ó ~9­\c@ saddlmZmZddlmZddlmZddlmZm Z m Z m Z m Z m Z ddlmZddlmZddlmZddlmZdd lmZdd lmZdd lmZdd lmZmZdd lm Z m!Z!m"Z"ddl#m$Z$ddl%m&Z&m'Z'ddl(m)Z)m*Z*ddl+m,Z,ddl-m.Z.ddl/m0Z0ddl1m2Z2defd„ƒYZ3de ee3ƒfd„ƒYZ4de3fd„ƒYZ5de5fd„ƒYZ6de5fd„ƒYZ7de3fd „ƒYZ8e9d!„Z:d"e3fd#„ƒYZ;d$e;fd%„ƒYZ<d&e;fd'„ƒYZ=d(S()iÿÿÿÿ(tprint_functiontdivision(tBasic(tcacheit(tranget integer_typestwith_metaclasst is_sequencetiterabletordered(tTuple(tcall_highest_priority(tglobal_evaluate(tUndefinedFunction(tMul(tInteger(tEq(tSt Singleton(tDummytSymboltWild(tsympify(tlcmtfactor(tIntervalt Intersection(tsimplify(tIdx(tflatten(texpandtSeqBasecB s[eZdZeZdZed„ƒZd„Ze d„ƒZ e d„ƒZ e d„ƒZ e d„ƒZ e d„ƒZe d „ƒZe d „ƒZed „ƒZd „Zd „Zd„Zd„Zd„Zd„Zedƒd„ƒZd„Zedƒd„ƒZd„Zd„Zedƒd„ƒZd„Z d„Z!ddd„Z#RS(sBase class for sequencesicC s7y |j}Wn#tttfk r2tj}nX|S(s[Return start (if possible) else S.Infinity. adapted from Set._infimum_key (tstarttNotImplementedErrortAttributeErrort ValueErrorRtInfinity(texprR ((s5lib/python2.7/site-packages/sympy/series/sequences.pyt _start_key$s   cC s%t|j|jƒ}|j|jfS(sTReturns start and stop. Takes intersection over the two intervals. (Rtintervaltinftsup(tselftotherR'((s5lib/python2.7/site-packages/sympy/series/sequences.pyt_intersect_interval1scC std|ƒ‚dS(s&Returns the generator for the sequences(%s).genN(R!(R*((s5lib/python2.7/site-packages/sympy/series/sequences.pytgen9scC std|ƒ‚dS(s-The interval on which the sequence is defineds (%s).intervalN(R!(R*((s5lib/python2.7/site-packages/sympy/series/sequences.pyR'>scC std|ƒ‚dS(s:The starting point of the sequence. This point is includeds (%s).startN(R!(R*((s5lib/python2.7/site-packages/sympy/series/sequences.pyR CscC std|ƒ‚dS(s8The ending point of the sequence. This point is includeds (%s).stopN(R!(R*((s5lib/python2.7/site-packages/sympy/series/sequences.pytstopHscC std|ƒ‚dS(sLength of the sequences (%s).lengthN(R!(R*((s5lib/python2.7/site-packages/sympy/series/sequences.pytlengthMscC sdS(s-Returns a tuple of variables that are bounded(((R*((s5lib/python2.7/site-packages/sympy/series/sequences.pyt variablesRsc st‡fd†ˆjDƒƒS(sG This method returns the symbols in the object, excluding those that take on a specific value (i.e. the dummy symbols). Examples ======== >>> from sympy import SeqFormula >>> from sympy.abc import n, m >>> SeqFormula(m*n**2, (n, 0, 5)).free_symbols {m} c3 s1|]'}|jjˆjƒD] }|VqqdS(N(t free_symbolst differenceR0(t.0titj(R*(s5lib/python2.7/site-packages/sympy/series/sequences.pys es (tsettargs(R*((R*s5lib/python2.7/site-packages/sympy/series/sequences.pyR1WscC sG||jks||jkr:td||jfƒ‚n|j|ƒS(s#Returns the coefficient at point ptsIndex %s out of bounds %s(R R.t IndexErrorR't _eval_coeff(R*tpt((s5lib/python2.7/site-packages/sympy/series/sequences.pytcoeffhscC std|jƒ‚dS(NshThe _eval_coeff method should be added to%s to return coefficient so it is availablewhen coeff calls it.(R!tfunc(R*R:((s5lib/python2.7/site-packages/sympy/series/sequences.pyR9oscC sT|jtjkr|j}n |j}|jtjkrBd}nd}|||S(s†Returns the i'th point of a sequence. If start point is negative infinity, point is returned from the end. Assumes the first point to be indexed zero. Examples ========= >>> from sympy import oo >>> from sympy.series.sequences import SeqPer bounded >>> SeqPer((1, 2, 3), (-10, 10))._ith_point(0) -10 >>> SeqPer((1, 2, 3), (-10, 10))._ith_point(5) -5 End is at infinity >>> SeqPer((1, 2, 3), (0, oo))._ith_point(5) 5 Starts at negative infinity >>> SeqPer((1, 2, 3), (-oo, 0))._ith_point(5) -5 iÿÿÿÿi(R RtNegativeInfinityR.(R*R4tinitialtstep((s5lib/python2.7/site-packages/sympy/series/sequences.pyt _ith_pointus   cC sdS(s  Should only be used internally. self._add(other) returns a new, term-wise added sequence if self knows how to add with other, otherwise it returns ``None``. ``other`` should only be a sequence object. Used within :class:`SeqAdd` class. N(tNone(R*R+((s5lib/python2.7/site-packages/sympy/series/sequences.pyt_addžs cC sdS(s* Should only be used internally. self._mul(other) returns a new, term-wise multiplied sequence if self knows how to multiply with other, otherwise it returns ``None``. ``other`` should only be a sequence object. Used within :class:`SeqMul` class. N(RA(R*R+((s5lib/python2.7/site-packages/sympy/series/sequences.pyt_mul«s cC s t||ƒS(s¤ Should be used when ``other`` is not a sequence. Should be defined to define custom behaviour. Examples ======== >>> from sympy import S, oo, SeqFormula >>> from sympy.abc import n >>> SeqFormula(n**2).coeff_mul(2) SeqFormula(2*n**2, (n, 0, oo)) Notes ===== '*' defines multiplication of sequences with sequences only. (R(R*R+((s5lib/python2.7/site-packages/sympy/series/sequences.pyt coeff_mul¸scC s5t|tƒs(tdt|ƒƒ‚nt||ƒS(s;Returns the term-wise addition of 'self' and 'other'. ``other`` should be a sequence. Examples ======== >>> from sympy import S, oo, SeqFormula >>> from sympy.abc import n >>> SeqFormula(n**2) + SeqFormula(n**3) SeqFormula(n**3 + n**2, (n, 0, oo)) scannot add sequence and %s(t isinstanceRt TypeErrorttypetSeqAdd(R*R+((s5lib/python2.7/site-packages/sympy/series/sequences.pyt__add__Ìs RIcC s||S(N((R*R+((s5lib/python2.7/site-packages/sympy/series/sequences.pyt__radd__ÝscC s6t|tƒs(tdt|ƒƒ‚nt|| ƒS(s:Returns the term-wise subtraction of 'self' and 'other'. ``other`` should be a sequence. Examples ======== >>> from sympy import S, oo, SeqFormula >>> from sympy.abc import n >>> SeqFormula(n**2) - (SeqFormula(n)) SeqFormula(n**2 - n, (n, 0, oo)) scannot subtract sequence and %s(RERRFRGRH(R*R+((s5lib/python2.7/site-packages/sympy/series/sequences.pyt__sub__ás RKcC s | |S(N((R*R+((s5lib/python2.7/site-packages/sympy/series/sequences.pyt__rsub__òscC s |jdƒS(sÚNegates the sequence. Examples ======== >>> from sympy import S, oo, SeqFormula >>> from sympy.abc import n >>> -SeqFormula(n**2) SeqFormula(-n**2, (n, 0, oo)) iÿÿÿÿ(RD(R*((s5lib/python2.7/site-packages/sympy/series/sequences.pyt__neg__ös cC s5t|tƒs(tdt|ƒƒ‚nt||ƒS(s‚Returns the term-wise multiplication of 'self' and 'other'. ``other`` should be a sequence. For ``other`` not being a sequence see :func:`coeff_mul` method. Examples ======== >>> from sympy import S, oo, SeqFormula >>> from sympy.abc import n >>> SeqFormula(n**2) * (SeqFormula(n)) SeqFormula(n**3, (n, 0, oo)) scannot multiply sequence and %s(RERRFRGtSeqMul(R*R+((s5lib/python2.7/site-packages/sympy/series/sequences.pyt__mul__sROcC s||S(N((R*R+((s5lib/python2.7/site-packages/sympy/series/sequences.pyt__rmul__scc s;x4t|jƒD]#}|j|ƒ}|j|ƒVqWdS(N(RR/R@R;(R*R4R:((s5lib/python2.7/site-packages/sympy/series/sequences.pyt__iter__scC s¼t|tƒr+|j|ƒ}|j|ƒSt|tƒr¸|j|j}}|dkrbd}n|dkrz|j}ngt |||j p’dƒD]}|j|j|ƒƒ^q–SdS(Nii( RERR@R;tsliceR R.RAR/RR?(R*tindexR R.R4((s5lib/python2.7/site-packages/sympy/series/sequences.pyt __getitem__s     cC sÚddlm}g|| D]}tt|ƒƒ^q}t|ƒ}|dkr^|d}nt||dƒ}g} xEtd|dƒD]0} d| } g} x,t| ƒD]} | j|| | | !ƒq®W|| ƒ}|j ƒdkr‹t|j ||| | !ƒƒƒ}|| kr9t |ddd…ƒ} Png} x3t| || ƒD]} | j|| | | !ƒqSW|| ƒ}||||| ƒkr»t |ddd…ƒ} Pq»q‹q‹W|dkrÏ| St| ƒ} | dkrñgdfS|| d|| dd| | d|| }}xt| dƒD]{}|||||7}xBt| |dƒD],}|| ||||||d8}qhW|| |||d8}q7W| tt |ƒt |ƒƒfSdS(s› Finds the shortest linear recurrence that satisfies the first n terms of sequence of order `\leq` n/2 if possible. If d is specified, find shortest linear recurrence of order `\leq` min(d, n/2) if possible. Returns list of coefficients ``[b(1), b(2), ...]`` corresponding to the recurrence relation ``x(n) = b(1)*x(n-1) + b(2)*x(n-2) + ...`` Returns ``[]`` if no recurrence is found. If gfvar is specified, also returns ordinary generating function as a function of gfvar. Examples ======== >>> from sympy import sequence, sqrt, oo, lucas >>> from sympy.abc import n, x, y >>> sequence(n**2).find_linear_recurrence(10, 2) [] >>> sequence(n**2).find_linear_recurrence(10) [3, -3, 1] >>> sequence(2**n).find_linear_recurrence(10) [2] >>> sequence(23*n**4+91*n**2).find_linear_recurrence(10) [5, -10, 10, -5, 1] >>> sequence(sqrt(5)*(((1 + sqrt(5))/2)**n - (-(1 + sqrt(5))/2)**(-n))/5).find_linear_recurrence(10) [1, 1] >>> sequence(x+y*(-2)**(-n), (n, 0, oo)).find_linear_recurrence(30) [1/2, 1/2] >>> sequence(3*5**n + 12).find_linear_recurrence(20,gfvar=x) ([6, -5], 3*(5 - 21*x)/((x - 1)*(5*x - 1))) >>> sequence(lucas(n)).find_linear_recurrence(15,gfvar=x) ([1, 1], (x - 2)/(x**2 + x - 1)) iÿÿÿÿ(tMatrixiiiN( tsympy.matricesRURRtlenRAtminRtappendtdettLUsolveRR(R*tntdtgfvarRUtttxtlxtrtcoeffstltl2tmlisttktmtyR4R5((s5lib/python2.7/site-packages/sympy/series/sequences.pytfind_linear_recurrence+sJ")     "       5*N($t__name__t __module__t__doc__tTruetis_commutativet _op_priorityt staticmethodR&R,tpropertyR-R'R R.R/R0R1RR;R9R@RBRCRDRIR RJRKRLRMRORPRQRTRARj(((s5lib/python2.7/site-packages/sympy/series/sequences.pyRs8   )      t EmptySequencecB s>eZdZed„ƒZed„ƒZd„Zd„ZRS(sÍRepresents an empty sequence. The empty sequence is available as a singleton as ``S.EmptySequence``. Examples ======== >>> from sympy import S, SeqPer, oo >>> from sympy.abc import x >>> S.EmptySequence EmptySequence() >>> SeqPer((1, 2), (x, 0, 10)) + S.EmptySequence SeqPer((1, 2), (x, 0, 10)) >>> SeqPer((1, 2)) * S.EmptySequence EmptySequence() >>> S.EmptySequence.coeff_mul(-1) EmptySequence() cC stjS(N(RtEmptySet(R*((s5lib/python2.7/site-packages/sympy/series/sequences.pyR'‹scC stjS(N(RtZero(R*((s5lib/python2.7/site-packages/sympy/series/sequences.pyR/scC s|S(s"See docstring of SeqBase.coeff_mul((R*R;((s5lib/python2.7/site-packages/sympy/series/sequences.pyRD“scC s tgƒS(N(titer(R*((s5lib/python2.7/site-packages/sympy/series/sequences.pyRQ—s(RkRlRmRrR'R/RDRQ(((s5lib/python2.7/site-packages/sympy/series/sequences.pyRsvs  tSeqExprcB sheZdZed„ƒZed„ƒZed„ƒZed„ƒZed„ƒZed„ƒZ RS(s¯Sequence expression class. Various sequences should inherit from this class. Examples ======== >>> from sympy.series.sequences import SeqExpr >>> from sympy.abc import x >>> s = SeqExpr((1, 2, 3), (x, 0, 10)) >>> s.gen (1, 2, 3) >>> s.interval Interval(0, 10) >>> s.length 11 See Also ======== sympy.series.sequences.SeqPer sympy.series.sequences.SeqFormula cC s |jdS(Ni(R7(R*((s5lib/python2.7/site-packages/sympy/series/sequences.pyR-´scC s#t|jdd|jddƒS(Nii(RR7(R*((s5lib/python2.7/site-packages/sympy/series/sequences.pyR'¸scC s |jjS(N(R'R((R*((s5lib/python2.7/site-packages/sympy/series/sequences.pyR ¼scC s |jjS(N(R'R)(R*((s5lib/python2.7/site-packages/sympy/series/sequences.pyR.ÀscC s|j|jdS(Ni(R.R (R*((s5lib/python2.7/site-packages/sympy/series/sequences.pyR/ÄscC s|jddfS(Nii(R7(R*((s5lib/python2.7/site-packages/sympy/series/sequences.pyR0Ès( RkRlRmRrR-R'R R.R/R0(((s5lib/python2.7/site-packages/sympy/series/sequences.pyRw›stSeqPercB s\eZdZdd„Zed„ƒZed„ƒZd„Zd„Z d„Z d„Z RS( sÝRepresents a periodic sequence. The elements are repeated after a given period. Examples ======== >>> from sympy import SeqPer, oo >>> from sympy.abc import k >>> s = SeqPer((1, 2, 3), (0, 5)) >>> s.periodical (1, 2, 3) >>> s.period 3 For value at a particular point >>> s.coeff(3) 1 supports slicing >>> s[:] [1, 2, 3, 1, 2, 3] iterable >>> list(s) [1, 2, 3, 1, 2, 3] sequence starts from negative infinity >>> SeqPer((1, 2, 3), (-oo, 0))[0:6] [1, 2, 3, 1, 2, 3] Periodic formulas >>> SeqPer((k, k**2, k**3), (k, 0, oo))[0:6] [0, 1, 8, 3, 16, 125] See Also ======== sympy.series.sequences.SeqFormula cC s°t|ƒ}d„}d \}}}|dkrP||ƒdtj}}}nt|tƒr³t|ƒdkrƒ|\}}}q³t|ƒdkr³||ƒ}|\}}q³nt|tt fƒ sá|dksá|dkrút dt |ƒƒ‚n|tj kr'|tjkr't dƒ‚nt|||fƒ}t|tƒrftt t|ƒƒƒ}nt d|ƒ‚t|d|dƒtjkrtjStj|||ƒS( NcS s6|j}t|jƒdkr(|jƒStdƒSdS(NiRg(R1RWtpopR(t periodicaltfree((s5lib/python2.7/site-packages/sympy/series/sequences.pyt_find_xs  iiisInvalid limits given: %ss/Both the start and end valuecannot be unboundeds6invalid period %s should be something like e.g (1, 2) i(NNN(RRARR$RR RWRERRR#tstrR=ttupleRRRtRsRt__new__(tclsRztlimitsR|R`R R.((s5lib/python2.7/site-packages/sympy/series/sequences.pyRýs.     .  cC s t|jƒS(N(RWR-(R*((s5lib/python2.7/site-packages/sympy/series/sequences.pytperiod%scC s|jS(N(R-(R*((s5lib/python2.7/site-packages/sympy/series/sequences.pyRz)scC s[|jtjkr)|j||j}n||j|j}|j|j|jd|ƒS(Ni(R RR=R.R‚RztsubsR0(R*R:tidx((s5lib/python2.7/site-packages/sympy/series/sequences.pyR9-sc C sÄt|tƒrÀ|j|j}}|j|j}}t||ƒ}g}xAt|ƒD]3}|||} |||} |j| | ƒqWW|j|ƒ\} } t||jd| | fƒSdS(sSee docstring of SeqBase._addiN( RERxRzR‚RRRYR,R0( R*R+tper1tlper1tper2tlper2t per_lengthtnew_perR`tele1tele2R R.((s5lib/python2.7/site-packages/sympy/series/sequences.pyRB4sc C sÄt|tƒrÀ|j|j}}|j|j}}t||ƒ}g}xAt|ƒD]3}|||} |||} |j| | ƒqWW|j|ƒ\} } t||jd| | fƒSdS(sSee docstring of SeqBase._muliN( RERxRzR‚RRRYR,R0( R*R+R…R†R‡RˆR‰RŠR`R‹RŒR R.((s5lib/python2.7/site-packages/sympy/series/sequences.pyRCEscC s@t|ƒ}g|jD]}||^q}t||jdƒS(s"See docstring of SeqBase.coeff_muli(RRzRxR7(R*R;R`tper((s5lib/python2.7/site-packages/sympy/series/sequences.pyRDVs  N( RkRlRmRARRrR‚RzR9RBRCRD(((s5lib/python2.7/site-packages/sympy/series/sequences.pyRxÍs. (   t SeqFormulacB sMeZdZdd„Zed„ƒZd„Zd„Zd„Z d„Z RS(saRepresents sequence based on a formula. Elements are generated using a formula. Examples ======== >>> from sympy import SeqFormula, oo, Symbol >>> n = Symbol('n') >>> s = SeqFormula(n**2, (n, 0, 5)) >>> s.formula n**2 For value at a particular point >>> s.coeff(3) 9 supports slicing >>> s[:] [0, 1, 4, 9, 16, 25] iterable >>> list(s) [0, 1, 4, 9, 16, 25] sequence starts from negative infinity >>> SeqFormula(n**2, (-oo, 0))[0:6] [0, 1, 4, 9, 16, 25] See Also ======== sympy.series.sequences.SeqPer cC svt|ƒ}d„}d\}}}|dkrP||ƒdtj}}}nt|tƒr³t|ƒdkrƒ|\}}}q³t|ƒdkr³||ƒ}|\}}q³nt|tt fƒ sá|dksá|dkrút dt |ƒƒ‚n|tj kr'|tjkr't dƒ‚nt|||fƒ}t |d|dƒtjkrctjStj|||ƒS( NcS sI|j}t|ƒdkr%|jƒS|s5tdƒStd|ƒ‚dS(NiRgs¦ specify dummy variables for %s. If the formula contains more than one free symbol, a dummy variable should be supplied explicitly e.g., SeqFormula(m*n**2, (n, 0, 5))(R1RWRyRR#(tformulaR{((s5lib/python2.7/site-packages/sympy/series/sequences.pyR|ˆs   iiisInvalid limits given: %ss0Both the start and end value cannot be unboundedi(NNN(RRARR$RR RWRERRR#R}R=RRtRsRR(R€RRR|R`R R.((s5lib/python2.7/site-packages/sympy/series/sequences.pyR…s&     . cC s|jS(N(R-(R*((s5lib/python2.7/site-packages/sympy/series/sequences.pyR¬scC s |jd}|jj||ƒS(Ni(R0RRƒ(R*R:R]((s5lib/python2.7/site-packages/sympy/series/sequences.pyR9°s c C s‚t|tƒr~|j|jd}}|j|jd}}||j||ƒ}|j|ƒ\}}t||||fƒSdS(sSee docstring of SeqBase._addiN(RERŽRR0RƒR,( R*R+tform1tv1tform2tv2RR R.((s5lib/python2.7/site-packages/sympy/series/sequences.pyRB´s c C s‚t|tƒr~|j|jd}}|j|jd}}||j||ƒ}|j|ƒ\}}t||||fƒSdS(sSee docstring of SeqBase._muliN(RERŽRR0RƒR,( R*R+RR‘R’R“RR R.((s5lib/python2.7/site-packages/sympy/series/sequences.pyRC½s cC s-t|ƒ}|j|}t||jdƒS(s"See docstring of SeqBase.coeff_muli(RRRŽR7(R*R;R((s5lib/python2.7/site-packages/sympy/series/sequences.pyRDÆs  N( RkRlRmRARRrRR9RBRCRD(((s5lib/python2.7/site-packages/sympy/series/sequences.pyRŽ]s& '  t RecursiveSeqcB skeZdZd dd„Zed„ƒZed„ƒZed„ƒZd„Z d„Z ed„ƒZ RS( sFA finite degree recursive sequence. That is, a sequence a(n) that depends on a fixed, finite number of its previous values. The general form is a(n) = f(a(n - 1), a(n - 2), ..., a(n - d)) for some fixed, positive integer d, where f is some function defined by a SymPy expression. Parameters ========== recurrence : SymPy expression defining recurrence This is *not* an equality, only the expression that the nth term is equal to. For example, if :code:`a(n) = f(a(n - 1), ..., a(n - d))`, then the expression should be :code:`f(a(n - 1), ..., a(n - d))`. y : function The name of the recursively defined sequence without argument, e.g., :code:`y` if the recurrence function is :code:`y(n)`. n : symbolic argument The name of the variable that the recurrence is in, e.g., :code:`n` if the recurrence function is :code:`y(n)`. initial : iterable with length equal to the degree of the recurrence The initial values of the recurrence. start : start value of sequence (inclusive) Examples ======== >>> from sympy import Function, symbols >>> from sympy.series.sequences import RecursiveSeq >>> y = Function("y") >>> n = symbols("n") >>> fib = RecursiveSeq(y(n - 1) + y(n - 2), y, n, [0, 1]) >>> fib.coeff(3) # Value at a particular point 2 >>> fib[:6] # supports slicing [0, 1, 1, 2, 3, 5] >>> fib.recurrence # inspect recurrence Eq(y(n), y(n - 2) + y(n - 1)) >>> fib.degree # automatically determine degree 2 >>> for x in zip(range(10), fib): # supports iteration ... print(x) (0, 0) (1, 1) (2, 1) (3, 2) (4, 3) (5, 5) (6, 8) (7, 13) (8, 21) (9, 34) See Also ======== sympy.series.sequences.SeqFormula ic  s tˆtƒs'tdjˆƒƒ‚nt|tƒ sA|j rYtdj|ƒƒ‚ntdd|fƒ}d}|jˆƒ}x |D]˜} t| j ƒdkr´tdƒ‚n| j dj ||ƒ|} | j ƒoð| j oð| dks tdj| ƒƒ‚n| |krŠ| }qŠqŠW|s]gt |ƒD]}td j|ƒƒ^q9}nt|ƒ|kr~td ƒ‚nt|ƒ}tˆƒ‰td „|DƒŒ}tj||ˆ|ƒ|ˆƒ} ‡‡fd †t|ƒDƒ| _ˆ| _|| _ˆ| _|| _|| _| S( Ns=recurrence sequence must be an undefined function, found `{}`s0recurrence variable must be a symbol, found `{}`Rgtexcludeiis)Recurrence should be in a single variablesDRecurrence should have constant, negative, integer shifts (found {})sc_{}s)Number of initial terms must equal degreecs s|]}t|ƒVqdS(N(R(R3R`((s5lib/python2.7/site-packages/sympy/series/sequences.pys =sc s)i|]\}}|ˆˆ|ƒ“qS(((R3Rgtinit(R Ri(s5lib/python2.7/site-packages/sympy/series/sequences.pys As (RER RFtformatRt is_symbolRtfindRWR7tmatcht is_constantt is_integerRRR#RRR Rt enumeratetcachet_starttdegreeRiR\t _recurrence( R€t recurrenceRiR\R>R RgR tprev_ystprev_ytshifttseq((R Ris5lib/python2.7/site-packages/sympy/series/sequences.pyRsB   !  1  !"     cC s|jS(s:The starting point of the sequence. This point is included(RŸ(R*((s5lib/python2.7/site-packages/sympy/series/sequences.pyR JscC stjS(s&The ending point of the sequence. (oo)(RR$(R*((s5lib/python2.7/site-packages/sympy/series/sequences.pyR.OscC s|jtjfS(s&Interval on which sequence is defined.(RŸRR$(R*((s5lib/python2.7/site-packages/sympy/series/sequences.pyR'TscC sÃ||jt|jƒkr0|j|j|ƒSxutt|jƒ|dƒD]W}|j|}|jji||j6ƒ}|j|jƒ}||j|j|ƒ>> from sympy import sequence, SeqPer, SeqFormula >>> from sympy.abc import n >>> sequence(n**2, (n, 0, 5)) SeqFormula(n**2, (n, 0, 5)) >>> sequence((1, 2, 3), (n, 0, 5)) SeqPer((1, 2, 3), (n, 0, 5)) See Also ======== sympy.series.sequences.SeqPer sympy.series.sequences.SeqFormula N(RRR RxRŽ(R¦R((s5lib/python2.7/site-packages/sympy/series/sequences.pytsequencets  t SeqExprOpcB sheZdZed„ƒZed„ƒZed„ƒZed„ƒZed„ƒZed„ƒZ RS(sÜBase class for operations on sequences. Examples ======== >>> from sympy.series.sequences import SeqExprOp, sequence >>> from sympy.abc import n >>> s1 = sequence(n**2, (n, 0, 10)) >>> s2 = sequence((1, 2, 3), (n, 5, 10)) >>> s = SeqExprOp(s1, s2) >>> s.gen (n**2, (1, 2, 3)) >>> s.interval Interval(5, 10) >>> s.length 6 See Also ======== sympy.series.sequences.SeqAdd sympy.series.sequences.SeqMul cC std„|jDƒƒS(sjGenerator for the sequence. returns a tuple of generators of all the argument sequences. cs s|]}|jVqdS(N(R-(R3ta((s5lib/python2.7/site-packages/sympy/series/sequences.pys µs(R~R7(R*((s5lib/python2.7/site-packages/sympy/series/sequences.pyR-¯scC std„|jDƒŒS(seSequence is defined on the intersection of all the intervals of respective sequences cs s|]}|jVqdS(N(R'(R3R®((s5lib/python2.7/site-packages/sympy/series/sequences.pys ¼s(RR7(R*((s5lib/python2.7/site-packages/sympy/series/sequences.pyR'·scC s |jjS(N(R'R((R*((s5lib/python2.7/site-packages/sympy/series/sequences.pyR ¾scC s |jjS(N(R'R)(R*((s5lib/python2.7/site-packages/sympy/series/sequences.pyR.ÂscC s)ttg|jD]}|j^qƒƒS(s%Cumulative of all the bound variables(R~RR7R0(R*R®((s5lib/python2.7/site-packages/sympy/series/sequences.pyR0ÆscC s|j|jdS(Ni(R.R (R*((s5lib/python2.7/site-packages/sympy/series/sequences.pyR/Ës( RkRlRmRrR-R'R R.R0R/(((s5lib/python2.7/site-packages/sympy/series/sequences.pyR­—sRHcB s/eZdZd„Zed„ƒZd„ZRS(s’Represents term-wise addition of sequences. Rules: * The interval on which sequence is defined is the intersection of respective intervals of sequences. * Anything + :class:`EmptySequence` remains unchanged. * Other rules are defined in ``_add`` methods of sequence classes. Examples ======== >>> from sympy import S, oo, SeqAdd, SeqPer, SeqFormula >>> from sympy.abc import n >>> SeqAdd(SeqPer((1, 2), (n, 0, oo)), S.EmptySequence) SeqPer((1, 2), (n, 0, oo)) >>> SeqAdd(SeqPer((1, 2), (n, 0, 5)), SeqPer((1, 2), (n, 6, 10))) EmptySequence() >>> SeqAdd(SeqPer((1, 2), (n, 0, oo)), SeqFormula(n**2, (n, 0, oo))) SeqAdd(SeqFormula(n**2, (n, 0, oo)), SeqPer((1, 2), (n, 0, oo))) >>> SeqAdd(SeqFormula(n**3), SeqFormula(n**2)) SeqFormula(n**3 + n**2, (n, 0, oo)) See Also ======== sympy.series.sequences.SeqMul c sÓ|jdtdƒ}t|ƒ}‡fd†‰ˆ|ƒ}g|D]}|tjk rD|^qD}|srtjStd„|DƒŒtjkr˜tjS|r«tj|ƒStt |t j ƒƒ}t j ||ŒS(Ntevaluateic sst|tƒrAt|tƒr7ttˆ|jƒgƒS|gSnt|ƒrcttˆ|ƒgƒStdƒ‚dS(Ns2Input must be Sequences or iterables of Sequences(RERRHtsumtmapR7RRF(targ(t_flatten(s5lib/python2.7/site-packages/sympy/series/sequences.pyR³ôs  cs s|]}|jVqdS(N(R'(R3R®((s5lib/python2.7/site-packages/sympy/series/sequences.pys s(tgetR tlistRRsRRtRHtreduceR RR&RR(R€R7tkwargsR¯R®((R³s5lib/python2.7/site-packages/sympy/series/sequences.pyRís  ( cC süt}xÃ|rËx¶t|ƒD]¨\}}t}xƒt|ƒD]u\}}||krYq;n|j|ƒ}|dk r;g|D]}|||fkr{|^q{}|j|ƒPq;q;W|r|}PqqWq Wt|ƒdkrè|jƒSt|dtƒSdS(sSimplify :class:`SeqAdd` using known rules. Iterates through all pairs and ask the constituent sequences if they can simplify themselves with any other constituent. Notes ===== adapted from ``Union.reduce`` iR¯N( RnRtFalseRBRARYRWRyRH(R7tnew_argstid1tstid2R_tnew_seqR®((s5lib/python2.7/site-packages/sympy/series/sequences.pyR¶s$    +   c st‡fd†|jDƒƒS(s9adds up the coefficients of all the sequences at point ptc3 s|]}|jˆƒVqdS(N(R;(R3R®(R:(s5lib/python2.7/site-packages/sympy/series/sequences.pys 7s(R°R7(R*R:((R:s5lib/python2.7/site-packages/sympy/series/sequences.pyR95s(RkRlRmRRqR¶R9(((s5lib/python2.7/site-packages/sympy/series/sequences.pyRHÐs $$RNcB s/eZdZd„Zed„ƒZd„ZRS(sRepresents term-wise multiplication of sequences. Handles multiplication of sequences only. For multiplication with other objects see :func:`SeqBase.coeff_mul`. Rules: * The interval on which sequence is defined is the intersection of respective intervals of sequences. * Anything \* :class:`EmptySequence` returns :class:`EmptySequence`. * Other rules are defined in ``_mul`` methods of sequence classes. Examples ======== >>> from sympy import S, oo, SeqMul, SeqPer, SeqFormula >>> from sympy.abc import n >>> SeqMul(SeqPer((1, 2), (n, 0, oo)), S.EmptySequence) EmptySequence() >>> SeqMul(SeqPer((1, 2), (n, 0, 5)), SeqPer((1, 2), (n, 6, 10))) EmptySequence() >>> SeqMul(SeqPer((1, 2), (n, 0, oo)), SeqFormula(n**2)) SeqMul(SeqFormula(n**2, (n, 0, oo)), SeqPer((1, 2), (n, 0, oo))) >>> SeqMul(SeqFormula(n**3), SeqFormula(n**2)) SeqFormula(n**5, (n, 0, oo)) See Also ======== sympy.series.sequences.SeqAdd c s«|jdtdƒ}t|ƒ}‡fd†‰ˆ|ƒ}|sJtjStd„|DƒŒtjkrptjS|rƒtj|ƒStt |t j ƒƒ}t j ||ŒS(NR¯ic sst|tƒrAt|tƒr7ttˆ|jƒgƒS|gSn"t|ƒrcttˆ|ƒgƒStdƒ‚dS(Ns2Input must be Sequences or iterables of Sequences(RERRNR°R±R7RRF(R²(R³(s5lib/python2.7/site-packages/sympy/series/sequences.pyR³as  cs s|]}|jVqdS(N(R'(R3R®((s5lib/python2.7/site-packages/sympy/series/sequences.pys qs(R´R RµRRsRRtRNR¶R RR&RR(R€R7R·R¯((R³s5lib/python2.7/site-packages/sympy/series/sequences.pyRZs   cC süt}xÃ|rËx¶t|ƒD]¨\}}t}xƒt|ƒD]u\}}||krYq;n|j|ƒ}|dk r;g|D]}|||fkr{|^q{}|j|ƒPq;q;W|r|}PqqWq Wt|ƒdkrè|jƒSt|dtƒSdS(sSimplify a :class:`SeqMul` using known rules. Iterates through all pairs and ask the constituent sequences if they can simplify themselves with any other constituent. Notes ===== adapted from ``Union.reduce`` iR¯N( RnRR¸RCRARYRWRyRN(R7R¹RºR»R¼R_R½R®((s5lib/python2.7/site-packages/sympy/series/sequences.pyR¶|s$    +   cC s1d}x$|jD]}||j|ƒ9}qW|S(s<multiplies the coefficients of all the sequences at point pti(R7R;(R*R:tvalR®((s5lib/python2.7/site-packages/sympy/series/sequences.pyR9 s(RkRlRmRRqR¶R9(((s5lib/python2.7/site-packages/sympy/series/sequences.pyRN:s "$N(>t __future__RRtsympy.core.basicRtsympy.core.cacheRtsympy.core.compatibilityRRRRRR tsympy.core.containersR tsympy.core.decoratorsR tsympy.core.evaluateR tsympy.core.functionR tsympy.core.mulRtsympy.core.numbersRtsympy.core.relationalRtsympy.core.singletonRRtsympy.core.symbolRRRtsympy.core.sympifyRt sympy.polysRRtsympy.sets.setsRRtsympy.simplifyRtsympy.tensor.indexedRtsympy.utilities.iterablesRtsympyRRRsRwRxRŽR”RAR¬R­RHRN(((s5lib/python2.7/site-packages/sympy/series/sequences.pyts<.ÿY%2p§ #9j