B [@sHdZddlmZmZddlmZmZmZddlm Z m Z m Z ddZ dS)z Recurrences )print_functiondivision)SSymbolsympify)as_intrangeiterablecs|s tjSt|stdt|s*tdt|}|dkrBtddd|Ddd|D}tt|kr|tdn|tjgt|7}fd d fd d |kr||Std dt||DS)a Evaluation of univariate linear recurrences of homogeneous type having coefficients independent of the recurrence variable. Parameters ========== coeffs : iterable Coefficients of the recurrence init : iterable Initial values of the recurrence n : Integer Point of evaluation for the recurrence Notes ===== Let `y(n)` be the recurrence of given type, ``c`` be the sequence of coefficients, ``b`` be the sequence of initial/base values of the recurrence and ``k`` (equal to ``len(c)``) be the order of recurrence. Then, .. math :: y(n) = \begin{cases} b_n & 0 \le n < k \\ c_0 y(n-1) + c_1 y(n-2) + \cdots + c_{k-1} y(n-k) & n > k \end{cases} Let `x_0, x_1, \ldots, x_n` be a sequence and consider the transformation that maps each polynomial `f(x)` to `T(f(x))` where each power `x^i` is replaced by the corresponding value `x_i`. The sequence is then a solution of the recurrence if and only if `T(x^i p(x)) = 0` for each `i \ge 0` where `p(x) = x^k - c_0 x^(k-1) - \cdots - c_{k-1}` is the characteristic polynomial. Then `T(f(x)p(x)) = 0` for each polynomial `f(x)` (as it is a linear combination of powers `x^i`). Now, if `x^n` is congruent to `g(x) = a_0 x^0 + a_1 x^1 + \cdots + a_{k-1} x^{k-1}` modulo `p(x)`, then `T(x^n) = x_n` is equal to `T(g(x)) = a_0 x_0 + a_1 x_1 + \cdots + a_{k-1} x_{k-1}`. Computation of `x^n`, given `x^k = c_0 x^{k-1} + c_1 x^{k-2} + \cdots + c_{k-1}` is performed using exponentiation by squaring (refer to [1]_) with an additional reduction step performed to retain only first `k` powers of `x` in the representation of `x^n`. Examples ======== >>> from sympy.discrete.recurrences import linrec >>> from sympy.abc import x, y, z >>> linrec(coeffs=[1, 1], init=[0, 1], n=10) 55 >>> linrec(coeffs=[1, 1], init=[x, y], n=10) 34*x + 55*y >>> linrec(coeffs=[x, y], init=[0, 1], n=5) x**2*y + x*(x**3 + 2*x*y) + y**2 >>> linrec(coeffs=[1, 2, 3, 0, 0, 4], init=[x, y, z], n=16) 13576*x + 5676*y + 2356*z References ========== .. [1] https://en.wikipedia.org/wiki/Exponentiation_by_squaring .. [2] https://en.wikipedia.org/w/index.php?title=Modular_exponentiation§ion=6#Matrices See Also ======== sympy.polys.agca.extensions.ExtensionElement.__pow__ z6Expected a sequence of coefficients for the recurrencezFExpected a sequence of values for the initialization of the recurrencerz@Point of evaluation of recurrence must be a non-negative integercSsg|] }t|qS)r).0argr r 9lib/python3.7/site-packages/sympy/discrete/recurrences.py fszlinrec..cSsg|] }t|qSr )r)r r r r r rgszECount of initial values should not exceed the order of the recurrencecstjgdt|d|}xFt|D]:\}}x0t|D]$\}}||||||7<q8Wq&WxVtt|dddD]:}x4tD](}|||d|||7<qWq~W|dS)N)rZerolen enumerater)uoffsetwipjq)ckr r _square_and_reduceps$,z"linrec.._square_and_reducecsJ|kr0tjg|tjgtjg|dS|d|dSdS)Nrr)rrZOne)n) _final_coeffsrrr r r s(zlinrec.._final_coeffscss|]\}}||VqdS)Nr )r rvr r r szlinrec..) rrr TypeErrorr ValueErrorrsumzip)ZcoeffsZinitrbr )r rrrr linrec s$M   r(N) __doc__Z __future__rrZ sympy.corerrrZsympy.core.compatibilityrrr r(r r r r s