\h-c$SrSSKJr SSKJr SSKJr SSKJrJ r SSK J r J r J r SSKJr SSKJr SS KJr SS KJr SS KJr SS KJrJrJr SS KJrJr SSKJ r J!r!J"r"J#r#J$r$J%r% SSK&J'r'J(r(J)r)J*r* SSK+J,r,J-r- SSK.J/r/ SSjr0Sr1Sr2SSjr3g)a1 This module is intended for solving recurrences or, in other words, difference equations. Currently supported are linear, inhomogeneous equations with polynomial or rational coefficients. The solutions are obtained among polynomials, rational functions, hypergeometric terms, or combinations of hypergeometric term which are pairwise dissimilar. ``rsolve_X`` functions were meant as a low level interface for ``rsolve`` which would use Mathematica's syntax. Given a recurrence relation: .. math:: a_{k}(n) y(n+k) + a_{k-1}(n) y(n+k-1) + ... + a_{0}(n) y(n) = f(n) where `k > 0` and `a_{i}(n)` are polynomials in `n`. To use ``rsolve_X`` we need to put all coefficients in to a list ``L`` of `k+1` elements the following way: ``L = [a_{0}(n), ..., a_{k-1}(n), a_{k}(n)]`` where ``L[i]``, for `i=0, \ldots, k`, maps to `a_{i}(n) y(n+i)` (`y(n+i)` is implicit). For example if we would like to compute `m`-th Bernoulli polynomial up to a constant (example was taken from rsolve_poly docstring), then we would use `b(n+1) - b(n) = m n^{m-1}` recurrence, which has solution `b(n) = B_m + C`. Then ``L = [-1, 1]`` and `f(n) = m n^(m-1)` and finally for `m=4`: >>> from sympy import Symbol, bernoulli, rsolve_poly >>> n = Symbol('n', integer=True) >>> rsolve_poly([-1, 1], 4*n**3, n) C0 + n**4 - 2*n**3 + n**2 >>> bernoulli(4, n) n**4 - 2*n**3 + n**2 - 1/30 For the sake of completeness, `f(n)` can be: [1] a polynomial -> rsolve_poly [2] a rational function -> rsolve_ratio [3] a hypergeometric function -> rsolve_hyper ) defaultdict)product)S)RationalI)SymbolWildDummy)Equality)Add)Mul)default_sort_key)sympify)simplify hypersimp hypersimilar)solvesolve_undetermined_coeffs)Polyquogcdlcmroots resultant)binomial factorialFallingFactorialRisingFactorial)Matrix casoratian)numbered_symbolsc ^^,^-^.^/[U5nURT5(dgURn[U5S- nUVs/sHn[ UT5PM nn[ ST5/US--n[ R [ R4/US--n [US-5Hqn [XS-5H,n X==X [X5RT5-- ss'M. XR(aMVXR5uupX|4X'Ms U SS=p[SUS-5H/n XSU :aXSn XSU - U:dM%XSU - nM1 [U 5[U5p[S5n[ R n[US-5H)n XSU - U:XdMUXS[X5-- nM+ [[!UUSSS9R#55nU(a [%U5/nO/nU(a UU*S- /- nO+UURT5R'5U- U*S- /- n[[%U55nUS:aAU(a9UR)SS 5(a[ R /4$[ R $gUU::a/m,[ R =nn[US-5H9n T,R+[-S [/X-5-55 UT,U TU --- nM; [US-5H.n UX R15UR3TTU -5-- nM0 [5UU- T,T5nUb0T,nT,Vs/sH nUU;dM UPM snm,UR3U5nGO;gUnUU-U-S-n[[!XSS S9R#55nU/:wa[%U5S-m/O[ R m/S nS nU/U4Sjm.0n[U*U S-5Hn U"U S-5n[SU S-5HnUUS- X-U- S--U- UU'M [ R UU '[US-5HUn [U S-5H@n[UX-5n T."XR15U5n!UU ==UUU -U!-- ss'MB MW M [7UUS5n"U(a[UU5Hn U"U5n#[SUU-S-5HRnU U- S:a OHUUU- R3XU- 5n [U5Hn U#U ==U U"U U- U 4-- ss'M MT UU*R3X5n$[U5Hn U#U *U$- U"X4'M M OU"U5n%[UU5Hn U"U5n#[ R n&[SUU-S-5H`nU U- S:a OVUUU- R3XU- 5n [U5Hn U#U ==U U"U U- U 4-- ss'M U&U U%U U- -- n&Mb UU*R3X5n$[U5Hn U#U *U$- U"X4'M T."XU- 5U&- U$- U%U 'M U"U5U"U5sn'm-[SU5H+n U'U S- TT/- U - S--U - R95U'U 'M- [U5HFn [;[=U"SS2U 4U'5V#V(s/sHun#n(U#U(-R95PM sn(n#6T-U 'MH U(d9[;[=W%U'5V&V(s/sHun&n(U&U(-R95PM sn(n&6n)[U5V s/sHn [-S [/X-5-5PM sn m,U,U-U.4Sjn&U(a&[US-U5V s/sH n U&"U 5PM nn O/[US-U5V s/sHn U&"U 5T."W)U 5-PM nn U/:waU[?U/T,Q76nU(dAU(a9UR)SS 5(a[ R /4$[ R $gO0nU(a[ R nOW)nT,SSn[[=T,T-55H>unn*UU;aUUU*-n+T,RAU5 OUU*-n+UU+R95- nM@ T,U:wa2URC[E[=T,U555nUS[T,5m,UR)SS 5(aUT,4$U$s snfs snfs sn(n#fs sn(n&fs sn fs sn fs sn f)a Given linear recurrence operator `\operatorname{L}` of order `k` with polynomial coefficients and inhomogeneous equation `\operatorname{L} y = f`, where `f` is a polynomial, we seek for all polynomial solutions over field `K` of characteristic zero. The algorithm performs two basic steps: (1) Compute degree `N` of the general polynomial solution. (2) Find all polynomials of degree `N` or less of `\operatorname{L} y = f`. There are two methods for computing the polynomial solutions. If the degree bound is relatively small, i.e. it's smaller than or equal to the order of the recurrence, then naive method of undetermined coefficients is being used. This gives a system of algebraic equations with `N+1` unknowns. In the other case, the algorithm performs transformation of the initial equation to an equivalent one for which the system of algebraic equations has only `r` indeterminates. This method is quite sophisticated (in comparison with the naive one) and was invented together by Abramov, Bronstein and Petkovsek. It is possible to generalize the algorithm implemented here to the case of linear q-difference and differential equations. Lets say that we would like to compute `m`-th Bernoulli polynomial up to a constant. For this we can use `b(n+1) - b(n) = m n^{m-1}` recurrence, which has solution `b(n) = B_m + C`. For example: >>> from sympy import Symbol, rsolve_poly >>> n = Symbol('n', integer=True) >>> rsolve_poly([-1, 1], 4*n**3, n) C0 + n**4 - 2*n**3 + n**2 References ========== .. [1] S. A. Abramov, M. Bronstein and M. Petkovsek, On polynomial solutions of linear operator equations, in: T. Levelt, ed., Proc. ISSAC '95, ACM Press, New York, 1995, 290-296. .. [2] M. Petkovsek, Hypergeometric solutions of linear recurrences with polynomial coefficients, J. Symbolic Computation, 14 (1992), 243-264. .. [3] M. Petkovsek, H. S. Wilf, D. Zeilberger, A = B, 1996. NrxZc US:$Nrrs L/var/www/auris/envauris/lib/python3.13/site-packages/sympy/solvers/recurr.pyrsolve_poly..AFfilter predicatesymbolsFCc US:$r'r(r)s r+r,r-sQr/c*[R/U-$NrZeroks r+ _zero_vector!rsolve_poly.._zero_vectorsFF8a< r/c*[R/U-$r7)rOner:s r+ _one_vector rsolve_poly.._one_vectorsEE7Q; r/c>[RnURTTU-5n[SUS-5H4nU[ XA- S- U5-nX2URTTU-U- 5-- nM6 U$)Nr#)rr?subsranger)pr;BDians r+_deltarsolve_poly.._deltasrAq!a% A1a!e_Xaeai++1q519---%Hr/c[X:H5$r7)int)rHjs r+r,r-s c!&kr/c p>[[TT5VVs/sHupUT"X 5-PM snn6$s snnfr7)r zip)rHcqr4QrKs r+r,r-3s,cs1ayAytqAfQlNyABAs2 )#r is_polynomialis_zerolenrrr9NegativeInfinityrDras_polyLTrNr rlistrkeysmaxdegreegetappendrstras_exprrCrrexpandr rQrremovexreplacedict)0coeffsfrJshifthints homogeneousr*coeffpolystermsrHrOexpdbr$ degree_poly nni_rootsNyE solutions_CrRresultAUr<r@alpharr;rFrGVvdenomGgPrEhrSsr4rTrKrIs0 ` @@@@r+ rsolve_polyrEsh  A ??1  ))K F aA*0 1&d5!n&F 1 !QZL!a% Effa(( ) *AE 2E 1q5\qa%A H 8A>#9#9!#<= =H!x!HKKMMFS|EH  !HQKA 1a!e_ 8A;? A 8A;?Q  aA  q63q6q c A&&K 1q5\ 8A;?a  58A;'7'== =KU;#"$$(DF,I ^   qb1fX  aiil!!#a'!a 00 CF A1u yyE**|#vv Av Aq1uA HHVC#ai.01 2 11 Aq1uA ""$QVVAq1u%55 5A.a!eQ:  B6Aq!9"4A6AVVI&F  EAIMux&((,0  ?I"AA   r1q5!AAE"A1a!e_Qx1519q=1!3!%vvE!H1q5\q1uA AE*Aux//115A!H!Qq(H &"" 1a1 2 1a[ Oq!a%!),A1uqya!e ))!U3A"1X!Aa!eQhK/& -qb q,qA teemAadG"!"QA1a[ OFFq!a%!),A1uqya!e ))!U3A"1X!Aa!eQhK/&Qq1uX%A-qb q,qA teemAadG"qa%(1,5!)!,1~|A1q!Aa!eHA A .q088:AaDqAC!Q$ODODAq!A#ODEAaDQ;Aqslln;o1q! $oA> 7a ! IyyE22 !|+ vv I VVFF qTQODAqI~aLN aC ahhj F$ Bwc!Rj!12 wAK yyE""{ {2N7FE< < 0>s0d! d /d 0 d% 7 d+ )#d1/d6d;c [U5nURU5(dg[[[U55n[ U5S- nXUSpeUR X"U- 5R 5n[S5n[XVR X"U-5U5nURU5(dUR5up[XU5n[[XSSS9R55n U (d [XU40UD6$[R[R /US--p[#[%['U 55SS5Hn[)XVR X"U-5U5n[X_U5n[XoR X"U- 5U5nU [+[#US-5Vs/sHnUR X"U- 5PM sn6-n M [#US-5Vs/sHoR X"U-5PM nn[#US-5H9n[)XUUU5n[XUU5X'[UUUU5UU'M; [#US-5H!nX==[+USUUUS-S-6-ss'M# [X[+U6-U40UD6nUb;UR-S S 5(a[/USU - 5US4$[/UU - 5$gs snfs snf) aC Given linear recurrence operator `\operatorname{L}` of order `k` with polynomial coefficients and inhomogeneous equation `\operatorname{L} y = f`, where `f` is a polynomial, we seek for all rational solutions over field `K` of characteristic zero. This procedure accepts only polynomials, however if you are interested in solving recurrence with rational coefficients then use ``rsolve`` which will pre-process the given equation and run this procedure with polynomial arguments. The algorithm performs two basic steps: (1) Compute polynomial `v(n)` which can be used as universal denominator of any rational solution of equation `\operatorname{L} y = f`. (2) Construct new linear difference equation by substitution `y(n) = u(n)/v(n)` and solve it for `u(n)` finding all its polynomial solutions. Return ``None`` if none were found. The algorithm implemented here is a revised version of the original Abramov's algorithm, developed in 1989. The new approach is much simpler to implement and has better overall efficiency. This method can be easily adapted to the q-difference equations case. Besides finding rational solutions alone, this functions is an important part of Hyper algorithm where it is used to find a particular solution for the inhomogeneous part of a recurrence. Examples ======== >>> from sympy.abc import x >>> from sympy.solvers.recurr import rsolve_ratio >>> rsolve_ratio([-2*x**3 + x**2 + 2*x - 1, 2*x**3 + x**2 - 6*x, ... - 2*x**3 - 11*x**2 - 18*x - 9, 2*x**3 + 13*x**2 + 22*x + 8], 0, x) C0*(2*x - 3)/(2*(x**2 - 1)) References ========== .. [1] S. A. Abramov, Rational solutions of linear difference and q-difference equations with polynomial coefficients, in: T. Levelt, ed., Proc. ISSAC '95, ACM Press, New York, 1995, 285-289 See Also ======== rsolve_hyper Nr#rrr%c US:$r'r(r)s r+r,rsolve_ratio..r.r/r0r3F)rrUr[maprWrCrcr ras_numer_denomrrr\rrr?r9rDrNr]rr r_r)rgrhrJrjr*rzrFrresrErSrsr4numersrHrprOdenomsrrys r+ rsolve_ratiorbsj  A ??1   #gv& 'F F aA 9fQiq qa%!A c A AvvaQ' +C   Q  !!#!lU3#"$$(DF,I 6a1511EEAFF8QU+6s3y>*B3AAvvaQ'+AA! AAvvaQ'+A q1u>AqvvaQ'>? ?A 4-2!a%L9Lq&&E"L9q1uAFIvay!,AFIq!,FIF1Iq!,F1I  q1uA IvbqzF1q56N:< >> from sympy.solvers import rsolve_hyper >>> from sympy.abc import x >>> rsolve_hyper([-1, -1, 1], 0, x) C0*(1/2 - sqrt(5)/2)**x + C1*(1/2 + sqrt(5)/2)**x >>> rsolve_hyper([-1, 1], 1 + x, x) C0 + x*(x + 1)/2 References ========== .. [1] M. Petkovsek, Hypergeometric solutions of linear recurrences with polynomial coefficients, J. Symbolic Computation, 14 (1992), 243-264. .. [2] M. Petkovsek, H. S. Wilf, D. Zeilberger, A = B, 1996. r#NTr3rr%r4F)zero)keyr3)4r[rrrWsetrVis_Addrcargsis_hypergeometricr\rrr9items enumerater?rrDrCrr rrQrr rr r is_integerrr`rYr^r]nthrbrracollectrlhasrrrrrr sortrr!r_)-rgrhrJrjr*kernelr3similarrr inhomogeneousrHrlrmrrrOrErSRsymsryr% p_factors q_factorsfactorsrzrFdegreesrGrIrqpolyrpz recurr_coeffssolr4ration0n_rootKskkerr;s- r+ rsolve_hyperrs n#gv& 'F AVq"cewA 999 88GXXZ__**1-- A#A!,, a ( "#GJ%07}}?tqQUM?M   # #CMm,DA55&)5eeWa!e_F!A1a!e_q519--++-1aA q %1q5\aC&!*va!ef~"=??" UCL!TBA}4s4!SY78AsF|Q7 A% -(Ff%FE-H c A !9finnQA 2qAU1a[%%'(IU1a[%%'(Iquu~G A|| aQUAEN++ "++A!a%+A!*+A!%%Q+A'kAoG1Rw affQA " "q1uAa91affQA&9:Aa < 1affQA& <=Avy{1}a+D LLa )<<<ux01 'lAFFtAtq1uAHLLOEAFF"1 $  tQ$$&Ayy>CAElKl1X--/14lMKAv!sq!a%$QA]1%5a%7$QRRcCG $5567' q!S\SWX Tkk$'-12Tsyy|T2Aqa% 0014q8 #E$8$8$:1$=qAFFHFzz!}}# +4#aZ I EAr1q5>255$4oFF  AfslAE:a?MM!$#'3t KK$K% c"3'0 1B3!c' yyE"""%"$!QA"%%W && M@x ,+:<.L$Q3:&s68_0__ _# >_( ;"_-<_2%_7/_<Nc  ^[U[5(aURUR- nURSm[ ST4S9nUR 5RUR[ SSS955n[[5n/n[R"U5HnURUR5upxU(dURU5 M:UHn U R(a_U RUR:XaEU RSR!TU-5n U b!U[#X5RU5 Ms[%SUR<S T<S U <S35e M UHn[XC6XC'M S Ul[U6nUR)5Hup7[+U5XC'M [,R.n UR0(dgUR3T5(dQUR4(a2[7U4S jUR 5R55(d[%S U-5eUR95H\nUR;T5(a7UR=T5(d[?XRA5ST5n MNMP[%SU-5e URA5upU R=T5(a [?XT5n U [,R.LaKUR)5H(up7URA5upU[CXT5-XC'M* U [CXT5-n[EURG55nUS:a[IU5n[S5nURKTTU-5R 5nU RKTTU-5R 5n UR)5H.up7URKTTU-5R 5UUU-'M0 OUn[MURG55n[OUS-5Vs/sHnUUPM nn[QUU*TSS9n U cgU unnU0/4;aSnU(aUb[U[5(a([O[SU55Vs0sH nUUU_M nn/nUR)5Hqunn[#U5nURKTU5U- nURW[,RX5(aUR[TU5U- nURU5 Ms []U/UQ76n U (dgURKU 5nU$s snfs snf![TaT UR(a4URUR:Xa[#URS5nN[%SU-5ef=f)a Solve univariate recurrence with rational coefficients. Given `k`-th order linear recurrence `\operatorname{L} y = f`, or equivalently: .. math:: a_{k}(n) y(n+k) + a_{k-1}(n) y(n+k-1) + \cdots + a_{0}(n) y(n) = f(n) where `a_{i}(n)`, for `i=0, \ldots, k`, are polynomials or rational functions in `n`, and `f` is a hypergeometric function or a sum of a fixed number of pairwise dissimilar hypergeometric terms in `n`, finds all solutions or returns ``None``, if none were found. Initial conditions can be given as a dictionary in two forms: (1) ``{ n_0 : v_0, n_1 : v_1, ..., n_m : v_m}`` (2) ``{y(n_0) : v_0, y(n_1) : v_1, ..., y(n_m) : v_m}`` or as a list ``L`` of values: ``L = [v_0, v_1, ..., v_m]`` where ``L[i] = v_i``, for `i=0, \ldots, m`, maps to `y(n_i)`. Examples ======== Lets consider the following recurrence: .. math:: (n - 1) y(n + 2) - (n^2 + 3 n - 2) y(n + 1) + 2 n (n + 1) y(n) = 0 >>> from sympy import Function, rsolve >>> from sympy.abc import n >>> y = Function('y') >>> f = (n - 1)*y(n + 2) - (n**2 + 3*n - 2)*y(n + 1) + 2*n*(n + 1)*y(n) >>> rsolve(f, y(n)) 2**n*C0 + C1*factorial(n) >>> rsolve(f, y(n), {y(0):0, y(1):3}) 3*2**n - 3*factorial(n) See Also ======== rsolve_poly, rsolve_ratio, rsolve_hyper rr;)excludemT)integerN'(z + k)' expected, got 'cgr'r(r(r/r+r,rsolve..sQr/c3D># UHoRT5v M g7fr7)r).0r$rJs r+ rsolve..s"XCWa#6#6q#9#9CWs zJThe independent term should be a sum of hypergeometric functions, got '%s'r#z2Polynomial or rational function expected, got '%s'c"[R$r7r8r(r/r+r,rsQVVr/rz"Integer or term expected, got '%s')/ isinstancer lhsrhsrr rcrfuncrr[r make_args as_coeff_mulr` is_FunctionmatchrN ValueErrordefault_factoryrrrr?rVrrallvaluesis_rational_functionrUrrrminr\absrCr]rDrrW TypeErrorrNaNlimitr)rhruinitr;h_parti_partrrldeprrycommoni_numeri_denomnumerrK_minrH_partK_maxrHrgsolutionr3 equationsr~eqrJs @r+rsolversh!X EEAEEM q A S1$A  166$sD"9:;A  F F ]]1 ^^AFF+  MM% A}}166!1Q/%3vy>*11%856VVQBD D  O &F &\FLLNUO #UUF >>&":":1"="= MMc"X6==?CWCW"XYYehnnoo  % %a ( (&&q))V%9%9%;A%>B*DuLN N !,,.GQVa( QUU HA //1LEc&33FI'Va00  E qy J^,QA&--/QA&--/ HA!JJq!a%0779F1q5M'  E!&uqy!1 2!1AfQi!1F 2 &6'1d ;F ~Hg Bx4# dD ! !(-c$i(89(81AtAwJ(8D9 JJLDAq OFq!$q(Bvvaee}}^^Aq)A-   R !y+7+}}V,H OO3: O==QVVqvv%5AFF1IA$%IA%MNN  Os=U##U( U--A W <W )rr7)4__doc__ collectionsrsympy.concretersympy.core.singletonrsympy.core.numbersrrsympy.core.symbolrr r sympy.core.relationalr sympy.core.addr sympy.core.mulr sympy.core.sortingrsympy.core.sympifyrsympy.simplifyrrr sympy.solversrr sympy.polysrrrrrrsympy.functionsrrrrsympy.matricesrr sympy.utilities.iterablesr!rrrrr(r/r+rsd/`$""*11*/&<<:==RR-6Zzl^Rjer/