The most canonical example of this is symbolic integration. The two most powerful algorithms implemented in SymPy, Risch and Meijer G, work completely differently from how a human would compute an integral (there are also some table lookup algorithms and manualintegrate which do work how a human would). Risch works by converting the expression to a tower of differential extensions (roughly, it converts the integrand into a rational function where the variables have a derivative which can be expressed in terms of the other variables). It then performs what basically amounts to several polynomial manipulation algorithms on this to compute the integral. To imagine the difference, consider how many times you compute a polynomial gcd (the Euclidean algorithm) as part of computing an indefinite integral by hand. It's probably zero. By contrast, the Risch algorithm computes gcds hundreds of times over the course of calculating an integral.
The Meijer G algorithm works (very roughly) by converting the integral into a Meijer G function, integrating it, and then converting it back. The usual by-hand integration methods are sort of heuristics. There are several methods which apply to specific expressions, and if you can pattern match your expression to a known technique then you can apply it. Risch and the Meijer G algorithm are quite general (Risch is extremely general, to the point where if it can't compute an integral for an elementary function then it is a proof that none exists). There are other good examples from the polys module. For instance, the algorithm to factorize polynomials (again here, a human "algorithm" would be heuristics, which might not always apply, whereas the computer algorithm is based on some deeper math that lets you always compute the factorization for any polynomial). In general, I would say that human methods tend to use a lot of little tricks, which are designed to minimize brute calculation. This is because humans are bad at brute calculation, but very good at the pattern matching and intuition required to apply and invent such methods. Computers on the other hand are very good at brute calculation, but very bad at pattern matching and intuition. So even algorithms that are essentially done the same way by a human and a computer are somewhat different for actual computation because humans know when they can "skip" certain steps, for instance. A human isn't going to run through the full Euclidean algorithm to compute gcd(1, x), for instance, and even for gcd(x, x**2 + x) a human can see the answer right away without running through any polynomial long division. Aaron Meurer On Wed, Oct 29, 2014 at 3:40 AM, Christophe Bal <projet...@gmail.com> wrote: > Hello. > > I'm looking for formal outputs that are not calculated as a human can do > (this is for a free french "book"). Do you know such kind of examples ? > > Christophe BAL > > PS : this questions has been posted on both Sage and Sympy lists. > > -- > 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 sympy+unsubscr...@googlegroups.com. > To post to this group, send email to sy...@googlegroups.com. > Visit this group at http://groups.google.com/group/sympy. > To view this discussion on the web visit > https://groups.google.com/d/msgid/sympy/CAAb4jGkNc8QcUAxhYnhoRN017tEXfDZrtABVr4w_OTQEhfXSqQ%40mail.gmail.com. > For more options, visit https://groups.google.com/d/optout. -- 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 sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/d/optout.
