From 290aac57b83c8cb6ad3c8de6d3036e3f63cd926b Mon Sep 17 00:00:00 2001 From: Antonio Jimenez Pastor Date: Wed, 5 Dec 2018 16:02:13 +0100 Subject: [PATCH] Added new guessing functionality Added three new methods to toDiffAlgebraic.py - A method that, if y(x) = exp(int(u(x))), computes the differential polynomials U_n(u) such that y^(n)(x) = y(x)U_n(u). (Exponential_polynomials) - A method that, given an homogeneous differentially algebraic equation suppose that the solution is of the form y(x) = exp(int(u(x))) and computes a differentially algebraic equation for u(x) which will have less order than y(x). (simplify_homogeneous) - A method that, given an homogeneous differentially algebraic equation use the method simplify_homogeneous and in case that a linear equation is achieved, then a Dn-finite function is returned as an iteration of exponential functions (guess_homogeenous_DNfinite). --- ajpastor/dd_functions/toDiffAlgebraic.py | 223 +++++++++++++++++++-- releases/diff_defined_functions__0.6.zip | Bin 124876 -> 126299 bytes ...f_defined_functions__0.6__18.12.05_16:01:57.zip | Bin 0 -> 126609 bytes ...f_defined_functions__0.6__18.12.05_16:02:13.zip | Bin 0 -> 126299 bytes 4 files changed, 205 insertions(+), 18 deletions(-) create mode 100644 releases/old/diff_defined_functions__0.6__18.12.05_16:01:57.zip create mode 100644 releases/old/diff_defined_functions__0.6__18.12.05_16:02:13.zip diff --git a/ajpastor/dd_functions/toDiffAlgebraic.py b/ajpastor/dd_functions/toDiffAlgebraic.py index 92cb654..60d4013 100644 --- a/ajpastor/dd_functions/toDiffAlgebraic.py +++ b/ajpastor/dd_functions/toDiffAlgebraic.py @@ -6,13 +6,14 @@ from sage.rings.polynomial.multi_polynomial_ring import is_MPolynomialRing; from sage.rings.fraction_field import is_FractionField; from ajpastor.dd_functions.ddFunction import *; +from ajpastor.dd_functions.ddExamples import *; from sage.rings.polynomial.infinite_polynomial_ring import InfinitePolynomialRing; from ajpastor.misc.matrix import matrix_of_dMovement; ################################################################################ ### -### Second approach: use an infinite polynomial field +### Utilities functions for InfinitePolynomialRings ### ################################################################################ def is_InfinitePolynomialRing(parent): @@ -109,6 +110,11 @@ def infinite_derivative(element, times=1, var=x): monomials = [parent(el) for el in element.monomials()]; return sum([infinite_derivative(coefficients[i])*monomials[i] + coefficients[i]*infinite_derivative(monomials[i]) for i in range(len(monomials))]); +################################################################################ +### +### Changes from and to InfinitePolynomialRings and MultivariatePolynomialRings +### +################################################################################ def fromInfinityPolynomial_toFinitePolynomial(poly): if(not is_InfinitePolynomialRing(poly.parent())): return poly; @@ -133,15 +139,26 @@ def fromFinitePolynomial_toInfinitePolynomial(poly): if(not all(el.find(prefix) == 0 for el in names)): return poly; R = InfinitePolynomialRing(parent.base(), prefix); - return R(str(poly)); + return R(poly); -def toDifferentiallyAlgebraic_Below(poly, _debug=False): +################################################################################ +### +### Methods for computing Diff. Algebraic equations with simpler coefficients +### +################################################################################ +def toDifferentiallyAlgebraic_Below(poly, _infinite=False, _debug=False): ''' Method that receives a polynomial with variables y_0,...,y_m with coefficients in some ring DD(R) and reduce it to a polynomial with coefficients in R. The optional input '_debug' allow the user to print extra information as the matrix that the determinant is computed. ''' + ## Processing the input _infinite + if(_infinite): + r_method = fromFinitePolynomial_toInfinitePolynomial; + else: + r_method = fromInfinityPolynomial_toFinitePolynomial; + ### Preprocessing the input parent = poly.parent(); if(not is_InfinitePolynomialRing(parent)): @@ -171,7 +188,7 @@ def toDifferentiallyAlgebraic_Below(poly, _debug=False): ### Now poly is in a InfinitePolynomialRing with one generator "y_*" and some base ring. if(not isinstance(parent.base(), DDRing)): print "The base ring is not a DDRing. Returning the original polynomial (reached the bottom)"; - return fromInfinityPolynomial_toFinitePolynomial(poly); + return r_method(poly); up_ddring = parent.base() dw_ddring = up_ddring.base(); @@ -223,9 +240,9 @@ def toDifferentiallyAlgebraic_Below(poly, _debug=False): M = Matrix(rows); if(_debug): print M; - return M.determinant().numerator(); + return r_method(M.determinant().numerator()); -def diff_to_diffalg(func, _debug=False): +def diff_to_diffalg(func, _infinite=False, _debug=False): ''' Method that compute an differentially algebraic equation for the obect func. This objet may be any element in QQ(x) or an element in some DDRing. In the first case @@ -237,22 +254,37 @@ def diff_to_diffalg(func, _debug=False): The optional argument _debug allows the user to see during execution more data of the computation. ''' + ## Processing the input _infinite + if(_infinite): + r_method = fromFinitePolynomial_toInfinitePolynomial; + else: + r_method = fromInfinityPolynomial_toFinitePolynomial; + + ## Computations try: parent = func.parent(); except AttributeError: - return PolynomialRing(QQ,"y_0")("y_0 + %s" %func); + return r_method(PolynomialRing(QQ,"y_0")("y_0 + %s" %func)); if(isinstance(parent, DDRing)): R = InfinitePolynomialRing(parent.base(), "y"); p = sum([R("y_%d" %i)*func[i] for i in range(func.getOrder()+1)], R.zero()); for i in range(parent.depth()-1): - p = toDifferentiallyAlgebraic_Below(p, _debug); - return p; + p = toDifferentiallyAlgebraic_Below(p, _infinite=True, _debug=_debug); + return r_method(p); else: R = PolynomialRing(PolynomialRing(QQ,x).fraction_field, "y_0"); - return R.gens()[0] - func; + return r_method(R.gens()[0] - func); -def inverse_DA(poly, vars=None): +################################################################################ +### +### Methods for computing Diff. Algebraic equation for inverses +### +### - Multiplicative inverse from DA equation +### - Functional inverse from Dn-finite equation +### +################################################################################ +def inverse_DA(poly, vars=None, _infinite=False): ''' Method that computes the DA equation for the multiplicative inverse of the solutions of a DA equation. @@ -262,6 +294,12 @@ def inverse_DA(poly, vars=None): If vars is not provided, we take all the variables from the polynomial that is given. ''' + ## Processing the input _infinite + if(_infinite): + r_method = fromFinitePolynomial_toInfinitePolynomial; + else: + r_method = fromInfinityPolynomial_toFinitePolynomial; + ## Checking that poly is a polynomial parent = poly.parent(); @@ -269,7 +307,7 @@ def inverse_DA(poly, vars=None): parent = parent.base(); if(is_InfinitePolynomialRing(parent)): poly = fromInfinityPolynomial_toFinitePolynomial(poly); - return inverse_DA(poly, vars); + return inverse_DA(poly, vars, _infinite=_infinite); if(not (is_PolynomialRing(parent) or is_MPolynomialRing(parent))): raise TypeError("No polynomial is given"); poly = parent(poly); @@ -287,9 +325,19 @@ def inverse_DA(poly, vars=None): ))]; ## Computing the final equation - return poly(**{str(g[i]): derivatives[i] for i in range(len(g))}).numerator(); + return r_method(poly(**{str(g[i]): derivatives[i] for i in range(len(g))}).numerator()); -def func_inverse_DA(func, _debug=False): +def func_inverse_DA(func, _inifnite=False, _debug=False): + ''' + TODO: Add documentation + ''' + ## Processing the input _infinite + if(_infinite): + r_method = fromFinitePolynomial_toInfinitePolynomial; + else: + r_method = fromInfinityPolynomial_toFinitePolynomial; + + ## Computations equation = fromFinitePolynomial_toInfinitePolynomial(diff_to_diffalg(func, _debug=_debug)); if(_debug): print equation; @@ -312,7 +360,20 @@ def func_inverse_DA(func, _debug=False): if(_debug): print sol; if(hasattr(sol, "numerator")): return parent(sol.numerator()); - return parent(sol); + return r_method(parent(sol)); + + +################################################################################ +### +### Methods for compute Dn-finite equations from DA-equations +### +################################################################################ +def guess_Dnfinite(poly, init): + ## TODO: Method not public + res = guess_homogeneous_DNfinite(poly, init); + if(is_DDFunction(res)): + return res; + return guess_DA_DDfinite(poly, init); def guess_DA_DDfinite(poly, init=[], bS=None, bd = None, all=True): ''' @@ -329,6 +390,8 @@ def guess_DA_DDfinite(poly, init=[], bS=None, bd = None, all=True): - db: bound to the order of the DD-finite equation - all: compute all the posibles equations ''' + poly = fromInfinityPolynomial_toFinitePolynomial(poly); + if(not poly.is_homogeneous()): raise TypeError("We require a homogeneous polynomial"); @@ -415,8 +478,117 @@ def guess_DA_DDfinite(poly, init=[], bS=None, bd = None, all=True): return sols; +def guess_homogeneous_DNfinite(poly, init): + ''' + Method that tries to compute a Dn-finite differential equation + for a DA-function defined from an homogeneous equation. + + This method simplifies the equation supposing that the solution + is of the form y(x) = exp(int(u(x))), getting a differential equation + for u(x) of less order. + + If we end up with a linear equation, then we can return a Dn-finite function + where n depends on how many iterations we needed. + + If the final algebraic equation is not linear, then we return this + last equation together with the number of iterations done. + + INPUT: + - poly: polynomial with the differential equation we want to mimic with DD-finite elements + - init: initial values of the solution of poly (must be enough). + ''' + new_poly, depth = simplify_homogeneous(poly); + parent = new_poly.parent(); base = parent.base(); y = parent.gens()[0]; + + order = max(get_InfinitePolynomialRingGen(parent, v, True)[1] for v in new_poly.variables()); + + if(new_poly.degree() == 1 and new_poly.is_homogeneous()): ## We can build something + coeffs = []; + for i in range(order+1): + if(y[i] in new_poly.variables()): + coeffs += new_poly.coefficients()[new_poly.monomials().index(y[i])]; + else: + coeffs += [0]; + + inhom = new_poly.constant_coefficient(); + if(is_DDRing(base)): + base_field = base.base_field; + base = base.to_depth(base.depth()+1); + else: + base_field = base.fraction_field(); + base = DDRing(PolynomialRing(base_field, 'x')); + + ## We can build a D(depth+1)-finite function + deep_init = build_initial_from_homogeneous(tuple(init), order, depth, base_field); + + res = base.element(coeffs, deep_init, inhomogeneous=inhom); + for i in range(depth): + res = Exp(res.integrate(0)); + return res; + + else: + return (new_poly, depth); + + +@cached_function +def build_initial_from_homogeneous(init, n, depth, base): + ## Checking we have enough data + if(len(init) < n+depth): + raise TypeError("Not enought initial data was provided (required %d)" %(n+depth)); + elif(len(init) > n+depth): + return build_initial_from_homogeneous(tuple(list(init)[:n+depth]), n, depth, base); + + ## Base case with depth = 0 + if(depth == 0): + return init[:n]; + + ## Casting the input to the desired ring + init = [base(el) for el in init]; + + ## Checking the initial condition init[0] + if(init[0] == 0): + raise ValueError("Initial condition is zero. Impossible to compute a solution of the form e(int(u(x)))"); + + ## Computing the initial conditions for depth "depth-1" + new_init = [init[1]/init[0]]; + + parent = Exponential_polynomials(1,base).parent(); y = parent.gens()[0]; + + for i in range(1,n+depth-1): + poly = fromInfinityPolynomial_toFinitePolynomial(-(Exponential_polynomials(i+1, parent) - y[i])); + new_init += [base(poly(**{str(y[j]) : new_init[j] for j in range(i)})) + init[i+1]/init[0]]; + + ## Returning the recurive call with one less depth + return build_initial_from_homogeneous(tuple(new_init), n, depth-1, base); + +@cached_function +def simplify_homogeneous(poly): + ''' + Given an homogeneous differential polynomial 'poly', we reduce the order of the equation by 1 + using the change of variables y(x) = exp(int(u(x))). + + This leads to a differentially algebraic equation of order 1 less than 'poly'. If we obtain a + new homogeneous equation, we can iterate. The return of this function is this final result toether + with the number of steps performed. + ''' + poly = fromFinitePolynomial_toInfinitePolynomial(poly); + parent = poly.parent(); y = parent.gens()[0]; + + if(not(poly.is_homogeneous()) or (poly.degree() == 1)): + return (poly,0); + + d = {str(y[0]) : parent.one()}; + for v in poly.variables(): + if(v != y[0]): + d[str(v)] = Exponential_polynomials(get_InfinitePolynomialRingGen(parent, v, True)[1], parent); + + new_poly = poly(**d); + result,n = simplify_homogeneous(new_poly); + + return (result, n+1); + ################################################################################################### -### FAA DI BRUNO functions +### Polynomial functions ################################################################################################### @cached_function def FaaDiBruno_polynomials(n, parent): @@ -441,8 +613,23 @@ def FaaDiBruno_polynomials(n, parent): prev = [FaaDiBruno_polynomials(k, parent) for k in range(n)]; ele = -sum(prev[k][0]*bell_polynomial(n,k)(**{"x%d" %i : y[i+1] for i in range(n-k+1)})/prev[k][1] for k in range(1,n))/(y[1]**n); return (ele.numerator(), ele.denominator()); - S + +@cached_function +def Exponential_polynomials(n, parent): + if(n <= 0): + raise ValueError("No Exponential polynomial can be computed for non-positive index"); + elif(not(is_InfinitePolynomialRing(parent))): + return Exponential_polynomials(n, InfinitePolynomialRing(parent, "y")); + + y = parent.gens()[0]; + + if(n == 1): + return y[0]; + else: + prev = Exponential_polynomials(n-1,parent); + return infinite_derivative(prev) + y[0]*prev; + #################################################################################################### #### PACKAGE ENVIRONMENT VARIABLES #################################################################################################### -__all__ = ["is_InfinitePolynomialRing", "get_InfinitePolynomialRingGen", "get_InfinitePolynomialRingVaribale", "infinite_derivative", 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