". I think there are > ways > >> >> to > >> >> convince maxima to use bfloats more pervasively. Perhaps a global > >> >> precision > >> >> setting somewhere? > >> >> > >> >> > >> >>> I wish there was a more accessible full implementation of Risch > >> >>> algorithm... > >> >>> > >> >> > >> >> This is a rational function, so a first calculus course would > already > >> >> teach > >> >> you the relevant part of the Risch algorithm. It's a little more > tricky > >> >> to > >> > > >> > Risch, as implemented in Axiom, does not do factorisation (i.e. no > >> > partial fractions). > >> > In this example at least it produces much nicer looking > antiderivative, > >> > no huge integers. > >> > http://axiom-wiki.newsynthesis.org/ExampleIntegration > >> > > >> > Dima > >> > >> For what it's worth, here's how to mostly do that Axiom session, but > >> in a SageMathCloud worksheet... > >> > >> > >> > https://cloud.sagemath.com/projects/4a5f0542-5873-4eed-a85c-a18c706e8bcd/files/support/2015-03-11-093745-axiom-integral.sagews > > > > > > > here is a better version (all the stuff works): > > > https://cloud.sagemath.com/projects/bb6fd6ca-6304-4dda-be31-bd2dd5eb3d98/files/support/2015-03-11-093745-axiom-integral.sagews > > > > > Thanks! I replaced mine by yours. > > William > > > Dima > > > >> > >> > >> William > >> > >> > > >> >> get an ostensibly real-valued function as an antiderivative. Anyway, > >> >> sympy > >> >> produces a reasonable-looking antiderivative. > >> >> > >> >> Interestingly, we have: > >> >> > >> >> sage: I=integral(x/(x^3-x+1), x, 1, 2, algorithm='sympy') > >> >> sage: RIF(I) > >> >> TypeError: unable to simplify to a real interval approximation > >> >> > >> >> The offending subexpression seems to be: > >> >> > >> >> sage: A=(299838966359964800*69^(5/6)*2^(2/3) - > >> >> 11515081166050000*69^(2/3)*2^(1/3)*(25*sqrt(69) + 207)^(1/3) - > >> >> 99785894223312000*sqrt(69)*(25*sqrt(69) + 207)^(2/3) + > >> >> 2271318237097115625*69^(1/3)*2^(2/3) - > >> >> 99785894223312000*69^(1/6)*2^(1/3)*(25*sqrt(69) + 207)^(1/3) - > >> >> 497728835949744*9522^(1/3)*(25*sqrt(69) + 207)^(1/3) - > >> >> 828883890137982336*(25*sqrt(69) + 207)^(2/3) + > >> >> 219331275901257879*276^(1/3))^(QQ(-1)) > >> >> sage: RIF(A) > >> >> TypeError: unable to simplify to a real interval approximation > >> >> > >> >> Note the *rational* exponent -1. If that's an integer there's no > >> >> problem. > >> >> Using RealIntervalField(200) has the same problem. Using > RealField(...) > >> >> seems to work fine. > >> >> > >> > > >> > -- > >> > You received this message because you are subscribed to the Google > >> > Groups "sage-support" 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 http://groups.google.com/group/sage-support. > >> > For more options, visit https://groups.google.com/d/optout. > >> > >> > >> > >> -- > >> William (http://wstein.org) > > > > -- > > You received this message because you are subscribed to the Google > Groups > > "sage-support" 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 http://groups.google.com/group/sage-support. > > For more options, visit https://groups.google.com/d/optout. > > > > -- > William (http://wstein.org) > -- You received this message because you are subscribed to the Google Groups "sage-support" 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 http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/d/optout.
