On Thu, Aug 21, 2014 at 8:26 AM, clemens novak <[email protected]> wrote: > Am Mittwoch, 20. August 2014 23:45:55 UTC+2 schrieb Mateusz Paprocki: >> >> Hi, >> >> On 20 August 2014 21:28, Ondřej Čertík <[email protected]> wrote: >> > Hi Clemens, >> > >> > On Wed, Aug 20, 2014 at 5:15 AM, clemens novak <[email protected]> >> > wrote: >> >> I have a question regarding the apart function. I want to obtain the >> >> partial >> >> fractions for eq = z / (z**2-z-1) . >> >> >> >> The denominator has real roots solve(denom(eq)) yields [1/2 + sqrt(5)/2 >> >> , >> >> 1/2- sqrt(5)/2], but apart does not return the partial fractions; i,e. >> >> apart(eq, z) = z/(z**2 - z - 1) . >> >> >> >> Using apart(eq, z, full=True) yields RootSum(_w**2 - _w - 1, Lambda(_a, >> >> (_a/5 + 2/5)/(-_a + z))) which doesn't seem to be of much help (at >> >> least for >> >> me). >> >> >> >> Using apart with a fraction with "simpler" roots produces the desired >> >> partial fractions; e.g. apart (z / (z**2+z-2), z) yields 2/(3*(z + 2)) >> >> + >> >> 1/(3*(z - 1)) >> >> >> >> Are there any additional options I can provide to apart to obtain the >> >> desired result? >> > >> > Good point. I checked Mathematica and it produces the same result as >> > SymPy. >> > However, clearly this can be decomposed, so I don't quite understand >> > if this is a bug in Mathematica as well, >> > or whether perhaps there is some reason not to do the decomposition >> > for this case. >> >> There is no bug in this case. apart() uses polynomial factorization >> routines to "decompose" the denominator. By default, sympy (but also >> many other computer algebra systems) work over the rationals and z**2 >> - z - 1 doesn't have rational zeros. If you invoke apart(f, z), then >> you will get partial fraction decomposition over rationals. To get the >> expected result, you have set the domain of computation properly, e.g. >> apart(f, z, extension=sqrt(5)) is helpful in this case. apart() >> accepts all options that factor() and other polynomial manipulation >> functions accept. >> >> In [1]: apart(z/(z**2 - z - 1), z, extension=sqrt(5)) >> Out[1]: >> ___ ___ >> ╲╱ 5 + 5 -5 + ╲╱ 5 >> ─────────────────── - ─────────────────── >> ⎛ ___ ⎞ ⎛ ___⎞ >> 5⋅⎝2⋅z - ╲╱ 5 - 1⎠ 5⋅⎝2⋅z - 1 + ╲╱ 5 ⎠ >> >> In [2]: simplify(_) >> Out[2]: >> z >> ────────── >> 2 >> z - z - 1 >> >> Another option is to use full=True (no need for extension, because >> this method doesn't use factorization at all). The result seems >> useless at first but you can use .doit() on the resulting RootSum to >> get more familiar result: >> >> In [1]: apart(z/(z**2 - z - 1), z, full=True) >> Out[1]: >> ⎛ a 2 ⎞ >> ⎜ ─ + ─ ⎟ >> ⎜ 2 5 5 ⎟ >> RootSum⎜w - w - 1, a ↦ ──────⎟ >> ⎝ -a + z⎠ >> >> In [2]: _.doit() >> Out[2]: >> ___ ___ >> ╲╱ 5 1 ╲╱ 5 1 >> ───── + ─ - ───── + ─ >> 10 2 10 2 >> ───────────── + ───────────── >> ___ ___ >> ╲╱ 5 1 1 ╲╱ 5 >> z - ───── - ─ z - ─ + ───── >> 2 2 2 2 >> >> In [3]: simplify(_) >> Out[3]: >> z >> ────────── >> 2 >> z - z - 1 >> >> It would be good to document this in apart()'s docstring (the current >> one is pretty weak), because it's a common misconception that apart() >> uses either roots() or solve() to decompose the denominator. >> >> Mateusz >> >> > Ondrej >> > >> >> >> >> Thanks & regards - Clemens >> >> >> >> -- >> > > Hello, > > thanks for quick answer and support. Both suggestions (apart with full=True > + doit() and apart with extension) work for me. > > I read factor's documentation > (http://docs.sympy.org/latest/modules/polys/reference.html#symbolic-root-finding-algorithms) > which says: > > "By default, the factorization is computed over the rationals. To factor > over other domain, e.g. an lgebraic or finite field, use appropriate > options: > ``extension``, ``modulus`` or ``domain``." > > I don't quite get it: > > "extension" allows providing a list of irrational numbers which shall be > considered during factorization: > > eq = x**2 - sqrt(2)*x + sqrt(3)*x - sqrt(6) > factor(eq, extension=[sqrt(2), sqrt(3)]) > > This works for complex numbers as well > > eq = x**2+x+5 > factor(eq, extension=sqrt(-19)) > > > "modulus allows factorization over a finite field (there is an example in > the "docstring) > > However, I don't get the use of "domain" - from the name I would assume that > you can restrict the domain (e.g. integer, ration, irrational, complex...) > from which the factors are obtained? What domains does sympy provide (and > where can I find that in the documentation)? > > If you want I can try to improve the apart (maybe also the factor) docstring > and send a corresponding PR (probably in the next few days).
That would be awesome --- I was going to suggest if you could send a PR with a better docstring. Those are best written by somebody new to sympy like you, who just figured out how it works and so can document it in a way that other new people will understand well. Ondrej > > Regards - Clemens > > -- > 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 http://groups.google.com/group/sympy. > To view this discussion on the web visit > https://groups.google.com/d/msgid/sympy/ed1fcf30-233e-4551-8b34-040e047e42c5%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]. To post to this group, send email to [email protected]. 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/CADDwiVDmcoSs8eUqEk8wQSTAmPZqFLxW7ab1_Brxkw-w5Tib1A%40mail.gmail.com. For more options, visit https://groups.google.com/d/optout.
