366 lines
11 KiB
Python
366 lines
11 KiB
Python
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"""Recurrence Operators"""
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from sympy.core.singleton import S
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from sympy.core.symbol import (Symbol, symbols)
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from sympy.printing import sstr
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from sympy.core.sympify import sympify
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def RecurrenceOperators(base, generator):
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"""
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Returns an Algebra of Recurrence Operators and the operator for
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shifting i.e. the `Sn` operator.
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The first argument needs to be the base polynomial ring for the algebra
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and the second argument must be a generator which can be either a
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noncommutative Symbol or a string.
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Examples
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========
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>>> from sympy import ZZ
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>>> from sympy import symbols
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>>> from sympy.holonomic.recurrence import RecurrenceOperators
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>>> n = symbols('n', integer=True)
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>>> R, Sn = RecurrenceOperators(ZZ.old_poly_ring(n), 'Sn')
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"""
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ring = RecurrenceOperatorAlgebra(base, generator)
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return (ring, ring.shift_operator)
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class RecurrenceOperatorAlgebra:
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"""
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A Recurrence Operator Algebra is a set of noncommutative polynomials
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in intermediate `Sn` and coefficients in a base ring A. It follows the
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commutation rule:
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Sn * a(n) = a(n + 1) * Sn
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This class represents a Recurrence Operator Algebra and serves as the parent ring
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for Recurrence Operators.
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Examples
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========
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>>> from sympy import ZZ
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>>> from sympy import symbols
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>>> from sympy.holonomic.recurrence import RecurrenceOperators
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>>> n = symbols('n', integer=True)
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>>> R, Sn = RecurrenceOperators(ZZ.old_poly_ring(n), 'Sn')
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>>> R
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Univariate Recurrence Operator Algebra in intermediate Sn over the base ring
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ZZ[n]
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See Also
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========
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RecurrenceOperator
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"""
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def __init__(self, base, generator):
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# the base ring for the algebra
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self.base = base
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# the operator representing shift i.e. `Sn`
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self.shift_operator = RecurrenceOperator(
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[base.zero, base.one], self)
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if generator is None:
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self.gen_symbol = symbols('Sn', commutative=False)
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else:
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if isinstance(generator, str):
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self.gen_symbol = symbols(generator, commutative=False)
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elif isinstance(generator, Symbol):
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self.gen_symbol = generator
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def __str__(self):
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string = 'Univariate Recurrence Operator Algebra in intermediate '\
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+ sstr(self.gen_symbol) + ' over the base ring ' + \
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(self.base).__str__()
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return string
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__repr__ = __str__
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def __eq__(self, other):
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if self.base == other.base and self.gen_symbol == other.gen_symbol:
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return True
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else:
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return False
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def _add_lists(list1, list2):
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if len(list1) <= len(list2):
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sol = [a + b for a, b in zip(list1, list2)] + list2[len(list1):]
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else:
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sol = [a + b for a, b in zip(list1, list2)] + list1[len(list2):]
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return sol
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class RecurrenceOperator:
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"""
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The Recurrence Operators are defined by a list of polynomials
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in the base ring and the parent ring of the Operator.
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Explanation
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===========
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Takes a list of polynomials for each power of Sn and the
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parent ring which must be an instance of RecurrenceOperatorAlgebra.
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A Recurrence Operator can be created easily using
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the operator `Sn`. See examples below.
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Examples
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========
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>>> from sympy.holonomic.recurrence import RecurrenceOperator, RecurrenceOperators
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>>> from sympy import ZZ
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>>> from sympy import symbols
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>>> n = symbols('n', integer=True)
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>>> R, Sn = RecurrenceOperators(ZZ.old_poly_ring(n),'Sn')
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>>> RecurrenceOperator([0, 1, n**2], R)
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(1)Sn + (n**2)Sn**2
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>>> Sn*n
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(n + 1)Sn
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>>> n*Sn*n + 1 - Sn**2*n
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(1) + (n**2 + n)Sn + (-n - 2)Sn**2
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See Also
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========
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DifferentialOperatorAlgebra
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"""
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_op_priority = 20
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def __init__(self, list_of_poly, parent):
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# the parent ring for this operator
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# must be an RecurrenceOperatorAlgebra object
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self.parent = parent
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# sequence of polynomials in n for each power of Sn
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# represents the operator
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# convert the expressions into ring elements using from_sympy
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if isinstance(list_of_poly, list):
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for i, j in enumerate(list_of_poly):
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if isinstance(j, int):
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list_of_poly[i] = self.parent.base.from_sympy(S(j))
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elif not isinstance(j, self.parent.base.dtype):
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list_of_poly[i] = self.parent.base.from_sympy(j)
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self.listofpoly = list_of_poly
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self.order = len(self.listofpoly) - 1
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def __mul__(self, other):
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"""
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Multiplies two Operators and returns another
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RecurrenceOperator instance using the commutation rule
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Sn * a(n) = a(n + 1) * Sn
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"""
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listofself = self.listofpoly
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base = self.parent.base
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if not isinstance(other, RecurrenceOperator):
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if not isinstance(other, self.parent.base.dtype):
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listofother = [self.parent.base.from_sympy(sympify(other))]
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else:
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listofother = [other]
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else:
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listofother = other.listofpoly
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# multiply a polynomial `b` with a list of polynomials
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def _mul_dmp_diffop(b, listofother):
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if isinstance(listofother, list):
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sol = []
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for i in listofother:
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sol.append(i * b)
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return sol
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else:
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return [b * listofother]
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sol = _mul_dmp_diffop(listofself[0], listofother)
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# compute Sn^i * b
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def _mul_Sni_b(b):
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sol = [base.zero]
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if isinstance(b, list):
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for i in b:
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j = base.to_sympy(i).subs(base.gens[0], base.gens[0] + S.One)
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sol.append(base.from_sympy(j))
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else:
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j = b.subs(base.gens[0], base.gens[0] + S.One)
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sol.append(base.from_sympy(j))
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return sol
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for i in range(1, len(listofself)):
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# find Sn^i * b in ith iteration
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listofother = _mul_Sni_b(listofother)
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# solution = solution + listofself[i] * (Sn^i * b)
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sol = _add_lists(sol, _mul_dmp_diffop(listofself[i], listofother))
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return RecurrenceOperator(sol, self.parent)
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def __rmul__(self, other):
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if not isinstance(other, RecurrenceOperator):
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if isinstance(other, int):
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other = S(other)
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if not isinstance(other, self.parent.base.dtype):
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other = (self.parent.base).from_sympy(other)
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sol = []
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for j in self.listofpoly:
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sol.append(other * j)
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return RecurrenceOperator(sol, self.parent)
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def __add__(self, other):
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if isinstance(other, RecurrenceOperator):
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sol = _add_lists(self.listofpoly, other.listofpoly)
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return RecurrenceOperator(sol, self.parent)
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else:
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if isinstance(other, int):
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other = S(other)
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list_self = self.listofpoly
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if not isinstance(other, self.parent.base.dtype):
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list_other = [((self.parent).base).from_sympy(other)]
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else:
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list_other = [other]
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sol = []
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sol.append(list_self[0] + list_other[0])
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sol += list_self[1:]
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return RecurrenceOperator(sol, self.parent)
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__radd__ = __add__
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def __sub__(self, other):
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return self + (-1) * other
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def __rsub__(self, other):
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return (-1) * self + other
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def __pow__(self, n):
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if n == 1:
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return self
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if n == 0:
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return RecurrenceOperator([self.parent.base.one], self.parent)
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# if self is `Sn`
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if self.listofpoly == self.parent.shift_operator.listofpoly:
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sol = []
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for i in range(0, n):
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sol.append(self.parent.base.zero)
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sol.append(self.parent.base.one)
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return RecurrenceOperator(sol, self.parent)
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else:
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if n % 2 == 1:
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powreduce = self**(n - 1)
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return powreduce * self
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elif n % 2 == 0:
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powreduce = self**(n / 2)
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return powreduce * powreduce
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def __str__(self):
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listofpoly = self.listofpoly
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print_str = ''
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for i, j in enumerate(listofpoly):
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if j == self.parent.base.zero:
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continue
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if i == 0:
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print_str += '(' + sstr(j) + ')'
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continue
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if print_str:
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print_str += ' + '
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if i == 1:
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print_str += '(' + sstr(j) + ')Sn'
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continue
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print_str += '(' + sstr(j) + ')' + 'Sn**' + sstr(i)
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return print_str
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__repr__ = __str__
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def __eq__(self, other):
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if isinstance(other, RecurrenceOperator):
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if self.listofpoly == other.listofpoly and self.parent == other.parent:
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return True
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else:
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return False
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else:
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if self.listofpoly[0] == other:
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for i in self.listofpoly[1:]:
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if i is not self.parent.base.zero:
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return False
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return True
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else:
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return False
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class HolonomicSequence:
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"""
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A Holonomic Sequence is a type of sequence satisfying a linear homogeneous
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recurrence relation with Polynomial coefficients. Alternatively, A sequence
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is Holonomic if and only if its generating function is a Holonomic Function.
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"""
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def __init__(self, recurrence, u0=[]):
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self.recurrence = recurrence
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if not isinstance(u0, list):
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self.u0 = [u0]
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else:
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self.u0 = u0
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if len(self.u0) == 0:
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self._have_init_cond = False
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else:
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self._have_init_cond = True
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self.n = recurrence.parent.base.gens[0]
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def __repr__(self):
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str_sol = 'HolonomicSequence(%s, %s)' % ((self.recurrence).__repr__(), sstr(self.n))
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if not self._have_init_cond:
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return str_sol
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else:
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cond_str = ''
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seq_str = 0
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for i in self.u0:
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cond_str += ', u(%s) = %s' % (sstr(seq_str), sstr(i))
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seq_str += 1
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sol = str_sol + cond_str
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return sol
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__str__ = __repr__
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def __eq__(self, other):
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if self.recurrence == other.recurrence:
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if self.n == other.n:
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if self._have_init_cond and other._have_init_cond:
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if self.u0 == other.u0:
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return True
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else:
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return False
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else:
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return True
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else:
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return False
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else:
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return False
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