o jg׊@sddlmZddlmZddlmZddlmZddlm Z ddl m Z m Z ddl mZddlmZdd lmZdd lmZmZdd lmZdd lmZmZmZdd lmZddlmZddl m!Z!m"Z"ddl#m$Z$m%Z%ddl&m'Z'ddl(m)Z)m*Z*m+Z+GdddeZ,Gddde,edZ-Gddde,Z.Gddde.Z/Gddde.Z0Gddde,Z1d)d!d"Z2Gd#d$d$e,Z3Gd%d&d&e3Z4Gd'd(d(e3Z5d S)*)Basic)cacheit)Tuple)call_highest_priority)global_parameters) AppliedUndefexpandMul)Integer)Eq)S Singleton)ordered)DummySymbolWildsympify)Matrix)lcmfactor)Interval Intersection)Idx)flatten is_sequenceiterablec@seZdZdZdZdZeddZddZe dd Z e d d Z e d d Z e ddZ e ddZe ddZe ddZeddZddZddZddZddZd d!Zd"d#Zed$d%d&Zd'd(Zed)d*d+Zd,d-Zd.d/Zed0d1d2Zd3d4Z d5d6Z!d:d8d9Z"d7S);SeqBasezBase class for sequencesTcCs(z|j}W|Stytj}Y|Sw)z[Return start (if possible) else S.Infinity. adapted from Set._infimum_key )startNotImplementedErrorr Infinity)exprr r$N/var/www/html/zoom/venv/lib/python3.10/site-packages/sympy/series/sequences.py _start_key s zSeqBase._start_keycCst|j|j}|j|jfS)zTReturns start and stop. Takes intersection over the two intervals. )rintervalinfsup)selfotherr'r$r$r%_intersect_interval,s zSeqBase._intersect_intervalcC td|)z&Returns the generator for the sequencez(%s).genr!r*r$r$r%gen4 z SeqBase.gencCr-)z-The interval on which the sequence is definedz (%s).intervalr.r/r$r$r%r'9r1zSeqBase.intervalcCr-):The starting point of the sequence. This point is includedz (%s).startr.r/r$r$r%r >r1z SeqBase.startcCr-)z8The ending point of the sequence. This point is includedz (%s).stopr.r/r$r$r%stopCr1z SeqBase.stopcCr-)zLength of the sequencez (%s).lengthr.r/r$r$r%lengthHr1zSeqBase.lengthcCdS)z-Returns a tuple of variables that are boundedr$r$r/r$r$r% variablesMszSeqBase.variablescsfddjDS)aG 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} cs$h|]}|jjD]}|q qSr$) free_symbols differencer6).0ijr/r$r% `s z'SeqBase.free_symbols..argsr/r$r/r%r7RszSeqBase.free_symbolscCs0||jks ||jkrtd||jf||S)z#Returns the coefficient at point ptzIndex %s out of bounds %s)r r3 IndexErrorr' _eval_coeffr*ptr$r$r%coeffcs z SeqBase.coeffcCstd|j)NzhThe _eval_coeff method should be added to%s to return coefficient so it is availablewhen coeff calls it.)r!funcrAr$r$r%r@jszSeqBase._eval_coeffcCs<|jtjur |j}n|j}|jtjurd}nd}|||S)aReturns the i'th point of a sequence. Explanation =========== 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 )r r NegativeInfinityr3)r*r:initialstepr$r$r% _ith_pointps   zSeqBase._ith_pointcCr5)aI Should only be used internally. Explanation =========== 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. Nr$r*r+r$r$r%_addz SeqBase._addcCr5)aS Should only be used internally. Explanation =========== self._mul(other) returns a new, term-wise multiplied sequence if self knows how to multiply with other, otherwise it returns ``None``. ``other`` should only be a sequence object. Used within :class:`SeqMul` class. Nr$rKr$r$r%_mulrMz SeqBase._mulcCs t||S)a Should be used when ``other`` is not a sequence. Should be defined to define custom behaviour. Examples ======== >>> from sympy import 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 rKr$r$r% coeff_muls zSeqBase.coeff_mulcC$t|ts tdt|t||S)a4Returns the term-wise addition of 'self' and 'other'. ``other`` should be a sequence. Examples ======== >>> from sympy import SeqFormula >>> from sympy.abc import n >>> SeqFormula(n**2) + SeqFormula(n**3) SeqFormula(n**3 + n**2, (n, 0, oo)) zcannot add sequence and %s isinstancer TypeErrortypeSeqAddrKr$r$r%__add__s  zSeqBase.__add__rVcCs||SNr$rKr$r$r%__radd__zSeqBase.__radd__cCs&t|ts tdt|t|| S)a7Returns the term-wise subtraction of ``self`` and ``other``. ``other`` should be a sequence. Examples ======== >>> from sympy import SeqFormula >>> from sympy.abc import n >>> SeqFormula(n**2) - (SeqFormula(n)) SeqFormula(n**2 - n, (n, 0, oo)) zcannot subtract sequence and %srQrKr$r$r%__sub__s  zSeqBase.__sub__rZcCs | |SrWr$rKr$r$r%__rsub__ zSeqBase.__rsub__cCs |dS)zNegates the sequence. Examples ======== >>> from sympy import SeqFormula >>> from sympy.abc import n >>> -SeqFormula(n**2) SeqFormula(-n**2, (n, 0, oo)) rE)rOr/r$r$r%__neg__s zSeqBase.__neg__cCrP)a{Returns the term-wise multiplication of 'self' and 'other'. ``other`` should be a sequence. For ``other`` not being a sequence see :func:`coeff_mul` method. Examples ======== >>> from sympy import SeqFormula >>> from sympy.abc import n >>> SeqFormula(n**2) * (SeqFormula(n)) SeqFormula(n**3, (n, 0, oo)) zcannot multiply sequence and %s)rRrrSrTSeqMulrKr$r$r%__mul__s  zSeqBase.__mul__r_cCs||SrWr$rKr$r$r%__rmul__rYzSeqBase.__rmul__ccs,t|jD] }||}||VqdSrW)ranger4rJrC)r*r:rBr$r$r%__iter__s  zSeqBase.__iter__cstt|tr|}|St|tr8|j|j}}|dur!d}|dur(j}fddt|||j p4dDSdS)Nrcsg|] }|qSr$)rCrJ)r9r:r/r$r% ,z'SeqBase.__getitem__..rF) rRintrJrCslicer r3r4rarI)r*indexr r3r$r/r% __getitem__"s     zSeqBase.__getitem__Ncs,ddlmfdd|d|D}t|}|dur |d}nt||d}g}td|dD]r}d|} g} t|D] } | || | |q>> from sympy import sequence, sqrt, oo, lucas >>> from sympy.abc import n, x, y >>> sequence(n**2).find_linear_recurrence(10, 2) [] >>> sequence(n**2).find_linear_recurrence(10) [3, -3, 1] >>> sequence(2**n).find_linear_recurrence(10) [2] >>> sequence(23*n**4+91*n**2).find_linear_recurrence(10) [5, -10, 10, -5, 1] >>> sequence(sqrt(5)*(((1 + sqrt(5))/2)**n - (-(1 + sqrt(5))/2)**(-n))/5).find_linear_recurrence(10) [1, 1] >>> sequence(x+y*(-2)**(-n), (n, 0, oo)).find_linear_recurrence(30) [1/2, 1/2] >>> sequence(3*5**n + 12).find_linear_recurrence(20,gfvar=x) ([6, -5], 3*(5 - 21*x)/((x - 1)*(5*x - 1))) >>> sequence(lucas(n)).find_linear_recurrence(15,gfvar=x) ([1, 1], (x - 2)/(x**2 + x - 1)) rsimplifycsg|]}t|qSr$)r)r9trir$r%rcRsz2SeqBase.find_linear_recurrence..NrFrE) sympy.simplifyrjlenminraappendrdetLUsolverr)r*ndgfvarxlxrcoeffsll2mlistkmyr:r;r$rir%find_linear_recurrence/sL "   2&zSeqBase.find_linear_recurrence)NN)#__name__ __module__ __qualname____doc__is_commutative _op_priority staticmethodr&r,propertyr0r'r r3r4r6r7rrCr@rJrLrNrOrVrrXrZr[r]r_r`rbrhrr$r$r$r%rsR          ,     rc@s8eZdZdZeddZeddZddZdd Zd S) EmptySequenceaRepresents an empty sequence. The empty sequence is also available as a singleton as ``S.EmptySequence``. Examples ======== >>> from sympy import EmptySequence, SeqPer >>> from sympy.abc import x >>> EmptySequence EmptySequence >>> SeqPer((1, 2), (x, 0, 10)) + EmptySequence SeqPer((1, 2), (x, 0, 10)) >>> SeqPer((1, 2)) * EmptySequence EmptySequence >>> EmptySequence.coeff_mul(-1) EmptySequence cCtjSrW)r EmptySetr/r$r$r%r'zEmptySequence.intervalcCrrW)r Zeror/r$r$r%r4rzEmptySequence.lengthcCs|S)"See docstring of SeqBase.coeff_mulr$)r*rCr$r$r%rOszEmptySequence.coeff_mulcCstgSrW)iterr/r$r$r%rbszEmptySequence.__iter__N) rrrrrr'r4rOrbr$r$r$r%rzs   r) metaclassc@XeZdZdZeddZeddZeddZedd Zed d Z ed d Z dS)SeqExpraSequence expression class. Various sequences should inherit from this class. Examples ======== >>> from sympy.series.sequences import SeqExpr >>> from sympy.abc import x >>> from sympy import Tuple >>> s = SeqExpr(Tuple(1, 2, 3), Tuple(x, 0, 10)) >>> s.gen (1, 2, 3) >>> s.interval Interval(0, 10) >>> s.length 11 See Also ======== sympy.series.sequences.SeqPer sympy.series.sequences.SeqFormula cC |jdSNrr=r/r$r$r%r0r\z SeqExpr.gencCst|jdd|jddS)NrFrl)rr>r/r$r$r%r'szSeqExpr.intervalcC|jjSrWr'r(r/r$r$r%r rYz SeqExpr.startcCrrWr'r)r/r$r$r%r3rYz SeqExpr.stopcC|j|jdSNrFr3r r/r$r$r%r4zSeqExpr.lengthcCs|jddfS)NrFrr=r/r$r$r%r6rzSeqExpr.variablesN) rrrrrr0r'r r3r4r6r$r$r$r%rs     rc@sReZdZdZdddZeddZeddZd d Zd d Z d dZ ddZ dS)SeqPera Represents 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 NcCs"t|}dd}d\}}}|dur||dtj}}}t|tr;t|dkr-|\}}}nt|dkr;||}|\}}t|ttfrJ|dusJ|durRt dt ||tj ur`|tjur`t dt|||f}t|trutt t |}nt d |t|d |dtjurtjSt|||S) NcSs$|j}t|jdkr|StdS)NrFr})r7rnpopr) periodicalfreer$r$r%_find_xszSeqPer.__new__.._find_xNNNrrlInvalid limits given: %sz/Both the start and end valuecannot be unboundedz6invalid period %s should be something like e.g (1, 2) rF)rr r"rrrnrRrr ValueErrorstrrGtuplerrrrr__new__)clsrlimitsrrvr r3r$r$r%rs0      zSeqPer.__new__cCs t|jSrW)rnr0r/r$r$r%period+r\z SeqPer.periodcC|jSrWr0r/r$r$r%r/rzSeqPer.periodicalcCsF|jtjur|j||j}n||j|j}|j||jd|Sr)r r rGr3rrsubsr6)r*rBidxr$r$r%r@3s zSeqPer._eval_coeffc Cst|trF|j|j}}|j|j}}t||}g}t|D]}|||} |||} || | q||\} } t||jd| | fSdSzSee docstring of SeqBase._addrN rRrrrrrarpr,r6 r*r+per1lper1per2lper2 per_lengthnew_perrvele1ele2r r3r$r$r%rL:     z SeqPer._addc Cst|trF|j|j}}|j|j}}t||}g}t|D]}|||} |||} || | q||\} } t||jd| | fSdSzSee docstring of SeqBase._mulrNrrr$r$r%rNKrz SeqPer._mulcs,tfdd|jD}t||jdS)rcsg|]}|qSr$r$r9rvrCr$r%rc_sz$SeqPer.coeff_mul..rF)rrrr>)r*rCperr$rr%rO\szSeqPer.coeff_mulrW) rrrrrrrrr@rLrNrOr$r$r$r%rs 0(   rc@sNeZdZdZdddZeddZddZd d Zd d Z d dZ ddZ dS) SeqFormulaaf Represents sequence based on a formula. Elements are generated using a formula. Examples ======== >>> from sympy import SeqFormula, oo, Symbol >>> n = Symbol('n') >>> s = SeqFormula(n**2, (n, 0, 5)) >>> s.formula n**2 For value at a particular point >>> s.coeff(3) 9 supports slicing >>> s[:] [0, 1, 4, 9, 16, 25] iterable >>> list(s) [0, 1, 4, 9, 16, 25] sequence starts from negative infinity >>> SeqFormula(n**2, (-oo, 0))[0:6] [0, 1, 4, 9, 16, 25] See Also ======== sympy.series.sequences.SeqPer NcCst|}dd}d\}}}|dur||dtj}}}t|tr;t|dkr-|\}}}nt|dkr;||}|\}}t|ttfrJ|dusJ|durRt dt ||tj ur`|tjur`t dt|||f}t |d |dtj urvtjSt|||S) NcSs2|j}t|dkr |S|stdStd|)NrFr}z specify dummy variables for %s. If the formula contains more than one free symbol, a dummy variable should be supplied explicitly e.g., SeqFormula(m*n**2, (n, 0, 5)))r7rnrrr)formularr$r$r%rs z#SeqFormula.__new__.._find_xrrrrlrz0Both the start and end value cannot be unboundedrF)rr r"rrrnrRrrrrrGrrrrr)rrrrrvr r3r$r$r%rs&     zSeqFormula.__new__cCrrWrr/r$r$r%rrzSeqFormula.formulacCs|jd}|j||Sr)r6rr)r*rBrtr$r$r%r@s zSeqFormula._eval_coeffc Cs`t|tr.|j|jd}}|j|jd}}||||}||\}}t||||fSdSrrRrrr6rr, r*r+form1v1form2v2rr r3r$r$r%rL zSeqFormula._addc Cs`t|tr.|j|jd}}|j|jd}}||||}||\}}t||||fSdSrrrr$r$r%rNrzSeqFormula._mulcCs"t|}|j|}t||jdS)rrF)rrrr>)r*rCrr$r$r%rOs zSeqFormula.coeff_mulcOs$tt|jg|Ri||jdSr)rrrr>)r*r>kwargsr$r$r%rs$zSeqFormula.expandrW) rrrrrrrr@rLrNrOrr$r$r$r%rcs ('   rc@seZdZdZdddZeddZedd Zed d Zed d Z eddZ eddZ eddZ eddZ eddZddZddZdS) RecursiveSeqa A finite degree recursive sequence. Explanation =========== 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))`. yn : applied undefined function Represents the nth term of the sequence as e.g. :code:`y(n)` where :code:`y` is an undefined function and `n` is the sequence index. 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), 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 Nrc s\t|ts td|t|tr|jstd||j|fkr%td|jtd|fd}d}| }|D]3} t | jdkrEtd| jd |||} | r\| j r\| dksctd | | |krk| }q8|swd d t|D}t ||krtd t|}ttd d|D}t|||||} fddt|D| _|| _| S)NzErecurrence sequence must be an applied undefined function, found `{}`z0recurrence variable must be a symbol, found `{}`z)recurrence sequence does not match symbolr})excluderrFz)Recurrence should be in a single variablezDRecurrence should have constant, negative, integer shifts (found {})cSsg|] }td|qS)zc_{})rformat)r9r}r$r$r%rcGz(RecursiveSeq.__new__..z)Number of initial terms must equal degreecss|]}t|VqdSrWrrr$r$r% Osz'RecursiveSeq.__new__..csi|] \}}||qSr$r$)r9r}initr rr$r% Ssz(RecursiveSeq.__new__..)rRrrSrr is_symbolr>rDrfindrnmatch is_constant is_integerrarr rrr enumeratecachedegree) r recurrenceynrsrHr r}rprev_ysprev_yshiftseqr$rr%r#sH     zRecursiveSeq.__new__cCrzEquation defining recurrence.rr=r/r$r$r% _recurrenceX zRecursiveSeq._recurrencecCst|j|jdSr)r rr>r/r$r$r%r]szRecursiveSeq.recurrencecCr)z*Applied function representing the nth termrFr=r/r$r$r%rbrzRecursiveSeq.yncCr)z3Undefined function for the nth term of the sequence)rrDr/r$r$r%rgszRecursiveSeq.ycCr)zSequence index symbolrlr=r/r$r$r%rslrzRecursiveSeq.ncCr)z"The initial values of the sequencerr=r/r$r$r%rHqrzRecursiveSeq.initialcCr)r2r=r/r$r$r%r vrzRecursiveSeq.startcCr)z&The ending point of the sequence. (oo))r r"r/r$r$r%r3{szRecursiveSeq.stopcCs |jtjfS)z&Interval on which sequence is defined.)r r r"r/r$r$r%r'r1zRecursiveSeq.intervalcCs||jt|jkr|j||Stt|j|dD]}|j|}|j|j|i}||j}||j||<q|j||j|Sr)r rnrrrarxreplacers)r*rgcurrent seq_indexcurrent_recurrencenew_termr$r$r%r@s  zRecursiveSeq._eval_coeffccs |j} ||V|d7}q)NTrF)r r@)r*rgr$r$r%rbs  zRecursiveSeq.__iter__r)rrrrrrrrrrrsrHr r3r'r@rbr$r$r$r%rs. L5          rNcCs&t|}t|trt||St||S)a Returns appropriate sequence object. Explanation =========== If ``seq`` is a SymPy sequence, returns :class:`SeqPer` object otherwise returns :class:`SeqFormula` object. Examples ======== >>> from sympy import sequence >>> from sympy.abc import n >>> sequence(n**2, (n, 0, 5)) SeqFormula(n**2, (n, 0, 5)) >>> sequence((1, 2, 3), (n, 0, 5)) SeqPer((1, 2, 3), (n, 0, 5)) See Also ======== sympy.series.sequences.SeqPer sympy.series.sequences.SeqFormula )rrrrr)rrr$r$r%sequences   rc@r) SeqExprOpa Base class for operations on sequences. Examples ======== >>> from sympy.series.sequences import SeqExprOp, sequence >>> from sympy.abc import n >>> s1 = sequence(n**2, (n, 0, 10)) >>> s2 = sequence((1, 2, 3), (n, 5, 10)) >>> s = SeqExprOp(s1, s2) >>> s.gen (n**2, (1, 2, 3)) >>> s.interval Interval(5, 10) >>> s.length 6 See Also ======== sympy.series.sequences.SeqAdd sympy.series.sequences.SeqMul cCstdd|jDS)zjGenerator for the sequence. returns a tuple of generators of all the argument sequences. cs|]}|jVqdSrWrr9ar$r$r%rz SeqExprOp.gen..)rr>r/r$r$r%r0sz SeqExprOp.gencCstdd|jDS)zeSequence is defined on the intersection of all the intervals of respective sequences csrrWr'rr$r$r%rrz%SeqExprOp.interval..)rr>r/r$r$r%r'szSeqExprOp.intervalcCrrWrr/r$r$r%r rYzSeqExprOp.startcCrrWrr/r$r$r%r3rYzSeqExprOp.stopcCsttdd|jDS)z%Cumulative of all the bound variablescSsg|]}|jqSr$)r6rr$r$r%rcsz'SeqExprOp.variables..)rrr>r/r$r$r%r6szSeqExprOp.variablescCrrrr/r$r$r%r4rzSeqExprOp.lengthN) rrrrrr0r'r r3r6r4r$r$r$r%rs     rc@,eZdZdZddZeddZddZdS) rUaRepresents term-wise addition of sequences. Rules: * The interval on which sequence is defined is the intersection of respective intervals of sequences. * Anything + :class:`EmptySequence` remains unchanged. * Other rules are defined in ``_add`` methods of sequence classes. Examples ======== >>> from sympy import EmptySequence, oo, SeqAdd, SeqPer, SeqFormula >>> from sympy.abc import n >>> SeqAdd(SeqPer((1, 2), (n, 0, oo)), EmptySequence) SeqPer((1, 2), (n, 0, oo)) >>> SeqAdd(SeqPer((1, 2), (n, 0, 5)), SeqPer((1, 2), (n, 6, 10))) EmptySequence >>> SeqAdd(SeqPer((1, 2), (n, 0, oo)), SeqFormula(n**2, (n, 0, oo))) SeqAdd(SeqFormula(n**2, (n, 0, oo)), SeqPer((1, 2), (n, 0, oo))) >>> SeqAdd(SeqFormula(n**3), SeqFormula(n**2)) SeqFormula(n**3 + n**2, (n, 0, oo)) See Also ======== sympy.series.sequences.SeqMul cs|dtj}t|}fdd|}dd|D}|s!tjStdd|Dtjur0tjS|r7t |Stt |t j }t j|g|RS)NevaluatecLt|trt|trtt|jgS|gSt|r"tt|gStdNz2Input must be Sequences or iterables of Sequences)rRrrUsummapr>rrSarg_flattenr$r%r   z SeqAdd.__new__.._flattencSsg|] }|tjur|qSr$)r rrr$r$r%rc,rz"SeqAdd.__new__..csrrWrrr$r$r%r2rz!SeqAdd.__new__..)getrrlistr rrrrUreducerrr&rrrr>rrr$rr%rs   zSeqAdd.__new__cd}|r?t|D]4\}d}t|D]#\}||krq}|dur5fdd|D}||nq|r<|}nq|st|dkrI|St|ddS)aSimplify :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`` TFNcg|] }|fvr|qSr$r$rsrkr$r%rcUrdz!SeqAdd.reduce..rFr)rrLrprnrrUr>new_argsid1id2new_seqr$rr%r=s*     z SeqAdd.reducecstfdd|jDS)z9adds up the coefficients of all the sequences at point ptc3s|]}|VqdSrWrrrBr$r%rcsz%SeqAdd._eval_coeff..)rr>rAr$rr%r@aszSeqAdd._eval_coeffNrrrrrrrr@r$r$r$r%rUs $  #rUc@r) r^a'Represents term-wise multiplication of sequences. Explanation =========== 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 EmptySequence, oo, SeqMul, SeqPer, SeqFormula >>> from sympy.abc import n >>> SeqMul(SeqPer((1, 2), (n, 0, oo)), 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 cs|dtj}t|}fdd|}|stjStdd|Dtjur)tjS|r0t |Stt |t j }t j|g|RS)Nrcrr)rRrr^rrr>rrSrrr$r%rrz SeqMul.__new__.._flattencsrrWrrr$r$r%rrz!SeqMul.__new__..)rrrrr rrrr^rrrr&rrrr$rr%rs   zSeqMul.__new__cr)a.Simplify a :class:`SeqMul` using known rules. Explanation =========== Iterates through all pairs and ask the constituent sequences if they can simplify themselves with any other constituent. Notes ===== adapted from ``Union.reduce`` TFNcrr$r$rrr$r%rcrdz!SeqMul.reduce..rFr)rrNrprnrr^rr$rr%rs*    z SeqMul.reducecCs"d}|jD] }|||9}q|S)zrC)r*rBvalrr$r$r%r@s zSeqMul._eval_coeffNr r$r$r$r%r^fs ""  &r^rW)6sympy.core.basicrsympy.core.cachersympy.core.containersrsympy.core.decoratorsrsympy.core.parametersrsympy.core.functionrrsympy.core.mulr sympy.core.numbersr sympy.core.relationalr sympy.core.singletonr rsympy.core.sortingrsympy.core.symbolrrrsympy.core.sympifyrsympy.matricesr sympy.polysrrsympy.sets.setsrrsympy.tensor.indexedrsympy.utilities.iterablesrrrrrrrrrrrrUr^r$r$r$r%s@           b%3s F':j