From 7de181917cc1740fb6c4687a71eb38c9d6d644c8 Mon Sep 17 00:00:00 2001 From: Antonio Jimenez Pastor Date: Tue, 12 Mar 2019 12:09:06 +0100 Subject: [PATCH] Fixed some errors in the files ddExamples and ddFunction Added the Example for Riccati differential equations Added a new zip file with the latest version of the code. This file will be linked from any source for download. --- Makefile | 1 + ajpastor/dd_functions/ddExamples.py | 66 ++++++- ajpastor/dd_functions/ddFunction.py | 198 +++++++++++++++++++-- releases/diff_defined_functions.zip | Bin 0 -> 142412 bytes releases/diff_defined_functions__0.6.zip | Bin 140013 -> 142412 bytes ...f_defined_functions__0.6__19.03.12_12:09:06.zip | Bin 0 -> 142412 bytes 6 files changed, 249 insertions(+), 16 deletions(-) create mode 100644 releases/diff_defined_functions.zip create mode 100644 releases/old/diff_defined_functions__0.6__19.03.12_12:09:06.zip diff --git a/Makefile b/Makefile index 99ef63f..7f61fe8 100644 --- a/Makefile +++ b/Makefile @@ -18,6 +18,7 @@ zip: clean_pyc @rm -f ./releases/$(ZIP)__$(VERSION).zip @zip -r ./releases/$(ZIP)__$(VERSION).zip $(BASE) type SPKG.txt setup.py package-version.txt Makefile dependencies @cp ./releases/$(ZIP)__$(VERSION).zip ./releases/old/$(ZIP)__$(VERSION)__`date +'%y.%m.%d_%H:%M:%S'`.zip + @cp ./releases/$(ZIP)__$(VERSION).zip ./releases/$(ZIP).zip clean_pyc: @echo "Cleaning the Python precompiled files (.pyc)" diff --git a/ajpastor/dd_functions/ddExamples.py b/ajpastor/dd_functions/ddExamples.py index 2af2424..eca1dfb 100644 --- a/ajpastor/dd_functions/ddExamples.py +++ b/ajpastor/dd_functions/ddExamples.py @@ -52,6 +52,8 @@ def ddExamples(functions = False, names=False): ** HYPERGEOMETRIC FUNCTIONS (see chapters 15, 16 in https://dlmf.nist.gov) - HypergeometricFunction - GenericHypergeometricFunction + ** RICCATI EQUATION (see https://en.wikipedia.org/wiki/Riccati_equation) + - RiccatiD ** MATHIEU TYPE FUNCTIONS (see chapter 28 in https://dlmf.nist.gov) - MathieuD - MathieuSin @@ -1159,6 +1161,66 @@ def GenericHypergeometricFunction(num=[],den=[],init=1): ## Return the cached element return __CACHED_HYPERGEOMETRIC[(numerator,denominator,initial)]; +###### RICCATI DIFFERENTIAL EQUATION +### Basic Riccati differential equation +@cached_function +def RiccatiD(a,b,c,init=None, ddR = None, full = False, name="w"): + ''' + Implementation using DDFunctions of the solutions for the Ricatti differential equation. + + References: + - https://en.wikipedia.org/wiki/Riccati_equation + + The Riccati differential equation is a non-linear differential equation of order one of the shape + u' = a + bu + cu^2 + + In particular, this represent all the monomials of degree 2 or less. It can be shown that if + a and b are Dn-finite and c is D(n-1)-finite, then u(x) will be the logarithmic derivative + of a D(n+1)-finite function (w), and hence, it is D(n+2)-finite. More precisely: + w'' - (b + c'/c) w' + acw = 0 + + Given the initial condition u(0) is enough to determined all the coefficients of the solution. + + INPUT: + - a: function to represent the constant term in the quadratic polynomial that is the derivative of u(x) + - b: function to represent the linear term in the quadratic polynomial that is the derivative of u(x) + - c: function to represent the leading term in the quadratic polynomial that is the dericative of u(x) + - init: initial condition u(0) of the solution. None is also valid + - ddR: basic DDRing where to put all the inputs a,b,c if they are not DDFunctions + - full: if True, it returns also the function w used to build the solution in a tuple (solution, w) + - name: name the system will give to the function w. By default this is "w" + ''' + ## Considering the three parameters + from sage.categories.pushout import pushout; + + if(is_DDFunction(a)): + da, dRa = (a.parent().depth(), a.parent()); + else: + a, dRa = __decide_parent(a, ddR); da = dRa.depth()-1; + if(is_DDFunction(b)): + db, dRb = (b.parent().depth(), b.parent()); + else: + b, dRb = __decide_parent(b, ddR); db = dRb.depth()-1; + if(is_DDFunction(c)): + dc, dRc = (c.parent().depth(), c.parent()); + else: + c, dRc = __decide_parent(c, ddR); dc = dRc.depth()-1; + + R = pushout(dRa, pushout(dRb, dRc)); + R = R.to_depth(max(da+1, db+1, dc+2)); + + x = R.variables()[0]; + + w = R.base().element([a*c, -b + c.derivative()/c,1], [1, -c(0)*init],name=DinamicString("_1(_2)", [name,repr(x)])); + solution = -w.derivative()*(c*w).inverse; + solution._DDFunction__name = DinamicString("Riccati(_1,_2,_3;_4)(_5)", [repr(a),repr(b),repr(c),str(init),repr(x)]); + + if(full): + return solution,w; + return solution; + + + ###### MATHIEU TYPE FUNCTIONS ### Mathieu's Functions @cached_function @@ -1967,7 +2029,9 @@ def __decide_parent(input, parent = None, depth = 1): dR = parent; if(dR is None): R = input.parent(); - if(isinstance(R, sage.symbolic.ring.SymbolicRing)): + if(input in QQ): + dR = DFinite.to_depth(depth); + elif(isinstance(R, sage.symbolic.ring.SymbolicRing)): parameters = set([str(el) for el in input.variables()])-set(['x']); if(len(parameters) > 0 ): dR = ParametrizedDDRing(DDRing(DFinite, depth=depth), parameters); diff --git a/ajpastor/dd_functions/ddFunction.py b/ajpastor/dd_functions/ddFunction.py index 0b0a66d..3e10024 100644 --- a/ajpastor/dd_functions/ddFunction.py +++ b/ajpastor/dd_functions/ddFunction.py @@ -1,9 +1,6 @@ from sage.all_cmdline import * # import sage library -_sage_const_1en10 = RealNumber('1e-10'); -_sage_const_0 = Integer(0); - # Python imports import warnings; @@ -60,7 +57,7 @@ def _aux_derivation(p): R = p.parent(); return derivative(p,R(x)); except Exception: - return _sage_const_0 ; + return 0 ; ##################################################### ### Class for DD-Rings @@ -71,14 +68,14 @@ def is_DDRing(ring): class DDRing (Ring_w_Sequence, IntegralDomain): Element = None; - _Default_Base = PolynomialRing(QQ,x); + #_Default_Base = PolynomialRing(QQ,x); _Default_Depth = 1 ; _Default_Base_Field = None; _Default_Invertibility = None; _Default_Derivation = None; _Default_Operator = None; _Default_Parameters = [ - ("base", _Default_Base), + ("base", None), ("depth", _Default_Depth), ("base_field", _Default_Base_Field), ("invertibility", _Default_Invertibility), @@ -193,7 +190,7 @@ class DDRing (Ring_w_Sequence, IntegralDomain): final_args += [[DDRing._Default_Parameters[i][0 ], args[i]]]; current = len(args); - for i in range(current, len(DDRing._Default_Parameters)): + for i in range(max(1,current), len(DDRing._Default_Parameters)): final_args += [[DDRing._Default_Parameters[i][0 ], kwds.get(DDRing._Default_Parameters[i][0 ],DDRing._Default_Parameters[i][1 ])]]; ## Managing the depth over DDRings @@ -217,7 +214,7 @@ class DDRing (Ring_w_Sequence, IntegralDomain): return DDRing._CACHED_DD_RINGS[cls][to_hash]; - def __init__(self, base=_Default_Base, depth = _Default_Depth, base_field = _Default_Base_Field, invertibility = _Default_Invertibility, derivation = _Default_Derivation, default_operator = _Default_Operator): + def __init__(self, base, depth = _Default_Depth, base_field = _Default_Base_Field, invertibility = _Default_Invertibility, derivation = _Default_Derivation, default_operator = _Default_Operator): ''' Class representing the Ring of Differentially Definable Functions with coefficients over a base ring. It can be built using different parameter configurations: - ``base``: is the ring where the coefficients of the differential equation will be. It is set by default as `\mathbb{Q}[x]`. @@ -677,6 +674,35 @@ class DDRing (Ring_w_Sequence, IntegralDomain): raise NotImplementedError("Not implemented evaluation of an element of this ring (%s) with the parameters %s and %s" %(self,repr(rx),input)); + def interlace_sequence(self, *functions): + ''' + Method that computes a function in 'self' which sequence is the interlacing of the sequences + given as input in 'functions'. These functions must be contained in 'self' or an error will be + raised. + + INPUT: + - functions: list of function in 'self' which we want to interlace + + OUTPUT: + - DDFunction 'f' in 'self' such that for all n + f.getSequenceElement(n) == functions[n%len(functions)].getSequenceElement(n//len(functions)) + + ERRORS: + - TypeError is raised if any of the functions can not be casted to 'self' + ''' + if(len(functions) == 1 and type(functions[0]) == list): + functions = functions[0]; + + ## Getting the main variable for the functions + x = self.variables()[-1]; + n = len(functions); + + ## Computing the dilated functions + functions = [self(el)(x=x**n) for el in functions]; + + ## Returning the resulting function + return sum(x**i*functions[i] for i in range(n)); + def get_recurrence(self, *args, **kwds): if(self.__get_recurrence is None): raise NotImplementedError("Recurrence method not implemented for %s" %self); @@ -1389,7 +1415,7 @@ class DDFunction (IntegralDomainElement): ''' ### We check some simplifications: if one of the functions is zero, then we can return directly 0 if ((other.is_null) or (self.is_null)): - return _sage_const_0 ; + return 0 ; if(self.is_one): return other; elif(other.is_one): @@ -1493,6 +1519,21 @@ class DDFunction (IntegralDomainElement): result.change_built("polynomial", (PolynomialRing(self.parent().base_field,[repr(X),'x1']).fraction_field()('x1/(%s^%d)' %(repr(X),n)), {repr(X):self.parent()(X),'x1':self})); return (n,result); + def min_coefficient(self): + ''' + Method that computes the first non-zero coefficient. IN case 'self' is zero, this method returns 0. + ''' + return self.zero_extraction[1](0); + + def contraction(self, level): + ''' + Method that tries to compute the contraction of a sequence jumping through the elements 0 (mod level). + + This is equivalent to compose self(pow(x,1/level)) if 'self' only has non-zero element in the positions 0 (mod level). + ''' + ## Computing the resulting differential equation + raise NotImplementedError("Method not yet implemented: contraction"); + def divide(self, other): if(self.is_null): return self.parent().zero(); @@ -1539,6 +1580,67 @@ class DDFunction (IntegralDomainElement): return self.parent()(X**min(se[0 ],oe[0 ])*gcd(se[1 ].getInitialValue(0 ),oe[1 ].getInitialValue(0 ))); ##################################### + ### Sequence methods + ##################################### + def split_sequence(self, parts=2): + ''' + Method that returns a list of 'parts' elements such that the interlacing of their sequence + is equal to the sequence of 'self'. + + INPUT: + - parts: positive integer + + OUTPUT: + - A list 'result' of 'parts' elements such that + 'result[0].interlace(result[1:]) == self' + + ERRORS: + - TypeError is raised if parts is not an integer + - ValueError is raised if 'parts' is smaller or equal to zero. + ''' + try: + parts = ZZ(parts); + except: + raise TypeError("Argument 'parts' is not an integer (split_sequence)"); + + if(parts <= 0): + raise ValueError("Argument 'parts' is smaller than 1 (split_sequence)"); + + if(parts == 1): + return self; + elif(parts == 2): + f = self; + f1 = (f + f(-x))/2; + f2 = ((f - f(-x))/2).zero_extraction[1]; + f = [f1,f2]; + else: + raise NotImplementedError("Method 'split_sequence' not implemented for 3 or more parts"); + + R = PolynomialRing(QQ[x], 'y'); y = R.gens()[0]; + p = y^parts - x; + + return [el.compose_algebraic(p, lambda n : el.getSequenceElement(parts*n)*factorial(n)) for el in f]; + + def interlace(self, *others): + ''' + Method that computes a functions which sequence is the interlacing between 'self' + and all the 'others' functions. + + See method 'interlace_sequence' on class DDRing for further information. + ''' + ## Checking the argument so 'others' is a list of elements + if(len(others) == 1 and type(others[0]) == list): + others = [others[0]]; + + ## Computing the common parent for all elements + po = self.parent(); + for el in others: + po = pushout(po, el.parent()); + + ## Calling the method in the DDRing level + return po.interlace_sequence([self]+list(others)); + + ##################################### ### Differential methods ##################################### def derivative(self, *args, **kwds): @@ -1664,7 +1766,6 @@ class DDFunction (IntegralDomainElement): ###################################### destiny_ring = None; ## First, compute the pushout fo the parents - from sage.categories.pushout import pushout; sp = self.parent(); op = other.parent(); push = pushout(sp, op); @@ -1714,6 +1815,73 @@ class DDFunction (IntegralDomainElement): new_name = din_m_replace(self.__name, {"x":repr(other)}, True); return destiny_ring.element(new_equation, new_init, name=new_name); + + def compose_algebraic(self, poly, init): + ''' + Method to compute the composition of 'self' with an algebraic function over some DDRing + + The method first compute the new ring where the composition will belong and then relies on the method 'compose_algebraic_solution' of the Operator class. + + Then, it uses the initial values given by init. + + INPUT: + - 'self': a DDFunction + - 'poly': polynomial in variable 'y' with coefficient in some DDRing or QQ[x] + - 'init': initial values. This can be given as a list or a method that receives integers and return a sequence + + OUTPUT: + - A new DDFunction 'f' representing the composition of 'self' with an algebraic function. + + ERRORS: + - ValueError: if other(0) != 0 + - TypeError: if we can not compute a destination ring + - Any error from Operator.compose_algebraic_solution + + WARNINGS: + - If the depth of the resulting DDFunction is greater than 3, a warning will be shown + ''' + ### Checking the input + ## Checking 'poly' is a polynomial + if(not(is_Polynomial(poly) or is_MPolynomial(poly))): + raise TypeError("'compose_algebraic': the input 'poly' is not a polynomial"); + + ## Checkig that 'y' is a variable + gens = [v for v in poly.parent().gens()]; + if(all(str(v) != 'y')): + raise TypeError("'compose_algebraic': the input poly has no variable 'y'"); + + if(len(gens) == 1): + R = poly.parent().base(); + if(is_DDRing(R)): + raise NotImplementedError("'compose_algebraic': composition with algebraic over DDFunction not implemented"); + else: + F = DFinite.base().fraction_field(); + Q = PolynomialRing(F, 'y'); + poly = Q(poly); + destiny_ring = self.parent(); + elif(len(gens) > 1): + # If multivariate, we only accept 2 variables + if(len(gens) > 2): + raise ValueError("'compose_algerbaic': the input 'poly' can only be bivariate polynomials"); + R = poly.parent().base(); + if(is_DDRing(R)): + raise NotImplementedError("'compose_algebraic': composition with algebraic over DDFunction not implemented"); + else: + y = gens[([str(v) for v in gens]).index('y')]; + gens = [v for v in gens if v != y]; + + F = PolynomialRing(R, gens).fraction_field(); + Q = PolynomialRing(F, 'y'); + poly = Q(poly); + destiny_ring = self.parent(); + + ## At this point, we have a polynomial in 'y' with coefficients either polynomial in 'x' or DDFunctions + # F: field of fractions of the coefficients + # Q = F[y] + # poly in Q + # destiny_ring: final DDRing where the result will belong + + raise NotImplementedError("Method 'compose_algebraic' is not yet implemented"); ##################################### ### Equality methods @@ -1828,7 +1996,7 @@ class DDFunction (IntegralDomainElement): ### If an error occur, then other can not be substracted from self, so they can not be equals return False; - def numerical_solution(self, value, delta=_sage_const_1en10 , max_depth=100 ): + def numerical_solution(self, value, delta=RealNumber('1e-10'), max_depth=100 ): try: ## We try to delegate the computation to the operator (in case of OreOperators) if(self.equation.getCoefficients()[-1 ](0 ) != 0 ): @@ -2177,7 +2345,7 @@ class DDFunction (IntegralDomainElement): ''' Missing method for DDFuntions. ''' - return _sage_const_0 ; + return 0 ; def __str__(self, detail=True): ''' @@ -2659,7 +2827,7 @@ def _get_derivation_matrix(poly, dic, simpl = False): ## TODO Saved function from the console def get_sequences(a,b): M = [[tensor_power(get_vector_prime(n+1),i+1) for i in range(a)] for n in range(b)] - return [[[list(M[i][j]).index(k^(j+1))+1 for k in [1] + [el for el in primes_first_n(i)]] for j in range(5)] for i in range(5)]; + return [[[list(M[i][j]).index(k**(j+1))+1 for k in [1] + [el for el in primes_first_n(i)]] for j in range(5)] for i in range(5)]; __CACHED_INIT_POLY = {}; @@ -2671,7 +2839,7 @@ def _get_initial_poly(poly, dic, m): ## Constant case if(poly.is_constant()): if(m > 0): - return _sage_const_0; + return 0; return poly.parent().base()(poly); ## Non-constant polynomial @@ -2770,7 +2938,7 @@ try: DFinite = DDRing(PolynomialRing(QQ,x), default_operator=w_OreOperator); except ImportError: ## No ore_algebra package found - warnings.warn("Package ore_algebra was not found. It is hightly recommended to get this package built by M. Kauers and M. Mezzaroba (http://marc.mezzarobba.net/code/ore_algebra-analytic/ or http://www.kauers.de/software.html)", NoOreAlgebraWarning); + warnings.warn("Package ore_algebra was not found. It is hightly recommended to get this package by M. Kauers and M. 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