K im VddlmZddlmZddlmZddlmZeeddfdZy) )S)Symbol)lcm)publicxFc#Kddlm}ddlm}tj gtj g}}tj gtj g}}t|rtd} || dk(r| dz } | t|k(ry|| dk(rtj || z g} t| dzt|D]C} d} t| | D]} | || dz| | | z dz zz} | j| || z E| }|ddc}|d<dgtt|| t|zz}dgtt|| t|zz}tt|D] } ||| z|| <t| | t|zD]} || xx|| | z z cc<tt|D] } ||| z|| <t| | t|zD]} || xx|| | z z cc<|ddk(r|j|ddk(r|ddk(r|j|ddk(r||}}||}}d|D]}t|||D]}t||tfdt|Dtfdt|Dz }|r ||n|t|rtyw) a Return a generator for consecutive Pade approximants for a series. It can also be used for computing the rational generating function of a series when possible, since the last approximant returned by the generator will be the generating function (if any). Explanation =========== The input list can contain more complex expressions than integer or rational numbers; symbols may also be involved in the computation. An example below show how to compute the generating function of the whole Pascal triangle. The generator can be asked to apply the sympy.simplify function on each generated term, which will make the computation slower; however it may be useful when symbols are involved in the expressions. Examples ======== >>> from sympy.series import approximants >>> from sympy import lucas, fibonacci, symbols, binomial >>> g = [lucas(k) for k in range(16)] >>> [e for e in approximants(g)] [2, -4/(x - 2), (5*x - 2)/(3*x - 1), (x - 2)/(x**2 + x - 1)] >>> h = [fibonacci(k) for k in range(16)] >>> [e for e in approximants(h)] [x, -x/(x - 1), (x**2 - x)/(2*x - 1), -x/(x**2 + x - 1)] >>> x, t = symbols("x,t") >>> p=[sum(binomial(k,i)*x**i for i in range(k+1)) for k in range(16)] >>> y = approximants(p, t) >>> for k in range(3): print(next(y)) 1 (x + 1)/((-x - 1)*(t*(x + 1) + (x + 1)/(-x - 1))) nan >>> y = approximants(p, t, simplify=True) >>> for k in range(3): print(next(y)) 1 -1/(t*(x + 1) - 1) nan See Also ======== sympy.concrete.guess.guess_generating_function_rational mpmath.pade r)simplify)denomNc3:K|]\}}|z|zzywN.0keXcs _/mnt/ssd/data/python-lab/Trading/venv/lib/python3.12/site-packages/sympy/series/approximants.py zapproximants..a!6AAaC1H6c3:K|]\}}|z|zzywrrrs rrzapproximants..brr)sympy.simplifyr sympy.simplify.radsimpr rOneZerolenrangeappendmaxpoprsum enumerate)lrr simpr p1q1p2q2bmrsjapqroutrs ` @r approximantsr4sh0,eeWqvvhBffXwB a& dAg FACF{dAgUU1Q4ZLqsCF# AA1a[ 'QqsVa!Ah&& ' HHQqtV    A$1Q4 C#c"gqRy) ) C#c"gqRy) )s2w ARU7AaD q!CG)$ A aDBqsGOD s2w ARU7AaD q!CG)$ A aDBqsGOD eQhaeQheQhaeQhQBQB  !AAuQx A ! !AAuQx A !616661667 s) IQ a&R s BKFK K>BKKN) sympy.core.singletonrsympy.core.symbolrsympy.polys.polytoolsrsympy.utilitiesrr4rrr:s."$%"SkE` ` r9