""" This module implements Holonomic Functions and various operations on them. """ from sympy.core import Add, Mul, Pow from sympy.core.numbers import (NaN, Infinity, NegativeInfinity, Float, I, pi, equal_valued) from sympy.core.singleton import S from sympy.core.sorting import ordered from sympy.core.symbol import Dummy, Symbol from sympy.core.sympify import sympify from sympy.functions.combinatorial.factorials import binomial, factorial, rf from sympy.functions.elementary.exponential import exp_polar, exp, log from sympy.functions.elementary.hyperbolic import (cosh, sinh) from sympy.functions.elementary.miscellaneous import sqrt from sympy.functions.elementary.trigonometric import (cos, sin, sinc) from sympy.functions.special.error_functions import (Ci, Shi, Si, erf, erfc, erfi) from sympy.functions.special.gamma_functions import gamma from sympy.functions.special.hyper import hyper, meijerg from sympy.integrals import meijerint from sympy.matrices import Matrix from sympy.polys.rings import PolyElement from sympy.polys.fields import FracElement from sympy.polys.domains import QQ, RR from sympy.polys.polyclasses import DMF from sympy.polys.polyroots import roots from sympy.polys.polytools import Poly from sympy.polys.matrices import DomainMatrix from sympy.printing import sstr from sympy.series.limits import limit from sympy.series.order import Order from sympy.simplify.hyperexpand import hyperexpand from sympy.simplify.simplify import nsimplify from sympy.solvers.solvers import solve from .recurrence import HolonomicSequence, RecurrenceOperator, RecurrenceOperators from .holonomicerrors import (NotPowerSeriesError, NotHyperSeriesError, SingularityError, NotHolonomicError) def _find_nonzero_solution(r, homosys): ones = lambda shape: DomainMatrix.ones(shape, r.domain) particular, nullspace = r._solve(homosys) nullity = nullspace.shape[0] nullpart = ones((1, nullity)) * nullspace sol = (particular + nullpart).transpose() return sol def DifferentialOperators(base, generator): r""" This function is used to create annihilators using ``Dx``. Explanation =========== Returns an Algebra of Differential Operators also called Weyl Algebra and the operator for differentiation i.e. the ``Dx`` operator. Parameters ========== base: Base polynomial ring for the algebra. The base polynomial ring is the ring of polynomials in :math:`x` that will appear as coefficients in the operators. generator: Generator of the algebra which can be either a noncommutative ``Symbol`` or a string. e.g. "Dx" or "D". Examples ======== >>> from sympy import ZZ >>> from sympy.abc import x >>> from sympy.holonomic.holonomic import DifferentialOperators >>> R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') >>> R Univariate Differential Operator Algebra in intermediate Dx over the base ring ZZ[x] >>> Dx*x (1) + (x)*Dx """ ring = DifferentialOperatorAlgebra(base, generator) return (ring, ring.derivative_operator) class DifferentialOperatorAlgebra: r""" An Ore Algebra is a set of noncommutative polynomials in the intermediate ``Dx`` and coefficients in a base polynomial ring :math:`A`. It follows the commutation rule: .. math :: Dxa = \sigma(a)Dx + \delta(a) for :math:`a \subset A`. Where :math:`\sigma: A \Rightarrow A` is an endomorphism and :math:`\delta: A \rightarrow A` is a skew-derivation i.e. :math:`\delta(ab) = \delta(a) b + \sigma(a) \delta(b)`. If one takes the sigma as identity map and delta as the standard derivation then it becomes the algebra of Differential Operators also called a Weyl Algebra i.e. an algebra whose elements are Differential Operators. This class represents a Weyl Algebra and serves as the parent ring for Differential Operators. Examples ======== >>> from sympy import ZZ >>> from sympy import symbols >>> from sympy.holonomic.holonomic import DifferentialOperators >>> x = symbols('x') >>> R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') >>> R Univariate Differential Operator Algebra in intermediate Dx over the base ring ZZ[x] See Also ======== DifferentialOperator """ def __init__(self, base, generator): # the base polynomial ring for the algebra self.base = base # the operator representing differentiation i.e. `Dx` self.derivative_operator = DifferentialOperator( [base.zero, base.one], self) if generator is None: self.gen_symbol = Symbol('Dx', commutative=False) else: if isinstance(generator, str): self.gen_symbol = Symbol(generator, commutative=False) elif isinstance(generator, Symbol): self.gen_symbol = generator def __str__(self): string = 'Univariate Differential Operator Algebra in intermediate '\ + sstr(self.gen_symbol) + ' over the base ring ' + \ (self.base).__str__() return string __repr__ = __str__ def __eq__(self, other): if self.base == other.base and self.gen_symbol == other.gen_symbol: return True else: return False class DifferentialOperator: """ Differential Operators are elements of Weyl Algebra. The Operators are defined by a list of polynomials in the base ring and the parent ring of the Operator i.e. the algebra it belongs to. Explanation =========== Takes a list of polynomials for each power of ``Dx`` and the parent ring which must be an instance of DifferentialOperatorAlgebra. A Differential Operator can be created easily using the operator ``Dx``. See examples below. Examples ======== >>> from sympy.holonomic.holonomic import DifferentialOperator, DifferentialOperators >>> from sympy import ZZ >>> from sympy import symbols >>> x = symbols('x') >>> R, Dx = DifferentialOperators(ZZ.old_poly_ring(x),'Dx') >>> DifferentialOperator([0, 1, x**2], R) (1)*Dx + (x**2)*Dx**2 >>> (x*Dx*x + 1 - Dx**2)**2 (2*x**2 + 2*x + 1) + (4*x**3 + 2*x**2 - 4)*Dx + (x**4 - 6*x - 2)*Dx**2 + (-2*x**2)*Dx**3 + (1)*Dx**4 See Also ======== DifferentialOperatorAlgebra """ _op_priority = 20 def __init__(self, list_of_poly, parent): """ Parameters ========== list_of_poly: List of polynomials belonging to the base ring of the algebra. parent: Parent algebra of the operator. """ # the parent ring for this operator # must be an DifferentialOperatorAlgebra object self.parent = parent base = self.parent.base self.x = base.gens[0] if isinstance(base.gens[0], Symbol) else base.gens[0][0] # sequence of polynomials in x for each power of Dx # the list should not have trailing zeroes # represents the operator # convert the expressions into ring elements using from_sympy for i, j in enumerate(list_of_poly): if not isinstance(j, base.dtype): list_of_poly[i] = base.from_sympy(sympify(j)) else: list_of_poly[i] = base.from_sympy(base.to_sympy(j)) self.listofpoly = list_of_poly # highest power of `Dx` self.order = len(self.listofpoly) - 1 def __mul__(self, other): """ Multiplies two DifferentialOperator and returns another DifferentialOperator instance using the commutation rule Dx*a = a*Dx + a' """ listofself = self.listofpoly if not isinstance(other, DifferentialOperator): if not isinstance(other, self.parent.base.dtype): listofother = [self.parent.base.from_sympy(sympify(other))] else: listofother = [other] else: listofother = other.listofpoly # multiplies a polynomial `b` with a list of polynomials def _mul_dmp_diffop(b, listofother): if isinstance(listofother, list): sol = [] for i in listofother: sol.append(i * b) return sol else: return [b * listofother] sol = _mul_dmp_diffop(listofself[0], listofother) # compute Dx^i * b def _mul_Dxi_b(b): sol1 = [self.parent.base.zero] sol2 = [] if isinstance(b, list): for i in b: sol1.append(i) sol2.append(i.diff()) else: sol1.append(self.parent.base.from_sympy(b)) sol2.append(self.parent.base.from_sympy(b).diff()) return _add_lists(sol1, sol2) for i in range(1, len(listofself)): # find Dx^i * b in ith iteration listofother = _mul_Dxi_b(listofother) # solution = solution + listofself[i] * (Dx^i * b) sol = _add_lists(sol, _mul_dmp_diffop(listofself[i], listofother)) return DifferentialOperator(sol, self.parent) def __rmul__(self, other): if not isinstance(other, DifferentialOperator): if not isinstance(other, self.parent.base.dtype): other = (self.parent.base).from_sympy(sympify(other)) sol = [] for j in self.listofpoly: sol.append(other * j) return DifferentialOperator(sol, self.parent) def __add__(self, other): if isinstance(other, DifferentialOperator): sol = _add_lists(self.listofpoly, other.listofpoly) return DifferentialOperator(sol, self.parent) else: list_self = self.listofpoly if not isinstance(other, self.parent.base.dtype): list_other = [((self.parent).base).from_sympy(sympify(other))] else: list_other = [other] sol = [] sol.append(list_self[0] + list_other[0]) sol += list_self[1:] return DifferentialOperator(sol, self.parent) __radd__ = __add__ def __sub__(self, other): return self + (-1) * other def __rsub__(self, other): return (-1) * self + other def __neg__(self): return -1 * self def __truediv__(self, other): return self * (S.One / other) def __pow__(self, n): if n == 1: return self if n == 0: return DifferentialOperator([self.parent.base.one], self.parent) # if self is `Dx` if self.listofpoly == self.parent.derivative_operator.listofpoly: sol = [self.parent.base.zero]*n sol.append(self.parent.base.one) return DifferentialOperator(sol, self.parent) # the general case else: if n % 2 == 1: powreduce = self**(n - 1) return powreduce * self elif n % 2 == 0: powreduce = self**(n / 2) return powreduce * powreduce def __str__(self): listofpoly = self.listofpoly print_str = '' for i, j in enumerate(listofpoly): if j == self.parent.base.zero: continue if i == 0: print_str += '(' + sstr(j) + ')' continue if print_str: print_str += ' + ' if i == 1: print_str += '(' + sstr(j) + ')*%s' %(self.parent.gen_symbol) continue print_str += '(' + sstr(j) + ')' + '*%s**' %(self.parent.gen_symbol) + sstr(i) return print_str __repr__ = __str__ def __eq__(self, other): if isinstance(other, DifferentialOperator): if self.listofpoly == other.listofpoly and self.parent == other.parent: return True else: return False else: if self.listofpoly[0] == other: for i in self.listofpoly[1:]: if i is not self.parent.base.zero: return False return True else: return False def is_singular(self, x0): """ Checks if the differential equation is singular at x0. """ base = self.parent.base return x0 in roots(base.to_sympy(self.listofpoly[-1]), self.x) class HolonomicFunction: r""" A Holonomic Function is a solution to a linear homogeneous ordinary differential equation with polynomial coefficients. This differential equation can also be represented by an annihilator i.e. a Differential Operator ``L`` such that :math:`L.f = 0`. For uniqueness of these functions, initial conditions can also be provided along with the annihilator. Explanation =========== Holonomic functions have closure properties and thus forms a ring. Given two Holonomic Functions f and g, their sum, product, integral and derivative is also a Holonomic Function. For ordinary points initial condition should be a vector of values of the derivatives i.e. :math:`[y(x_0), y'(x_0), y''(x_0) ... ]`. For regular singular points initial conditions can also be provided in this format: :math:`{s0: [C_0, C_1, ...], s1: [C^1_0, C^1_1, ...], ...}` where s0, s1, ... are the roots of indicial equation and vectors :math:`[C_0, C_1, ...], [C^0_0, C^0_1, ...], ...` are the corresponding initial terms of the associated power series. See Examples below. Examples ======== >>> from sympy.holonomic.holonomic import HolonomicFunction, DifferentialOperators >>> from sympy import QQ >>> from sympy import symbols, S >>> x = symbols('x') >>> R, Dx = DifferentialOperators(QQ.old_poly_ring(x),'Dx') >>> p = HolonomicFunction(Dx - 1, x, 0, [1]) # e^x >>> q = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]) # sin(x) >>> p + q # annihilator of e^x + sin(x) HolonomicFunction((-1) + (1)*Dx + (-1)*Dx**2 + (1)*Dx**3, x, 0, [1, 2, 1]) >>> p * q # annihilator of e^x * sin(x) HolonomicFunction((2) + (-2)*Dx + (1)*Dx**2, x, 0, [0, 1]) An example of initial conditions for regular singular points, the indicial equation has only one root `1/2`. >>> HolonomicFunction(-S(1)/2 + x*Dx, x, 0, {S(1)/2: [1]}) HolonomicFunction((-1/2) + (x)*Dx, x, 0, {1/2: [1]}) >>> HolonomicFunction(-S(1)/2 + x*Dx, x, 0, {S(1)/2: [1]}).to_expr() sqrt(x) To plot a Holonomic Function, one can use `.evalf()` for numerical computation. Here's an example on `sin(x)**2/x` using numpy and matplotlib. >>> import sympy.holonomic # doctest: +SKIP >>> from sympy import var, sin # doctest: +SKIP >>> import matplotlib.pyplot as plt # doctest: +SKIP >>> import numpy as np # doctest: +SKIP >>> var("x") # doctest: +SKIP >>> r = np.linspace(1, 5, 100) # doctest: +SKIP >>> y = sympy.holonomic.expr_to_holonomic(sin(x)**2/x, x0=1).evalf(r) # doctest: +SKIP >>> plt.plot(r, y, label="holonomic function") # doctest: +SKIP >>> plt.show() # doctest: +SKIP """ _op_priority = 20 def __init__(self, annihilator, x, x0=0, y0=None): """ Parameters ========== annihilator: Annihilator of the Holonomic Function, represented by a `DifferentialOperator` object. x: Variable of the function. x0: The point at which initial conditions are stored. Generally an integer. y0: The initial condition. The proper format for the initial condition is described in class docstring. To make the function unique, length of the vector `y0` should be equal to or greater than the order of differential equation. """ # initial condition self.y0 = y0 # the point for initial conditions, default is zero. self.x0 = x0 # differential operator L such that L.f = 0 self.annihilator = annihilator self.x = x def __str__(self): if self._have_init_cond(): str_sol = 'HolonomicFunction(%s, %s, %s, %s)' % (str(self.annihilator),\ sstr(self.x), sstr(self.x0), sstr(self.y0)) else: str_sol = 'HolonomicFunction(%s, %s)' % (str(self.annihilator),\ sstr(self.x)) return str_sol __repr__ = __str__ def unify(self, other): """ Unifies the base polynomial ring of a given two Holonomic Functions. """ R1 = self.annihilator.parent.base R2 = other.annihilator.parent.base dom1 = R1.dom dom2 = R2.dom if R1 == R2: return (self, other) R = (dom1.unify(dom2)).old_poly_ring(self.x) newparent, _ = DifferentialOperators(R, str(self.annihilator.parent.gen_symbol)) sol1 = [R1.to_sympy(i) for i in self.annihilator.listofpoly] sol2 = [R2.to_sympy(i) for i in other.annihilator.listofpoly] sol1 = DifferentialOperator(sol1, newparent) sol2 = DifferentialOperator(sol2, newparent) sol1 = HolonomicFunction(sol1, self.x, self.x0, self.y0) sol2 = HolonomicFunction(sol2, other.x, other.x0, other.y0) return (sol1, sol2) def is_singularics(self): """ Returns True if the function have singular initial condition in the dictionary format. Returns False if the function have ordinary initial condition in the list format. Returns None for all other cases. """ if isinstance(self.y0, dict): return True elif isinstance(self.y0, list): return False def _have_init_cond(self): """ Checks if the function have initial condition. """ return bool(self.y0) def _singularics_to_ord(self): """ Converts a singular initial condition to ordinary if possible. """ a = list(self.y0)[0] b = self.y0[a] if len(self.y0) == 1 and a == int(a) and a > 0: y0 = [] a = int(a) for i in range(a): y0.append(S.Zero) y0 += [j * factorial(a + i) for i, j in enumerate(b)] return HolonomicFunction(self.annihilator, self.x, self.x0, y0) def __add__(self, other): # if the ground domains are different if self.annihilator.parent.base != other.annihilator.parent.base: a, b = self.unify(other) return a + b deg1 = self.annihilator.order deg2 = other.annihilator.order dim = max(deg1, deg2) R = self.annihilator.parent.base K = R.get_field() rowsself = [self.annihilator] rowsother = [other.annihilator] gen = self.annihilator.parent.derivative_operator # constructing annihilators up to order dim for i in range(dim - deg1): diff1 = (gen * rowsself[-1]) rowsself.append(diff1) for i in range(dim - deg2): diff2 = (gen * rowsother[-1]) rowsother.append(diff2) row = rowsself + rowsother # constructing the matrix of the ansatz r = [] for expr in row: p = [] for i in range(dim + 1): if i >= len(expr.listofpoly): p.append(K.zero) else: p.append(K.new(expr.listofpoly[i].rep)) r.append(p) # solving the linear system using gauss jordan solver r = DomainMatrix(r, (len(row), dim+1), K).transpose() homosys = DomainMatrix.zeros((dim+1, 1), K) sol = _find_nonzero_solution(r, homosys) # if a solution is not obtained then increasing the order by 1 in each # iteration while sol.is_zero_matrix: dim += 1 diff1 = (gen * rowsself[-1]) rowsself.append(diff1) diff2 = (gen * rowsother[-1]) rowsother.append(diff2) row = rowsself + rowsother r = [] for expr in row: p = [] for i in range(dim + 1): if i >= len(expr.listofpoly): p.append(K.zero) else: p.append(K.new(expr.listofpoly[i].rep)) r.append(p) # solving the linear system using gauss jordan solver r = DomainMatrix(r, (len(row), dim+1), K).transpose() homosys = DomainMatrix.zeros((dim+1, 1), K) sol = _find_nonzero_solution(r, homosys) # taking only the coefficients needed to multiply with `self` # can be also be done the other way by taking R.H.S and multiplying with # `other` sol = sol.flat()[:dim + 1 - deg1] sol1 = _normalize(sol, self.annihilator.parent) # annihilator of the solution sol = sol1 * (self.annihilator) sol = _normalize(sol.listofpoly, self.annihilator.parent, negative=False) if not (self._have_init_cond() and other._have_init_cond()): return HolonomicFunction(sol, self.x) # both the functions have ordinary initial conditions if self.is_singularics() == False and other.is_singularics() == False: # directly add the corresponding value if self.x0 == other.x0: # try to extended the initial conditions # using the annihilator y1 = _extend_y0(self, sol.order) y2 = _extend_y0(other, sol.order) y0 = [a + b for a, b in zip(y1, y2)] return HolonomicFunction(sol, self.x, self.x0, y0) else: # change the initial conditions to a same point selfat0 = self.annihilator.is_singular(0) otherat0 = other.annihilator.is_singular(0) if self.x0 == 0 and not selfat0 and not otherat0: return self + other.change_ics(0) elif other.x0 == 0 and not selfat0 and not otherat0: return self.change_ics(0) + other else: selfatx0 = self.annihilator.is_singular(self.x0) otheratx0 = other.annihilator.is_singular(self.x0) if not selfatx0 and not otheratx0: return self + other.change_ics(self.x0) else: return self.change_ics(other.x0) + other if self.x0 != other.x0: return HolonomicFunction(sol, self.x) # if the functions have singular_ics y1 = None y2 = None if self.is_singularics() == False and other.is_singularics() == True: # convert the ordinary initial condition to singular. _y0 = [j / factorial(i) for i, j in enumerate(self.y0)] y1 = {S.Zero: _y0} y2 = other.y0 elif self.is_singularics() == True and other.is_singularics() == False: _y0 = [j / factorial(i) for i, j in enumerate(other.y0)] y1 = self.y0 y2 = {S.Zero: _y0} elif self.is_singularics() == True and other.is_singularics() == True: y1 = self.y0 y2 = other.y0 # computing singular initial condition for the result # taking union of the series terms of both functions y0 = {} for i in y1: # add corresponding initial terms if the power # on `x` is same if i in y2: y0[i] = [a + b for a, b in zip(y1[i], y2[i])] else: y0[i] = y1[i] for i in y2: if i not in y1: y0[i] = y2[i] return HolonomicFunction(sol, self.x, self.x0, y0) def integrate(self, limits, initcond=False): """ Integrates the given holonomic function. Examples ======== >>> from sympy.holonomic.holonomic import HolonomicFunction, DifferentialOperators >>> from sympy import QQ >>> from sympy import symbols >>> x = symbols('x') >>> R, Dx = DifferentialOperators(QQ.old_poly_ring(x),'Dx') >>> HolonomicFunction(Dx - 1, x, 0, [1]).integrate((x, 0, x)) # e^x - 1 HolonomicFunction((-1)*Dx + (1)*Dx**2, x, 0, [0, 1]) >>> HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]).integrate((x, 0, x)) HolonomicFunction((1)*Dx + (1)*Dx**3, x, 0, [0, 1, 0]) """ # to get the annihilator, just multiply by Dx from right D = self.annihilator.parent.derivative_operator # if the function have initial conditions of the series format if self.is_singularics() == True: r = self._singularics_to_ord() if r: return r.integrate(limits, initcond=initcond) # computing singular initial condition for the function # produced after integration. y0 = {} for i in self.y0: c = self.y0[i] c2 = [] for j, cj in enumerate(c): if cj == 0: c2.append(S.Zero) # if power on `x` is -1, the integration becomes log(x) # TODO: Implement this case elif i + j + 1 == 0: raise NotImplementedError("logarithmic terms in the series are not supported") else: c2.append(cj / S(i + j + 1)) y0[i + 1] = c2 if hasattr(limits, "__iter__"): raise NotImplementedError("Definite integration for singular initial conditions") return HolonomicFunction(self.annihilator * D, self.x, self.x0, y0) # if no initial conditions are available for the function if not self._have_init_cond(): if initcond: return HolonomicFunction(self.annihilator * D, self.x, self.x0, [S.Zero]) return HolonomicFunction(self.annihilator * D, self.x) # definite integral # initial conditions for the answer will be stored at point `a`, # where `a` is the lower limit of the integrand if hasattr(limits, "__iter__"): if len(limits) == 3 and limits[0] == self.x: x0 = self.x0 a = limits[1] b = limits[2] definite = True else: definite = False y0 = [S.Zero] y0 += self.y0 indefinite_integral = HolonomicFunction(self.annihilator * D, self.x, self.x0, y0) if not definite: return indefinite_integral # use evalf to get the values at `a` if x0 != a: try: indefinite_expr = indefinite_integral.to_expr() except (NotHyperSeriesError, NotPowerSeriesError): indefinite_expr = None if indefinite_expr: lower = indefinite_expr.subs(self.x, a) if isinstance(lower, NaN): lower = indefinite_expr.limit(self.x, a) else: lower = indefinite_integral.evalf(a) if b == self.x: y0[0] = y0[0] - lower return HolonomicFunction(self.annihilator * D, self.x, x0, y0) elif S(b).is_Number: if indefinite_expr: upper = indefinite_expr.subs(self.x, b) if isinstance(upper, NaN): upper = indefinite_expr.limit(self.x, b) else: upper = indefinite_integral.evalf(b) return upper - lower # if the upper limit is `x`, the answer will be a function if b == self.x: return HolonomicFunction(self.annihilator * D, self.x, a, y0) # if the upper limits is a Number, a numerical value will be returned elif S(b).is_Number: try: s = HolonomicFunction(self.annihilator * D, self.x, a,\ y0).to_expr() indefinite = s.subs(self.x, b) if not isinstance(indefinite, NaN): return indefinite else: return s.limit(self.x, b) except (NotHyperSeriesError, NotPowerSeriesError): return HolonomicFunction(self.annihilator * D, self.x, a, y0).evalf(b) return HolonomicFunction(self.annihilator * D, self.x) def diff(self, *args, **kwargs): r""" Differentiation of the given Holonomic function. Examples ======== >>> from sympy.holonomic.holonomic import HolonomicFunction, DifferentialOperators >>> from sympy import ZZ >>> from sympy import symbols >>> x = symbols('x') >>> R, Dx = DifferentialOperators(ZZ.old_poly_ring(x),'Dx') >>> HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]).diff().to_expr() cos(x) >>> HolonomicFunction(Dx - 2, x, 0, [1]).diff().to_expr() 2*exp(2*x) See Also ======== integrate """ kwargs.setdefault('evaluate', True) if args: if args[0] != self.x: return S.Zero elif len(args) == 2: sol = self for i in range(args[1]): sol = sol.diff(args[0]) return sol ann = self.annihilator # if the function is constant. if ann.listofpoly[0] == ann.parent.base.zero and ann.order == 1: return S.Zero # if the coefficient of y in the differential equation is zero. # a shifting is done to compute the answer in this case. elif ann.listofpoly[0] == ann.parent.base.zero: sol = DifferentialOperator(ann.listofpoly[1:], ann.parent) if self._have_init_cond(): # if ordinary initial condition if self.is_singularics() == False: return HolonomicFunction(sol, self.x, self.x0, self.y0[1:]) # TODO: support for singular initial condition return HolonomicFunction(sol, self.x) else: return HolonomicFunction(sol, self.x) # the general algorithm R = ann.parent.base K = R.get_field() seq_dmf = [K.new(i.rep) for i in ann.listofpoly] # -y = a1*y'/a0 + a2*y''/a0 ... + an*y^n/a0 rhs = [i / seq_dmf[0] for i in seq_dmf[1:]] rhs.insert(0, K.zero) # differentiate both lhs and rhs sol = _derivate_diff_eq(rhs) # add the term y' in lhs to rhs sol = _add_lists(sol, [K.zero, K.one]) sol = _normalize(sol[1:], self.annihilator.parent, negative=False) if not self._have_init_cond() or self.is_singularics() == True: return HolonomicFunction(sol, self.x) y0 = _extend_y0(self, sol.order + 1)[1:] return HolonomicFunction(sol, self.x, self.x0, y0) def __eq__(self, other): if self.annihilator == other.annihilator: if self.x == other.x: if self._have_init_cond() and other._have_init_cond(): if self.x0 == other.x0 and self.y0 == other.y0: return True else: return False else: return True else: return False else: return False def __mul__(self, other): ann_self = self.annihilator if not isinstance(other, HolonomicFunction): other = sympify(other) if other.has(self.x): raise NotImplementedError(" Can't multiply a HolonomicFunction and expressions/functions.") if not self._have_init_cond(): return self else: y0 = _extend_y0(self, ann_self.order) y1 = [] for j in y0: y1.append((Poly.new(j, self.x) * other).rep) return HolonomicFunction(ann_self, self.x, self.x0, y1) if self.annihilator.parent.base != other.annihilator.parent.base: a, b = self.unify(other) return a * b ann_other = other.annihilator list_self = [] list_other = [] a = ann_self.order b = ann_other.order R = ann_self.parent.base K = R.get_field() for j in ann_self.listofpoly: list_self.append(K.new(j.rep)) for j in ann_other.listofpoly: list_other.append(K.new(j.rep)) # will be used to reduce the degree self_red = [-list_self[i] / list_self[a] for i in range(a)] other_red = [-list_other[i] / list_other[b] for i in range(b)] # coeff_mull[i][j] is the coefficient of Dx^i(f).Dx^j(g) coeff_mul = [[K.zero for i in range(b + 1)] for j in range(a + 1)] coeff_mul[0][0] = K.one # making the ansatz lin_sys_elements = [[coeff_mul[i][j] for i in range(a) for j in range(b)]] lin_sys = DomainMatrix(lin_sys_elements, (1, a*b), K).transpose() homo_sys = DomainMatrix.zeros((a*b, 1), K) sol = _find_nonzero_solution(lin_sys, homo_sys) # until a non trivial solution is found while sol.is_zero_matrix: # updating the coefficients Dx^i(f).Dx^j(g) for next degree for i in range(a - 1, -1, -1): for j in range(b - 1, -1, -1): coeff_mul[i][j + 1] += coeff_mul[i][j] coeff_mul[i + 1][j] += coeff_mul[i][j] if isinstance(coeff_mul[i][j], K.dtype): coeff_mul[i][j] = DMFdiff(coeff_mul[i][j]) else: coeff_mul[i][j] = coeff_mul[i][j].diff(self.x) # reduce the terms to lower power using annihilators of f, g for i in range(a + 1): if not coeff_mul[i][b].is_zero: for j in range(b): coeff_mul[i][j] += other_red[j] * \ coeff_mul[i][b] coeff_mul[i][b] = K.zero # not d2 + 1, as that is already covered in previous loop for j in range(b): if not coeff_mul[a][j] == 0: for i in range(a): coeff_mul[i][j] += self_red[i] * \ coeff_mul[a][j] coeff_mul[a][j] = K.zero lin_sys_elements.append([coeff_mul[i][j] for i in range(a) for j in range(b)]) lin_sys = DomainMatrix(lin_sys_elements, (len(lin_sys_elements), a*b), K).transpose() sol = _find_nonzero_solution(lin_sys, homo_sys) sol_ann = _normalize(sol.flat(), self.annihilator.parent, negative=False) if not (self._have_init_cond() and other._have_init_cond()): return HolonomicFunction(sol_ann, self.x) if self.is_singularics() == False and other.is_singularics() == False: # if both the conditions are at same point if self.x0 == other.x0: # try to find more initial conditions y0_self = _extend_y0(self, sol_ann.order) y0_other = _extend_y0(other, sol_ann.order) # h(x0) = f(x0) * g(x0) y0 = [y0_self[0] * y0_other[0]] # coefficient of Dx^j(f)*Dx^i(g) in Dx^i(fg) for i in range(1, min(len(y0_self), len(y0_other))): coeff = [[0 for i in range(i + 1)] for j in range(i + 1)] for j in range(i + 1): for k in range(i + 1): if j + k == i: coeff[j][k] = binomial(i, j) sol = 0 for j in range(i + 1): for k in range(i + 1): sol += coeff[j][k]* y0_self[j] * y0_other[k] y0.append(sol) return HolonomicFunction(sol_ann, self.x, self.x0, y0) # if the points are different, consider one else: selfat0 = self.annihilator.is_singular(0) otherat0 = other.annihilator.is_singular(0) if self.x0 == 0 and not selfat0 and not otherat0: return self * other.change_ics(0) elif other.x0 == 0 and not selfat0 and not otherat0: return self.change_ics(0) * other else: selfatx0 = self.annihilator.is_singular(self.x0) otheratx0 = other.annihilator.is_singular(self.x0) if not selfatx0 and not otheratx0: return self * other.change_ics(self.x0) else: return self.change_ics(other.x0) * other if self.x0 != other.x0: return HolonomicFunction(sol_ann, self.x) # if the functions have singular_ics y1 = None y2 = None if self.is_singularics() == False and other.is_singularics() == True: _y0 = [j / factorial(i) for i, j in enumerate(self.y0)] y1 = {S.Zero: _y0} y2 = other.y0 elif self.is_singularics() == True and other.is_singularics() == False: _y0 = [j / factorial(i) for i, j in enumerate(other.y0)] y1 = self.y0 y2 = {S.Zero: _y0} elif self.is_singularics() == True and other.is_singularics() == True: y1 = self.y0 y2 = other.y0 y0 = {} # multiply every possible pair of the series terms for i in y1: for j in y2: k = min(len(y1[i]), len(y2[j])) c = [] for a in range(k): s = S.Zero for b in range(a + 1): s += y1[i][b] * y2[j][a - b] c.append(s) if not i + j in y0: y0[i + j] = c else: y0[i + j] = [a + b for a, b in zip(c, y0[i + j])] return HolonomicFunction(sol_ann, self.x, self.x0, y0) __rmul__ = __mul__ def __sub__(self, other): return self + other * -1 def __rsub__(self, other): return self * -1 + other def __neg__(self): return -1 * self def __truediv__(self, other): return self * (S.One / other) def __pow__(self, n): if self.annihilator.order <= 1: ann = self.annihilator parent = ann.parent if self.y0 is None: y0 = None else: y0 = [list(self.y0)[0] ** n] p0 = ann.listofpoly[0] p1 = ann.listofpoly[1] p0 = (Poly.new(p0, self.x) * n).rep sol = [parent.base.to_sympy(i) for i in [p0, p1]] dd = DifferentialOperator(sol, parent) return HolonomicFunction(dd, self.x, self.x0, y0) if n < 0: raise NotHolonomicError("Negative Power on a Holonomic Function") if n == 0: Dx = self.annihilator.parent.derivative_operator return HolonomicFunction(Dx, self.x, S.Zero, [S.One]) if n == 1: return self else: if n % 2 == 1: powreduce = self**(n - 1) return powreduce * self elif n % 2 == 0: powreduce = self**(n / 2) return powreduce * powreduce def degree(self): """ Returns the highest power of `x` in the annihilator. """ sol = [i.degree() for i in self.annihilator.listofpoly] return max(sol) def composition(self, expr, *args, **kwargs): """ Returns function after composition of a holonomic function with an algebraic function. The method cannot compute initial conditions for the result by itself, so they can be also be provided. Examples ======== >>> from sympy.holonomic.holonomic import HolonomicFunction, DifferentialOperators >>> from sympy import QQ >>> from sympy import symbols >>> x = symbols('x') >>> R, Dx = DifferentialOperators(QQ.old_poly_ring(x),'Dx') >>> HolonomicFunction(Dx - 1, x).composition(x**2, 0, [1]) # e^(x**2) HolonomicFunction((-2*x) + (1)*Dx, x, 0, [1]) >>> HolonomicFunction(Dx**2 + 1, x).composition(x**2 - 1, 1, [1, 0]) HolonomicFunction((4*x**3) + (-1)*Dx + (x)*Dx**2, x, 1, [1, 0]) See Also ======== from_hyper """ R = self.annihilator.parent a = self.annihilator.order diff = expr.diff(self.x) listofpoly = self.annihilator.listofpoly for i, j in enumerate(listofpoly): if isinstance(j, self.annihilator.parent.base.dtype): listofpoly[i] = self.annihilator.parent.base.to_sympy(j) r = listofpoly[a].subs({self.x:expr}) subs = [-listofpoly[i].subs({self.x:expr}) / r for i in range (a)] coeffs = [S.Zero for i in range(a)] # coeffs[i] == coeff of (D^i f)(a) in D^k (f(a)) coeffs[0] = S.One system = [coeffs] homogeneous = Matrix([[S.Zero for i in range(a)]]).transpose() while True: coeffs_next = [p.diff(self.x) for p in coeffs] for i in range(a - 1): coeffs_next[i + 1] += (coeffs[i] * diff) for i in range(a): coeffs_next[i] += (coeffs[-1] * subs[i] * diff) coeffs = coeffs_next # check for linear relations system.append(coeffs) sol, taus = (Matrix(system).transpose() ).gauss_jordan_solve(homogeneous) if sol.is_zero_matrix is not True: break tau = list(taus)[0] sol = sol.subs(tau, 1) sol = _normalize(sol[0:], R, negative=False) # if initial conditions are given for the resulting function if args: return HolonomicFunction(sol, self.x, args[0], args[1]) return HolonomicFunction(sol, self.x) def to_sequence(self, lb=True): r""" Finds recurrence relation for the coefficients in the series expansion of the function about :math:`x_0`, where :math:`x_0` is the point at which the initial condition is stored. Explanation =========== If the point :math:`x_0` is ordinary, solution of the form :math:`[(R, n_0)]` is returned. Where :math:`R` is the recurrence relation and :math:`n_0` is the smallest ``n`` for which the recurrence holds true. If the point :math:`x_0` is regular singular, a list of solutions in the format :math:`(R, p, n_0)` is returned, i.e. `[(R, p, n_0), ... ]`. Each tuple in this vector represents a recurrence relation :math:`R` associated with a root of the indicial equation ``p``. Conditions of a different format can also be provided in this case, see the docstring of HolonomicFunction class. If it's not possible to numerically compute a initial condition, it is returned as a symbol :math:`C_j`, denoting the coefficient of :math:`(x - x_0)^j` in the power series about :math:`x_0`. Examples ======== >>> from sympy.holonomic.holonomic import HolonomicFunction, DifferentialOperators >>> from sympy import QQ >>> from sympy import symbols, S >>> x = symbols('x') >>> R, Dx = DifferentialOperators(QQ.old_poly_ring(x),'Dx') >>> HolonomicFunction(Dx - 1, x, 0, [1]).to_sequence() [(HolonomicSequence((-1) + (n + 1)Sn, n), u(0) = 1, 0)] >>> HolonomicFunction((1 + x)*Dx**2 + Dx, x, 0, [0, 1]).to_sequence() [(HolonomicSequence((n**2) + (n**2 + n)Sn, n), u(0) = 0, u(1) = 1, u(2) = -1/2, 2)] >>> HolonomicFunction(-S(1)/2 + x*Dx, x, 0, {S(1)/2: [1]}).to_sequence() [(HolonomicSequence((n), n), u(0) = 1, 1/2, 1)] See Also ======== HolonomicFunction.series References ========== .. [1] https://hal.inria.fr/inria-00070025/document .. [2] https://www3.risc.jku.at/publications/download/risc_2244/DIPLFORM.pdf """ if self.x0 != 0: return self.shift_x(self.x0).to_sequence() # check whether a power series exists if the point is singular if self.annihilator.is_singular(self.x0): return self._frobenius(lb=lb) dict1 = {} n = Symbol('n', integer=True) dom = self.annihilator.parent.base.dom R, _ = RecurrenceOperators(dom.old_poly_ring(n), 'Sn') # substituting each term of the form `x^k Dx^j` in the # annihilator, according to the formula below: # x^k Dx^j = Sum(rf(n + 1 - k, j) * a(n + j - k) * x^n, (n, k, oo)) # for explanation see [2]. for i, j in enumerate(self.annihilator.listofpoly): listofdmp = j.all_coeffs() degree = len(listofdmp) - 1 for k in range(degree + 1): coeff = listofdmp[degree - k] if coeff == 0: continue if (i - k, k) in dict1: dict1[(i - k, k)] += (dom.to_sympy(coeff) * rf(n - k + 1, i)) else: dict1[(i - k, k)] = (dom.to_sympy(coeff) * rf(n - k + 1, i)) sol = [] keylist = [i[0] for i in dict1] lower = min(keylist) upper = max(keylist) degree = self.degree() # the recurrence relation holds for all values of # n greater than smallest_n, i.e. n >= smallest_n smallest_n = lower + degree dummys = {} eqs = [] unknowns = [] # an appropriate shift of the recurrence for j in range(lower, upper + 1): if j in keylist: temp = S.Zero for k in dict1.keys(): if k[0] == j: temp += dict1[k].subs(n, n - lower) sol.append(temp) else: sol.append(S.Zero) # the recurrence relation sol = RecurrenceOperator(sol, R) # computing the initial conditions for recurrence order = sol.order all_roots = roots(R.base.to_sympy(sol.listofpoly[-1]), n, filter='Z') all_roots = all_roots.keys() if all_roots: max_root = max(all_roots) + 1 smallest_n = max(max_root, smallest_n) order += smallest_n y0 = _extend_y0(self, order) u0 = [] # u(n) = y^n(0)/factorial(n) for i, j in enumerate(y0): u0.append(j / factorial(i)) # if sufficient conditions can't be computed then # try to use the series method i.e. # equate the coefficients of x^k in the equation formed by # substituting the series in differential equation, to zero. if len(u0) < order: for i in range(degree): eq = S.Zero for j in dict1: if i + j[0] < 0: dummys[i + j[0]] = S.Zero elif i + j[0] < len(u0): dummys[i + j[0]] = u0[i + j[0]] elif not i + j[0] in dummys: dummys[i + j[0]] = Symbol('C_%s' %(i + j[0])) unknowns.append(dummys[i + j[0]]) if j[1] <= i: eq += dict1[j].subs(n, i) * dummys[i + j[0]] eqs.append(eq) # solve the system of equations formed soleqs = solve(eqs, *unknowns) if isinstance(soleqs, dict): for i in range(len(u0), order): if i not in dummys: dummys[i] = Symbol('C_%s' %i) if dummys[i] in soleqs: u0.append(soleqs[dummys[i]]) else: u0.append(dummys[i]) if lb: return [(HolonomicSequence(sol, u0), smallest_n)] return [HolonomicSequence(sol, u0)] for i in range(len(u0), order): if i not in dummys: dummys[i] = Symbol('C_%s' %i) s = False for j in soleqs: if dummys[i] in j: u0.append(j[dummys[i]]) s = True if not s: u0.append(dummys[i]) if lb: return [(HolonomicSequence(sol, u0), smallest_n)] return [HolonomicSequence(sol, u0)] def _frobenius(self, lb=True): # compute the roots of indicial equation indicialroots = self._indicial() reals = [] compl = [] for i in ordered(indicialroots.keys()): if i.is_real: reals.extend([i] * indicialroots[i]) else: a, b = i.as_real_imag() compl.extend([(i, a, b)] * indicialroots[i]) # sort the roots for a fixed ordering of solution compl.sort(key=lambda x : x[1]) compl.sort(key=lambda x : x[2]) reals.sort() # grouping the roots, roots differ by an integer are put in the same group. grp = [] for i in reals: intdiff = False if len(grp) == 0: grp.append([i]) continue for j in grp: if int(j[0] - i) == j[0] - i: j.append(i) intdiff = True break if not intdiff: grp.append([i]) # True if none of the roots differ by an integer i.e. # each element in group have only one member independent = True if all(len(i) == 1 for i in grp) else False allpos = all(i >= 0 for i in reals) allint = all(int(i) == i for i in reals) # if initial conditions are provided # then use them. if self.is_singularics() == True: rootstoconsider = [] for i in ordered(self.y0.keys()): for j in ordered(indicialroots.keys()): if equal_valued(j, i): rootstoconsider.append(i) elif allpos and allint: rootstoconsider = [min(reals)] elif independent: rootstoconsider = [i[0] for i in grp] + [j[0] for j in compl] elif not allint: rootstoconsider = [] for i in reals: if not int(i) == i: rootstoconsider.append(i) elif not allpos: if not self._have_init_cond() or S(self.y0[0]).is_finite == False: rootstoconsider = [min(reals)] else: posroots = [] for i in reals: if i >= 0: posroots.append(i) rootstoconsider = [min(posroots)] n = Symbol('n', integer=True) dom = self.annihilator.parent.base.dom R, _ = RecurrenceOperators(dom.old_poly_ring(n), 'Sn') finalsol = [] char = ord('C') for p in rootstoconsider: dict1 = {} for i, j in enumerate(self.annihilator.listofpoly): listofdmp = j.all_coeffs() degree = len(listofdmp) - 1 for k in range(degree + 1): coeff = listofdmp[degree - k] if coeff == 0: continue if (i - k, k - i) in dict1: dict1[(i - k, k - i)] += (dom.to_sympy(coeff) * rf(n - k + 1 + p, i)) else: dict1[(i - k, k - i)] = (dom.to_sympy(coeff) * rf(n - k + 1 + p, i)) sol = [] keylist = [i[0] for i in dict1] lower = min(keylist) upper = max(keylist) degree = max([i[1] for i in dict1]) degree2 = min([i[1] for i in dict1]) smallest_n = lower + degree dummys = {} eqs = [] unknowns = [] for j in range(lower, upper + 1): if j in keylist: temp = S.Zero for k in dict1.keys(): if k[0] == j: temp += dict1[k].subs(n, n - lower) sol.append(temp) else: sol.append(S.Zero) # the recurrence relation sol = RecurrenceOperator(sol, R) # computing the initial conditions for recurrence order = sol.order all_roots = roots(R.base.to_sympy(sol.listofpoly[-1]), n, filter='Z') all_roots = all_roots.keys() if all_roots: max_root = max(all_roots) + 1 smallest_n = max(max_root, smallest_n) order += smallest_n u0 = [] if self.is_singularics() == True: u0 = self.y0[p] elif self.is_singularics() == False and p >= 0 and int(p) == p and len(rootstoconsider) == 1: y0 = _extend_y0(self, order + int(p)) # u(n) = y^n(0)/factorial(n) if len(y0) > int(p): for i in range(int(p), len(y0)): u0.append(y0[i] / factorial(i)) if len(u0) < order: for i in range(degree2, degree): eq = S.Zero for j in dict1: if i + j[0] < 0: dummys[i + j[0]] = S.Zero elif i + j[0] < len(u0): dummys[i + j[0]] = u0[i + j[0]] elif not i + j[0] in dummys: letter = chr(char) + '_%s' %(i + j[0]) dummys[i + j[0]] = Symbol(letter) unknowns.append(dummys[i + j[0]]) if j[1] <= i: eq += dict1[j].subs(n, i) * dummys[i + j[0]] eqs.append(eq) # solve the system of equations formed soleqs = solve(eqs, *unknowns) if isinstance(soleqs, dict): for i in range(len(u0), order): if i not in dummys: letter = chr(char) + '_%s' %i dummys[i] = Symbol(letter) if dummys[i] in soleqs: u0.append(soleqs[dummys[i]]) else: u0.append(dummys[i]) if lb: finalsol.append((HolonomicSequence(sol, u0), p, smallest_n)) continue else: finalsol.append((HolonomicSequence(sol, u0), p)) continue for i in range(len(u0), order): if i not in dummys: letter = chr(char) + '_%s' %i dummys[i] = Symbol(letter) s = False for j in soleqs: if dummys[i] in j: u0.append(j[dummys[i]]) s = True if not s: u0.append(dummys[i]) if lb: finalsol.append((HolonomicSequence(sol, u0), p, smallest_n)) else: finalsol.append((HolonomicSequence(sol, u0), p)) char += 1 return finalsol def series(self, n=6, coefficient=False, order=True, _recur=None): r""" Finds the power series expansion of given holonomic function about :math:`x_0`. Explanation =========== A list of series might be returned if :math:`x_0` is a regular point with multiple roots of the indicial equation. Examples ======== >>> from sympy.holonomic.holonomic import HolonomicFunction, DifferentialOperators >>> from sympy import QQ >>> from sympy import symbols >>> x = symbols('x') >>> R, Dx = DifferentialOperators(QQ.old_poly_ring(x),'Dx') >>> HolonomicFunction(Dx - 1, x, 0, [1]).series() # e^x 1 + x + x**2/2 + x**3/6 + x**4/24 + x**5/120 + O(x**6) >>> HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]).series(n=8) # sin(x) x - x**3/6 + x**5/120 - x**7/5040 + O(x**8) See Also ======== HolonomicFunction.to_sequence """ if _recur is None: recurrence = self.to_sequence() else: recurrence = _recur if isinstance(recurrence, tuple) and len(recurrence) == 2: recurrence = recurrence[0] constantpower = 0 elif isinstance(recurrence, tuple) and len(recurrence) == 3: constantpower = recurrence[1] recurrence = recurrence[0] elif len(recurrence) == 1 and len(recurrence[0]) == 2: recurrence = recurrence[0][0] constantpower = 0 elif len(recurrence) == 1 and len(recurrence[0]) == 3: constantpower = recurrence[0][1] recurrence = recurrence[0][0] else: sol = [] for i in recurrence: sol.append(self.series(_recur=i)) return sol n = n - int(constantpower) l = len(recurrence.u0) - 1 k = recurrence.recurrence.order x = self.x x0 = self.x0 seq_dmp = recurrence.recurrence.listofpoly R = recurrence.recurrence.parent.base K = R.get_field() seq = [] for i, j in enumerate(seq_dmp): seq.append(K.new(j.rep)) sub = [-seq[i] / seq[k] for i in range(k)] sol = list(recurrence.u0) if l + 1 >= n: pass else: # use the initial conditions to find the next term for i in range(l + 1 - k, n - k): coeff = S.Zero for j in range(k): if i + j >= 0: coeff += DMFsubs(sub[j], i) * sol[i + j] sol.append(coeff) if coefficient: return sol ser = S.Zero for i, j in enumerate(sol): ser += x**(i + constantpower) * j if order: ser += Order(x**(n + int(constantpower)), x) if x0 != 0: return ser.subs(x, x - x0) return ser def _indicial(self): """ Computes roots of the Indicial equation. """ if self.x0 != 0: return self.shift_x(self.x0)._indicial() list_coeff = self.annihilator.listofpoly R = self.annihilator.parent.base x = self.x s = R.zero y = R.one def _pole_degree(poly): root_all = roots(R.to_sympy(poly), x, filter='Z') if 0 in root_all.keys(): return root_all[0] else: return 0 degree = [j.degree() for j in list_coeff] degree = max(degree) inf = 10 * (max(1, degree) + max(1, self.annihilator.order)) deg = lambda q: inf if q.is_zero else _pole_degree(q) b = deg(list_coeff[0]) for j in range(1, len(list_coeff)): b = min(b, deg(list_coeff[j]) - j) for i, j in enumerate(list_coeff): listofdmp = j.all_coeffs() degree = len(listofdmp) - 1 if - i - b <= 0 and degree - i - b >= 0: s = s + listofdmp[degree - i - b] * y y *= x - i return roots(R.to_sympy(s), x) def evalf(self, points, method='RK4', h=0.05, derivatives=False): r""" Finds numerical value of a holonomic function using numerical methods. (RK4 by default). A set of points (real or complex) must be provided which will be the path for the numerical integration. Explanation =========== The path should be given as a list :math:`[x_1, x_2, \dots x_n]`. The numerical values will be computed at each point in this order :math:`x_1 \rightarrow x_2 \rightarrow x_3 \dots \rightarrow x_n`. Returns values of the function at :math:`x_1, x_2, \dots x_n` in a list. Examples ======== >>> from sympy.holonomic.holonomic import HolonomicFunction, DifferentialOperators >>> from sympy import QQ >>> from sympy import symbols >>> x = symbols('x') >>> R, Dx = DifferentialOperators(QQ.old_poly_ring(x),'Dx') A straight line on the real axis from (0 to 1) >>> r = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1] Runge-Kutta 4th order on e^x from 0.1 to 1. Exact solution at 1 is 2.71828182845905 >>> HolonomicFunction(Dx - 1, x, 0, [1]).evalf(r) [1.10517083333333, 1.22140257085069, 1.34985849706254, 1.49182424008069, 1.64872063859684, 1.82211796209193, 2.01375162659678, 2.22553956329232, 2.45960141378007, 2.71827974413517] Euler's method for the same >>> HolonomicFunction(Dx - 1, x, 0, [1]).evalf(r, method='Euler') [1.1, 1.21, 1.331, 1.4641, 1.61051, 1.771561, 1.9487171, 2.14358881, 2.357947691, 2.5937424601] One can also observe that the value obtained using Runge-Kutta 4th order is much more accurate than Euler's method. """ from sympy.holonomic.numerical import _evalf lp = False # if a point `b` is given instead of a mesh if not hasattr(points, "__iter__"): lp = True b = S(points) if self.x0 == b: return _evalf(self, [b], method=method, derivatives=derivatives)[-1] if not b.is_Number: raise NotImplementedError a = self.x0 if a > b: h = -h n = int((b - a) / h) points = [a + h] for i in range(n - 1): points.append(points[-1] + h) for i in roots(self.annihilator.parent.base.to_sympy(self.annihilator.listofpoly[-1]), self.x): if i == self.x0 or i in points: raise SingularityError(self, i) if lp: return _evalf(self, points, method=method, derivatives=derivatives)[-1] return _evalf(self, points, method=method, derivatives=derivatives) def change_x(self, z): """ Changes only the variable of Holonomic Function, for internal purposes. For composition use HolonomicFunction.composition() """ dom = self.annihilator.parent.base.dom R = dom.old_poly_ring(z) parent, _ = DifferentialOperators(R, 'Dx') sol = [] for j in self.annihilator.listofpoly: sol.append(R(j.rep)) sol = DifferentialOperator(sol, parent) return HolonomicFunction(sol, z, self.x0, self.y0) def shift_x(self, a): """ Substitute `x + a` for `x`. """ x = self.x listaftershift = self.annihilator.listofpoly base = self.annihilator.parent.base sol = [base.from_sympy(base.to_sympy(i).subs(x, x + a)) for i in listaftershift] sol = DifferentialOperator(sol, self.annihilator.parent) x0 = self.x0 - a if not self._have_init_cond(): return HolonomicFunction(sol, x) return HolonomicFunction(sol, x, x0, self.y0) def to_hyper(self, as_list=False, _recur=None): r""" Returns a hypergeometric function (or linear combination of them) representing the given holonomic function. Explanation =========== Returns an answer of the form: `a_1 \cdot x^{b_1} \cdot{hyper()} + a_2 \cdot x^{b_2} \cdot{hyper()} \dots` This is very useful as one can now use ``hyperexpand`` to find the symbolic expressions/functions. Examples ======== >>> from sympy.holonomic.holonomic import HolonomicFunction, DifferentialOperators >>> from sympy import ZZ >>> from sympy import symbols >>> x = symbols('x') >>> R, Dx = DifferentialOperators(ZZ.old_poly_ring(x),'Dx') >>> # sin(x) >>> HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]).to_hyper() x*hyper((), (3/2,), -x**2/4) >>> # exp(x) >>> HolonomicFunction(Dx - 1, x, 0, [1]).to_hyper() hyper((), (), x) See Also ======== from_hyper, from_meijerg """ if _recur is None: recurrence = self.to_sequence() else: recurrence = _recur if isinstance(recurrence, tuple) and len(recurrence) == 2: smallest_n = recurrence[1] recurrence = recurrence[0] constantpower = 0 elif isinstance(recurrence, tuple) and len(recurrence) == 3: smallest_n = recurrence[2] constantpower = recurrence[1] recurrence = recurrence[0] elif len(recurrence) == 1 and len(recurrence[0]) == 2: smallest_n = recurrence[0][1] recurrence = recurrence[0][0] constantpower = 0 elif len(recurrence) == 1 and len(recurrence[0]) == 3: smallest_n = recurrence[0][2] constantpower = recurrence[0][1] recurrence = recurrence[0][0] else: sol = self.to_hyper(as_list=as_list, _recur=recurrence[0]) for i in recurrence[1:]: sol += self.to_hyper(as_list=as_list, _recur=i) return sol u0 = recurrence.u0 r = recurrence.recurrence x = self.x x0 = self.x0 # order of the recurrence relation m = r.order # when no recurrence exists, and the power series have finite terms if m == 0: nonzeroterms = roots(r.parent.base.to_sympy(r.listofpoly[0]), recurrence.n, filter='R') sol = S.Zero for j, i in enumerate(nonzeroterms): if i < 0 or int(i) != i: continue i = int(i) if i < len(u0): if isinstance(u0[i], (PolyElement, FracElement)): u0[i] = u0[i].as_expr() sol += u0[i] * x**i else: sol += Symbol('C_%s' %j) * x**i if isinstance(sol, (PolyElement, FracElement)): sol = sol.as_expr() * x**constantpower else: sol = sol * x**constantpower if as_list: if x0 != 0: return [(sol.subs(x, x - x0), )] return [(sol, )] if x0 != 0: return sol.subs(x, x - x0) return sol if smallest_n + m > len(u0): raise NotImplementedError("Can't compute sufficient Initial Conditions") # check if the recurrence represents a hypergeometric series is_hyper = True for i in range(1, len(r.listofpoly)-1): if r.listofpoly[i] != r.parent.base.zero: is_hyper = False break if not is_hyper: raise NotHyperSeriesError(self, self.x0) a = r.listofpoly[0] b = r.listofpoly[-1] # the constant multiple of argument of hypergeometric function if isinstance(a.rep[0], (PolyElement, FracElement)): c = - (S(a.rep[0].as_expr()) * m**(a.degree())) / (S(b.rep[0].as_expr()) * m**(b.degree())) else: c = - (S(a.rep[0]) * m**(a.degree())) / (S(b.rep[0]) * m**(b.degree())) sol = 0 arg1 = roots(r.parent.base.to_sympy(a), recurrence.n) arg2 = roots(r.parent.base.to_sympy(b), recurrence.n) # iterate through the initial conditions to find # the hypergeometric representation of the given # function. # The answer will be a linear combination # of different hypergeometric series which satisfies # the recurrence. if as_list: listofsol = [] for i in range(smallest_n + m): # if the recurrence relation doesn't hold for `n = i`, # then a Hypergeometric representation doesn't exist. # add the algebraic term a * x**i to the solution, # where a is u0[i] if i < smallest_n: if as_list: listofsol.append(((S(u0[i]) * x**(i+constantpower)).subs(x, x-x0), )) else: sol += S(u0[i]) * x**i continue # if the coefficient u0[i] is zero, then the # independent hypergeomtric series starting with # x**i is not a part of the answer. if S(u0[i]) == 0: continue ap = [] bq = [] # substitute m * n + i for n for k in ordered(arg1.keys()): ap.extend([nsimplify((i - k) / m)] * arg1[k]) for k in ordered(arg2.keys()): bq.extend([nsimplify((i - k) / m)] * arg2[k]) # convention of (k + 1) in the denominator if 1 in bq: bq.remove(1) else: ap.append(1) if as_list: listofsol.append(((S(u0[i])*x**(i+constantpower)).subs(x, x-x0), (hyper(ap, bq, c*x**m)).subs(x, x-x0))) else: sol += S(u0[i]) * hyper(ap, bq, c * x**m) * x**i if as_list: return listofsol sol = sol * x**constantpower if x0 != 0: return sol.subs(x, x - x0) return sol def to_expr(self): """ Converts a Holonomic Function back to elementary functions. Examples ======== >>> from sympy.holonomic.holonomic import HolonomicFunction, DifferentialOperators >>> from sympy import ZZ >>> from sympy import symbols, S >>> x = symbols('x') >>> R, Dx = DifferentialOperators(ZZ.old_poly_ring(x),'Dx') >>> HolonomicFunction(x**2*Dx**2 + x*Dx + (x**2 - 1), x, 0, [0, S(1)/2]).to_expr() besselj(1, x) >>> HolonomicFunction((1 + x)*Dx**3 + Dx**2, x, 0, [1, 1, 1]).to_expr() x*log(x + 1) + log(x + 1) + 1 """ return hyperexpand(self.to_hyper()).simplify() def change_ics(self, b, lenics=None): """ Changes the point `x0` to ``b`` for initial conditions. Examples ======== >>> from sympy.holonomic import expr_to_holonomic >>> from sympy import symbols, sin, exp >>> x = symbols('x') >>> expr_to_holonomic(sin(x)).change_ics(1) HolonomicFunction((1) + (1)*Dx**2, x, 1, [sin(1), cos(1)]) >>> expr_to_holonomic(exp(x)).change_ics(2) HolonomicFunction((-1) + (1)*Dx, x, 2, [exp(2)]) """ symbolic = True if lenics is None and len(self.y0) > self.annihilator.order: lenics = len(self.y0) dom = self.annihilator.parent.base.domain try: sol = expr_to_holonomic(self.to_expr(), x=self.x, x0=b, lenics=lenics, domain=dom) except (NotPowerSeriesError, NotHyperSeriesError): symbolic = False if symbolic and sol.x0 == b: return sol y0 = self.evalf(b, derivatives=True) return HolonomicFunction(self.annihilator, self.x, b, y0) def to_meijerg(self): """ Returns a linear combination of Meijer G-functions. Examples ======== >>> from sympy.holonomic import expr_to_holonomic >>> from sympy import sin, cos, hyperexpand, log, symbols >>> x = symbols('x') >>> hyperexpand(expr_to_holonomic(cos(x) + sin(x)).to_meijerg()) sin(x) + cos(x) >>> hyperexpand(expr_to_holonomic(log(x)).to_meijerg()).simplify() log(x) See Also ======== to_hyper """ # convert to hypergeometric first rep = self.to_hyper(as_list=True) sol = S.Zero for i in rep: if len(i) == 1: sol += i[0] elif len(i) == 2: sol += i[0] * _hyper_to_meijerg(i[1]) return sol def from_hyper(func, x0=0, evalf=False): r""" Converts a hypergeometric function to holonomic. ``func`` is the Hypergeometric Function and ``x0`` is the point at which initial conditions are required. Examples ======== >>> from sympy.holonomic.holonomic import from_hyper >>> from sympy import symbols, hyper, S >>> x = symbols('x') >>> from_hyper(hyper([], [S(3)/2], x**2/4)) HolonomicFunction((-x) + (2)*Dx + (x)*Dx**2, x, 1, [sinh(1), -sinh(1) + cosh(1)]) """ a = func.ap b = func.bq z = func.args[2] x = z.atoms(Symbol).pop() R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') # generalized hypergeometric differential equation xDx = x*Dx r1 = 1 for ai in a: # XXX gives sympify error if Mul is used with list of all factors r1 *= xDx + ai xDx_1 = xDx - 1 # r2 = Mul(*([Dx] + [xDx_1 + bi for bi in b])) # XXX gives sympify error r2 = Dx for bi in b: r2 *= xDx_1 + bi sol = r1 - r2 simp = hyperexpand(func) if simp in (Infinity, NegativeInfinity): return HolonomicFunction(sol, x).composition(z) def _find_conditions(simp, x, x0, order, evalf=False): y0 = [] for i in range(order): if evalf: val = simp.subs(x, x0).evalf() else: val = simp.subs(x, x0) # return None if it is Infinite or NaN if val.is_finite is False or isinstance(val, NaN): return None y0.append(val) simp = simp.diff(x) return y0 # if the function is known symbolically if not isinstance(simp, hyper): y0 = _find_conditions(simp, x, x0, sol.order) while not y0: # if values don't exist at 0, then try to find initial # conditions at 1. If it doesn't exist at 1 too then # try 2 and so on. x0 += 1 y0 = _find_conditions(simp, x, x0, sol.order) return HolonomicFunction(sol, x).composition(z, x0, y0) if isinstance(simp, hyper): x0 = 1 # use evalf if the function can't be simplified y0 = _find_conditions(simp, x, x0, sol.order, evalf) while not y0: x0 += 1 y0 = _find_conditions(simp, x, x0, sol.order, evalf) return HolonomicFunction(sol, x).composition(z, x0, y0) return HolonomicFunction(sol, x).composition(z) def from_meijerg(func, x0=0, evalf=False, initcond=True, domain=QQ): """ Converts a Meijer G-function to Holonomic. ``func`` is the G-Function and ``x0`` is the point at which initial conditions are required. Examples ======== >>> from sympy.holonomic.holonomic import from_meijerg >>> from sympy import symbols, meijerg, S >>> x = symbols('x') >>> from_meijerg(meijerg(([], []), ([S(1)/2], [0]), x**2/4)) HolonomicFunction((1) + (1)*Dx**2, x, 0, [0, 1/sqrt(pi)]) """ a = func.ap b = func.bq n = len(func.an) m = len(func.bm) p = len(a) z = func.args[2] x = z.atoms(Symbol).pop() R, Dx = DifferentialOperators(domain.old_poly_ring(x), 'Dx') # compute the differential equation satisfied by the # Meijer G-function. xDx = x*Dx xDx1 = xDx + 1 r1 = x*(-1)**(m + n - p) for ai in a: # XXX gives sympify error if args given in list r1 *= xDx1 - ai # r2 = Mul(*[xDx - bi for bi in b]) # gives sympify error r2 = 1 for bi in b: r2 *= xDx - bi sol = r1 - r2 if not initcond: return HolonomicFunction(sol, x).composition(z) simp = hyperexpand(func) if simp in (Infinity, NegativeInfinity): return HolonomicFunction(sol, x).composition(z) def _find_conditions(simp, x, x0, order, evalf=False): y0 = [] for i in range(order): if evalf: val = simp.subs(x, x0).evalf() else: val = simp.subs(x, x0) if val.is_finite is False or isinstance(val, NaN): return None y0.append(val) simp = simp.diff(x) return y0 # computing initial conditions if not isinstance(simp, meijerg): y0 = _find_conditions(simp, x, x0, sol.order) while not y0: x0 += 1 y0 = _find_conditions(simp, x, x0, sol.order) return HolonomicFunction(sol, x).composition(z, x0, y0) if isinstance(simp, meijerg): x0 = 1 y0 = _find_conditions(simp, x, x0, sol.order, evalf) while not y0: x0 += 1 y0 = _find_conditions(simp, x, x0, sol.order, evalf) return HolonomicFunction(sol, x).composition(z, x0, y0) return HolonomicFunction(sol, x).composition(z) x_1 = Dummy('x_1') _lookup_table = None domain_for_table = None from sympy.integrals.meijerint import _mytype def expr_to_holonomic(func, x=None, x0=0, y0=None, lenics=None, domain=None, initcond=True): """ Converts a function or an expression to a holonomic function. Parameters ========== func: The expression to be converted. x: variable for the function. x0: point at which initial condition must be computed. y0: One can optionally provide initial condition if the method is not able to do it automatically. lenics: Number of terms in the initial condition. By default it is equal to the order of the annihilator. domain: Ground domain for the polynomials in ``x`` appearing as coefficients in the annihilator. initcond: Set it false if you do not want the initial conditions to be computed. Examples ======== >>> from sympy.holonomic.holonomic import expr_to_holonomic >>> from sympy import sin, exp, symbols >>> x = symbols('x') >>> expr_to_holonomic(sin(x)) HolonomicFunction((1) + (1)*Dx**2, x, 0, [0, 1]) >>> expr_to_holonomic(exp(x)) HolonomicFunction((-1) + (1)*Dx, x, 0, [1]) See Also ======== sympy.integrals.meijerint._rewrite1, _convert_poly_rat_alg, _create_table """ func = sympify(func) syms = func.free_symbols if not x: if len(syms) == 1: x= syms.pop() else: raise ValueError("Specify the variable for the function") elif x in syms: syms.remove(x) extra_syms = list(syms) if domain is None: if func.has(Float): domain = RR else: domain = QQ if len(extra_syms) != 0: domain = domain[extra_syms].get_field() # try to convert if the function is polynomial or rational solpoly = _convert_poly_rat_alg(func, x, x0=x0, y0=y0, lenics=lenics, domain=domain, initcond=initcond) if solpoly: return solpoly # create the lookup table global _lookup_table, domain_for_table if not _lookup_table: domain_for_table = domain _lookup_table = {} _create_table(_lookup_table, domain=domain) elif domain != domain_for_table: domain_for_table = domain _lookup_table = {} _create_table(_lookup_table, domain=domain) # use the table directly to convert to Holonomic if func.is_Function: f = func.subs(x, x_1) t = _mytype(f, x_1) if t in _lookup_table: l = _lookup_table[t] sol = l[0][1].change_x(x) else: sol = _convert_meijerint(func, x, initcond=False, domain=domain) if not sol: raise NotImplementedError if y0: sol.y0 = y0 if y0 or not initcond: sol.x0 = x0 return sol if not lenics: lenics = sol.annihilator.order _y0 = _find_conditions(func, x, x0, lenics) while not _y0: x0 += 1 _y0 = _find_conditions(func, x, x0, lenics) return HolonomicFunction(sol.annihilator, x, x0, _y0) if y0 or not initcond: sol = sol.composition(func.args[0]) if y0: sol.y0 = y0 sol.x0 = x0 return sol if not lenics: lenics = sol.annihilator.order _y0 = _find_conditions(func, x, x0, lenics) while not _y0: x0 += 1 _y0 = _find_conditions(func, x, x0, lenics) return sol.composition(func.args[0], x0, _y0) # iterate through the expression recursively args = func.args f = func.func sol = expr_to_holonomic(args[0], x=x, initcond=False, domain=domain) if f is Add: for i in range(1, len(args)): sol += expr_to_holonomic(args[i], x=x, initcond=False, domain=domain) elif f is Mul: for i in range(1, len(args)): sol *= expr_to_holonomic(args[i], x=x, initcond=False, domain=domain) elif f is Pow: sol = sol**args[1] sol.x0 = x0 if not sol: raise NotImplementedError if y0: sol.y0 = y0 if y0 or not initcond: return sol if sol.y0: return sol if not lenics: lenics = sol.annihilator.order if sol.annihilator.is_singular(x0): r = sol._indicial() l = list(r) if len(r) == 1 and r[l[0]] == S.One: r = l[0] g = func / (x - x0)**r singular_ics = _find_conditions(g, x, x0, lenics) singular_ics = [j / factorial(i) for i, j in enumerate(singular_ics)] y0 = {r:singular_ics} return HolonomicFunction(sol.annihilator, x, x0, y0) _y0 = _find_conditions(func, x, x0, lenics) while not _y0: x0 += 1 _y0 = _find_conditions(func, x, x0, lenics) return HolonomicFunction(sol.annihilator, x, x0, _y0) ## Some helper functions ## def _normalize(list_of, parent, negative=True): """ Normalize a given annihilator """ num = [] denom = [] base = parent.base K = base.get_field() lcm_denom = base.from_sympy(S.One) list_of_coeff = [] # convert polynomials to the elements of associated # fraction field for i, j in enumerate(list_of): if isinstance(j, base.dtype): list_of_coeff.append(K.new(j.rep)) elif not isinstance(j, K.dtype): list_of_coeff.append(K.from_sympy(sympify(j))) else: list_of_coeff.append(j) # corresponding numerators of the sequence of polynomials num.append(list_of_coeff[i].numer()) # corresponding denominators denom.append(list_of_coeff[i].denom()) # lcm of denominators in the coefficients for i in denom: lcm_denom = i.lcm(lcm_denom) if negative: lcm_denom = -lcm_denom lcm_denom = K.new(lcm_denom.rep) # multiply the coefficients with lcm for i, j in enumerate(list_of_coeff): list_of_coeff[i] = j * lcm_denom gcd_numer = base((list_of_coeff[-1].numer() / list_of_coeff[-1].denom()).rep) # gcd of numerators in the coefficients for i in num: gcd_numer = i.gcd(gcd_numer) gcd_numer = K.new(gcd_numer.rep) # divide all the coefficients by the gcd for i, j in enumerate(list_of_coeff): frac_ans = j / gcd_numer list_of_coeff[i] = base((frac_ans.numer() / frac_ans.denom()).rep) return DifferentialOperator(list_of_coeff, parent) def _derivate_diff_eq(listofpoly): """ Let a differential equation a0(x)y(x) + a1(x)y'(x) + ... = 0 where a0, a1,... are polynomials or rational functions. The function returns b0, b1, b2... such that the differential equation b0(x)y(x) + b1(x)y'(x) +... = 0 is formed after differentiating the former equation. """ sol = [] a = len(listofpoly) - 1 sol.append(DMFdiff(listofpoly[0])) for i, j in enumerate(listofpoly[1:]): sol.append(DMFdiff(j) + listofpoly[i]) sol.append(listofpoly[a]) return sol def _hyper_to_meijerg(func): """ Converts a `hyper` to meijerg. """ ap = func.ap bq = func.bq ispoly = any(i <= 0 and int(i) == i for i in ap) if ispoly: return hyperexpand(func) z = func.args[2] # parameters of the `meijerg` function. an = (1 - i for i in ap) anp = () bm = (S.Zero, ) bmq = (1 - i for i in bq) k = S.One for i in bq: k = k * gamma(i) for i in ap: k = k / gamma(i) return k * meijerg(an, anp, bm, bmq, -z) def _add_lists(list1, list2): """Takes polynomial sequences of two annihilators a and b and returns the list of polynomials of sum of a and b. """ if len(list1) <= len(list2): sol = [a + b for a, b in zip(list1, list2)] + list2[len(list1):] else: sol = [a + b for a, b in zip(list1, list2)] + list1[len(list2):] return sol def _extend_y0(Holonomic, n): """ Tries to find more initial conditions by substituting the initial value point in the differential equation. """ if Holonomic.annihilator.is_singular(Holonomic.x0) or Holonomic.is_singularics() == True: return Holonomic.y0 annihilator = Holonomic.annihilator a = annihilator.order listofpoly = [] y0 = Holonomic.y0 R = annihilator.parent.base K = R.get_field() for i, j in enumerate(annihilator.listofpoly): if isinstance(j, annihilator.parent.base.dtype): listofpoly.append(K.new(j.rep)) if len(y0) < a or n <= len(y0): return y0 else: list_red = [-listofpoly[i] / listofpoly[a] for i in range(a)] if len(y0) > a: y1 = [y0[i] for i in range(a)] else: y1 = list(y0) for i in range(n - a): sol = 0 for a, b in zip(y1, list_red): r = DMFsubs(b, Holonomic.x0) if not getattr(r, 'is_finite', True): return y0 if isinstance(r, (PolyElement, FracElement)): r = r.as_expr() sol += a * r y1.append(sol) list_red = _derivate_diff_eq(list_red) return y0 + y1[len(y0):] def DMFdiff(frac): # differentiate a DMF object represented as p/q if not isinstance(frac, DMF): return frac.diff() K = frac.ring p = K.numer(frac) q = K.denom(frac) sol_num = - p * q.diff() + q * p.diff() sol_denom = q**2 return K((sol_num.rep, sol_denom.rep)) def DMFsubs(frac, x0, mpm=False): # substitute the point x0 in DMF object of the form p/q if not isinstance(frac, DMF): return frac p = frac.num q = frac.den sol_p = S.Zero sol_q = S.Zero if mpm: from mpmath import mp for i, j in enumerate(reversed(p)): if mpm: j = sympify(j)._to_mpmath(mp.prec) sol_p += j * x0**i for i, j in enumerate(reversed(q)): if mpm: j = sympify(j)._to_mpmath(mp.prec) sol_q += j * x0**i if isinstance(sol_p, (PolyElement, FracElement)): sol_p = sol_p.as_expr() if isinstance(sol_q, (PolyElement, FracElement)): sol_q = sol_q.as_expr() return sol_p / sol_q def _convert_poly_rat_alg(func, x, x0=0, y0=None, lenics=None, domain=QQ, initcond=True): """ Converts polynomials, rationals and algebraic functions to holonomic. """ ispoly = func.is_polynomial() if not ispoly: israt = func.is_rational_function() else: israt = True if not (ispoly or israt): basepoly, ratexp = func.as_base_exp() if basepoly.is_polynomial() and ratexp.is_Number: if isinstance(ratexp, Float): ratexp = nsimplify(ratexp) m, n = ratexp.p, ratexp.q is_alg = True else: is_alg = False else: is_alg = True if not (ispoly or israt or is_alg): return None R = domain.old_poly_ring(x) _, Dx = DifferentialOperators(R, 'Dx') # if the function is constant if not func.has(x): return HolonomicFunction(Dx, x, 0, [func]) if ispoly: # differential equation satisfied by polynomial sol = func * Dx - func.diff(x) sol = _normalize(sol.listofpoly, sol.parent, negative=False) is_singular = sol.is_singular(x0) # try to compute the conditions for singular points if y0 is None and x0 == 0 and is_singular: rep = R.from_sympy(func).rep for i, j in enumerate(reversed(rep)): if j == 0: continue else: coeff = list(reversed(rep))[i:] indicial = i break for i, j in enumerate(coeff): if isinstance(j, (PolyElement, FracElement)): coeff[i] = j.as_expr() y0 = {indicial: S(coeff)} elif israt: p, q = func.as_numer_denom() # differential equation satisfied by rational sol = p * q * Dx + p * q.diff(x) - q * p.diff(x) sol = _normalize(sol.listofpoly, sol.parent, negative=False) elif is_alg: sol = n * (x / m) * Dx - 1 sol = HolonomicFunction(sol, x).composition(basepoly).annihilator is_singular = sol.is_singular(x0) # try to compute the conditions for singular points if y0 is None and x0 == 0 and is_singular and \ (lenics is None or lenics <= 1): rep = R.from_sympy(basepoly).rep for i, j in enumerate(reversed(rep)): if j == 0: continue if isinstance(j, (PolyElement, FracElement)): j = j.as_expr() coeff = S(j)**ratexp indicial = S(i) * ratexp break if isinstance(coeff, (PolyElement, FracElement)): coeff = coeff.as_expr() y0 = {indicial: S([coeff])} if y0 or not initcond: return HolonomicFunction(sol, x, x0, y0) if not lenics: lenics = sol.order if sol.is_singular(x0): r = HolonomicFunction(sol, x, x0)._indicial() l = list(r) if len(r) == 1 and r[l[0]] == S.One: r = l[0] g = func / (x - x0)**r singular_ics = _find_conditions(g, x, x0, lenics) singular_ics = [j / factorial(i) for i, j in enumerate(singular_ics)] y0 = {r:singular_ics} return HolonomicFunction(sol, x, x0, y0) y0 = _find_conditions(func, x, x0, lenics) while not y0: x0 += 1 y0 = _find_conditions(func, x, x0, lenics) return HolonomicFunction(sol, x, x0, y0) def _convert_meijerint(func, x, initcond=True, domain=QQ): args = meijerint._rewrite1(func, x) if args: fac, po, g, _ = args else: return None # lists for sum of meijerg functions fac_list = [fac * i[0] for i in g] t = po.as_base_exp() s = t[1] if t[0] == x else S.Zero po_list = [s + i[1] for i in g] G_list = [i[2] for i in g] # finds meijerg representation of x**s * meijerg(a1 ... ap, b1 ... bq, z) def _shift(func, s): z = func.args[-1] if z.has(I): z = z.subs(exp_polar, exp) d = z.collect(x, evaluate=False) b = list(d)[0] a = d[b] t = b.as_base_exp() b = t[1] if t[0] == x else S.Zero r = s / b an = (i + r for i in func.args[0][0]) ap = (i + r for i in func.args[0][1]) bm = (i + r for i in func.args[1][0]) bq = (i + r for i in func.args[1][1]) return a**-r, meijerg((an, ap), (bm, bq), z) coeff, m = _shift(G_list[0], po_list[0]) sol = fac_list[0] * coeff * from_meijerg(m, initcond=initcond, domain=domain) # add all the meijerg functions after converting to holonomic for i in range(1, len(G_list)): coeff, m = _shift(G_list[i], po_list[i]) sol += fac_list[i] * coeff * from_meijerg(m, initcond=initcond, domain=domain) return sol def _create_table(table, domain=QQ): """ Creates the look-up table. For a similar implementation see meijerint._create_lookup_table. """ def add(formula, annihilator, arg, x0=0, y0=()): """ Adds a formula in the dictionary """ table.setdefault(_mytype(formula, x_1), []).append((formula, HolonomicFunction(annihilator, arg, x0, y0))) R = domain.old_poly_ring(x_1) _, Dx = DifferentialOperators(R, 'Dx') # add some basic functions add(sin(x_1), Dx**2 + 1, x_1, 0, [0, 1]) add(cos(x_1), Dx**2 + 1, x_1, 0, [1, 0]) add(exp(x_1), Dx - 1, x_1, 0, 1) add(log(x_1), Dx + x_1*Dx**2, x_1, 1, [0, 1]) add(erf(x_1), 2*x_1*Dx + Dx**2, x_1, 0, [0, 2/sqrt(pi)]) add(erfc(x_1), 2*x_1*Dx + Dx**2, x_1, 0, [1, -2/sqrt(pi)]) add(erfi(x_1), -2*x_1*Dx + Dx**2, x_1, 0, [0, 2/sqrt(pi)]) add(sinh(x_1), Dx**2 - 1, x_1, 0, [0, 1]) add(cosh(x_1), Dx**2 - 1, x_1, 0, [1, 0]) add(sinc(x_1), x_1 + 2*Dx + x_1*Dx**2, x_1) add(Si(x_1), x_1*Dx + 2*Dx**2 + x_1*Dx**3, x_1) add(Ci(x_1), x_1*Dx + 2*Dx**2 + x_1*Dx**3, x_1) add(Shi(x_1), -x_1*Dx + 2*Dx**2 + x_1*Dx**3, x_1) def _find_conditions(func, x, x0, order): y0 = [] for i in range(order): val = func.subs(x, x0) if isinstance(val, NaN): val = limit(func, x, x0) if val.is_finite is False or isinstance(val, NaN): return None y0.append(val) func = func.diff(x) return y0