716 lines
23 KiB
Python
716 lines
23 KiB
Python
#!/usr/bin/env python
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# encoding: utf-8
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from .expression import Expression
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from .explicable import Explicable
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from .operator import op
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from .generic import spe_zip, expand_list, isNumber, transpose_fill, flatten_list, isPolynom, isNumerand
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from .render import txt,tex
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from .random_expression import RdExpression
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from itertools import chain
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from functools import wraps
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__all__ = ["Polynom"]
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def power_cache(fun):
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"""Decorator which cache calculated powers of polynoms """
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cache = {}
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@wraps(fun)
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def cached_fun(self, power):
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#print("cache -> ", cache)
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if (tuple(self._coef), power) in cache.keys():
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return cache[(tuple(self._coef), power)]
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else:
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poly_powered = fun(self, power)
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cache[(tuple(self._coef), power)] = poly_powered
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return poly_powered
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return cached_fun
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class Polynom(Explicable):
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"""Docstring for Polynom. """
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@classmethod
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def random(self, coefs_form=[], conditions=[], letter = "x", degree = 0, name = "P"):
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""" Create a random polynom from coefs_form and conditions
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:param coefs_form: list of forms (one by coef) (ascending degree sorted)
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:param conditions: condition on variables
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:param letter: the letter for the polynom
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:param degree: degree of the polynom (can't be used with coefs_form, it will be overwrite) - can't be higher than 26 (number of letters in alphabet)
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/!\ variables need to be in brackets {}
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>>> Polynom.random(["{b}", "{a}"]) # doctest:+ELLIPSIS
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< Polynom ...
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>>> Polynom.random(degree = 2) # doctest:+ELLIPSIS
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< Polynom ...
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>>> Polynom.random(degree = 2, conditions=["{b**2-4*a*c}>0"]) # Polynom deg 2 with positive Delta (ax^2 + bx + c)
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< Polynom ...
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>>> Polynom.random(["{c}", "{b}", "{a}"], conditions=["{b**2-4*a*c}>0"]) # Same as above
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< Polynom ...
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"""
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if (degree > 0 and degree < 26):
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# Générer assez de lettre pour les coefs
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coefs_name = map(chr, range(97, 98+degree))
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coefs_form = ["{" + i + "}" for i in coefs_name][::-1]
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form = str(coefs_form)
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# On créé les valeurs toutes concaténées dans un string
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coefs = RdExpression(form, conditions)()
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# On "parse" ce string pour créer les coefs
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coefs = [eval(i) if type(i)==str else i for i in eval(coefs)]
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# Création du polynom
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return Polynom(coefs = coefs, letter = letter, name = name)
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def __init__(self, coefs = [1], letter = "x", name = "P"):
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"""Initiate the polynom
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:param coef: coefficients of the polynom (ascending degree sorted)
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3 possibles type of coefficent:
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- a : simple "number". [1,2] designate 1 + 2x
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- [a,b,c]: list of coeficient for same degree. [1,[2,3],4] designate 1 + 2x + 3x + 4x^2
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- a: a Expression. [1, Expression("2+3"), 4] designate 1 + (2+3)x + 4x^2
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:param letter: the string describing the unknown
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:param name: Name of the polynom
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>>> P = Polynom([1, 2, 3])
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>>> P.mainOp
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'+'
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>>> P.name
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'P'
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>>> P._letter
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'x'
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>>> Polynom([1]).mainOp
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'*'
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>>> Polynom([0, 0, 3]).mainOp
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'*'
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>>> Polynom([1, 2, 3])._letter
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'x'
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>>> Polynom([1, 2, 3], "y")._letter
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'y'
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>>> Polynom([1, 2, 3], name = "Q").name
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'Q'
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"""
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super(Polynom, self).__init__()
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self.feed_coef(coefs)
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self._letter = letter
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self.name = name
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if self.is_monom():
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self.mainOp = op.mul
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else:
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self.mainOp = op.add
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self._isPolynom = 1
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def __call__(self, value):
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""" Evaluate the polynom in value
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:returns: Expression ready to be simplify
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>>> P = Polynom([1, 2, 3])
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>>> P(2)
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17
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>>> for i in P(2).explain():
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... print(i)
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3 \\times 2^{ 2 } + 2 \\times 2 + 1
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3 \\times 4 + 4 + 1
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12 + 4 + 1
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16 + 1
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17
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>>> Q = P("1+h")
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>>> print(Q)
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3 h^{ 2 } + 8 h + 6
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>>> R = P(Q)
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"""
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if isNumerand(value) or Expression.isExpression(value):
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postfix_exp = [value if i==self._letter else i for i in self.postfix_tokens]
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else:
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postfix_exp = [Expression(value) if i==self._letter else i for i in self.postfix_tokens]
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return Expression(postfix_exp).simplify()
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def feed_coef(self, l_coef):
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"""Feed coef of the polynom. Manage differently whether it's a number or an expression
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:l_coef: list of coef
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"""
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self._coef = []
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for coef in l_coef:
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if type(coef) == list and len(coef)==1:
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self._coef.append(coef[0])
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else:
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self._coef.append(coef)
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@property
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def degree(self):
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"""Getting the degree fo the polynom
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:returns: the degree of the polynom
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>>> Polynom([1, 2, 3]).degree
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2
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>>> Polynom([1]).degree
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0
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"""
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return len(self._coef) - 1
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def is_monom(self):
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"""is the polynom a monom (only one coefficent)
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:returns: 1 if yes 0 otherwise
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>>> Polynom([1, 2, 3]).is_monom()
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0
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>>> Polynom([1]).is_monom()
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1
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"""
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if len([i for i in self._coef if i != 0])==1:
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return 1
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else:
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return 0
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def give_name(self, name):
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self.name = name
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def __str__(self):
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return str(Expression(self.postfix_tokens))
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def __repr__(self):
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return "< Polynom " + str(self._coef) + ">"
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def __txt__(self):
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return txt(self.postfix_tokens)
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def __tex__(self):
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return tex(self.postfix_tokens)
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def coef_postfix(self, a, i):
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"""Return the postfix display of a coeficient
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:param a: value for the coeficient (/!\ as a postfix list)
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:param i: power
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:returns: postfix tokens of coef
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>>> p = Polynom()
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>>> p.coef_postfix([3],2)
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[3, 'x', 2, '^', '*']
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>>> p.coef_postfix([0],1)
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[]
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>>> p.coef_postfix([3],0)
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[3]
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>>> p.coef_postfix([3],1)
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[3, 'x', '*']
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>>> p.coef_postfix([1],1)
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['x']
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>>> p.coef_postfix([1],2)
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['x', 2, '^']
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"""
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# TODO: Couille certaine avec txt à qui il fait donner des opérateurs tout beau! |mar. nov. 11 13:08:35 CET 2014
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ans =[]
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if a == [0]:
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pass
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elif i == 0:
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ans = a
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elif i == 1:
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ans = a * (a!=[1]) + [self._letter] + [op.mul] * (a!=[1])
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else:
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ans = a * (a!=[1]) + [self._letter, i, op.pw] + [op.mul] * (a!=[1])
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return ans
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@property
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def postfix_tokens(self):
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"""Return the postfix form of the polynom
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:returns: the postfix list of polynom's tokens
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>>> p = Polynom([1, 2])
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>>> p.postfix_tokens
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[2, 'x', '*', 1, '+']
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>>> p = Polynom([1, -2])
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>>> p.postfix_tokens
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[2, 'x', '*', '-', 1, '+']
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>>> p = Polynom([1,2,3])
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>>> p.postfix_tokens
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[3, 'x', 2, '^', '*', 2, 'x', '*', '+', 1, '+']
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>>> p = Polynom([1])
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>>> p.postfix_tokens
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[1]
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>>> p = Polynom([0])
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>>> p.postfix_tokens
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[0]
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>>> p = Polynom([1,[2,3]])
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>>> p.postfix_tokens
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[2, 'x', '*', 3, 'x', '*', '+', 1, '+']
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>>> p = Polynom([1,[2,-3]])
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>>> p.postfix_tokens
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[2, 'x', '*', 3, 'x', '*', '-', 1, '+']
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>>> p = Polynom([1,[-2,-3]])
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>>> p.postfix_tokens
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[2, 'x', '*', '-', 3, 'x', '*', '-', 1, '+']
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>>> from pymath.expression import Expression
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>>> from pymath.operator import op
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>>> e = Expression([2,3,op.add])
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>>> p = Polynom([1,e])
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>>> p.postfix_tokens
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[2, 3, '+', 'x', '*', 1, '+']
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"""
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if self == 0:
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return [0]
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# TODO: Faudrait factoriser un peu tout ça..! |dim. déc. 21 16:02:34 CET 2014
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postfix = []
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for (i,a) in list(enumerate(self._coef))[::-1]:
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operator = [op.add]
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operator_sub1 = []
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if type(a) == Expression:
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# case coef is an arithmetic expression
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c = self.coef_postfix(a.postfix_tokens,i)
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if c != []:
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postfix.append(c)
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if len(postfix) > 1:
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postfix += operator
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elif type(a) == list:
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# case need to repeat the x^i
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for b in a:
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operator = [op.add]
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operator_sub1 = []
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if len(postfix) == 0 and isNumber(b) and b < 0:
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try:
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b = [(-b)[-1]]
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except TypeError:
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b = [-b]
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operator_sub1 = [op.sub1]
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elif len(postfix) > 0 and isNumber(b) and b < 0:
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try:
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b = [(-b)[-1]]
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except TypeError:
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b = [-b]
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operator = [op.sub]
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else:
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b = [b]
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c = self.coef_postfix(b,i)
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if c != []:
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postfix.append(c)
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if len(postfix) > 1:
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postfix += operator_sub1
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postfix += operator
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postfix += operator_sub1
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elif a != 0:
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if len(postfix) == 0 and a < 0:
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try:
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a = [(-a)[-1]]
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except TypeError:
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a = [-a]
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operator_sub1 = [op.sub1]
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elif len(postfix) > 0 and a < 0:
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try:
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a = [(-a)[-1]]
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except TypeError:
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a = [-a]
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operator = [op.sub]
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else:
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a = [a]
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c = self.coef_postfix(a,i)
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if c != []:
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postfix.append(c)
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if len(postfix) > 1:
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postfix += operator_sub1
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postfix += operator
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postfix += operator_sub1
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return flatten_list(postfix)
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def conv2poly(self, other):
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"""Convert anything number into a polynom
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>>> P = Polynom([1,2,3])
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>>> P.conv2poly(1)
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< Polynom [1]>
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>>> P.conv2poly(0)
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< Polynom [0]>
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"""
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if isNumber(other) and not isPolynom(other):
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return Polynom([other], letter = self._letter)
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elif isPolynom(other):
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return other
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else:
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raise ValueError(type(other) + " can't be converted into a polynom")
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def reduce(self):
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"""Compute coefficients which have same degree
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:returns: new Polynom with numbers coefficients
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>>> P = Polynom([1,2,3])
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>>> Q = P.reduce()
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>>> Q
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< Polynom [1, 2, 3]>
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>>> Q.steps
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[]
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>>> P = Polynom([[1,2], [3,4,5], 6])
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>>> Q = P.reduce()
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>>> Q
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< Polynom [3, 12, 6]>
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>>> Q.steps
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[< Polynom [< <class 'pymath.expression.Expression'> [1, 2, '+'] >, < <class 'pymath.expression.Expression'> [3, 4, '+', 5, '+'] >, 6]>, < Polynom [3, < <class 'pymath.expression.Expression'> [7, 5, '+'] >, 6]>]
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"""
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# TODO: It doesn't not compute quick enough |ven. févr. 27 18:04:01 CET 2015
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# gather steps for every coeficients
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coefs_steps = []
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for coef in self._coef:
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coef_steps = []
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if type(coef) == list:
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# On converti en postfix avec une addition
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postfix_add = self.postfix_add([i for i in coef if i!=0])
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# On converti en Expression
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coef_exp = Expression(postfix_add)
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with Expression.tmp_render():
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coef_steps = list(coef_exp.simplify().explain())
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#print('\t 1.coef_steps -> ', coef_steps)
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elif type(coef) == Expression:
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with Expression.tmp_render():
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coef_steps = list(coef.simplify().explain())
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#print('\t 2.coef_steps -> ', coef_steps)
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else:
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try:
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coef_steps += coef.simplify().explaine()
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except AttributeError:
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coef_steps = [coef]
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#print('\t 3.coef_steps -> ', coef_steps)
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# On ajoute toutes ces étapes
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coefs_steps.append(coef_steps)
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#print('\t coefs_steps -> ', coefs_steps)
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# On retourne la matrice
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steps = []
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for coefs in transpose_fill(coefs_steps):
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steps.append(Polynom(coefs, self._letter))
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ans, steps = steps[-1], steps[:-1]
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ans.steps = steps
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return ans
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def derivate(self):
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""" Return the derivated polynom
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>>> P = Polynom([1, 2, 3])
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>>> Q = P.derivate()
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>>> Q
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< Polynom [2, 6]>
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>>> print(Q.name)
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P'
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>>> for i in Q.explain():
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... print(i)
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2 \\times 3 x + 1 \\times 2
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6 x + 2
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"""
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derv_coefs = []
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for (i,c) in enumerate(self._coef):
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derv_coefs += [Expression([i, c, op.mul])]
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ans = Polynom(derv_coefs[1:]).simplify()
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ans.name = self.name + "'"
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return ans
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@staticmethod
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def postfix_add(numbers):
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"""Convert a list of numbers into a postfix addition
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:numbers: list of numbers
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:returns: Postfix list of succecive attition of number
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>>> Polynom.postfix_add([1])
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[1]
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>>> Polynom.postfix_add([1, 2])
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[1, 2, '+']
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>>> Polynom.postfix_add([1, 2, 3])
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[1, 2, '+', 3, '+']
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>>> Polynom.postfix_add(1)
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[1]
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"""
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if not type(numbers) == list:
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return [numbers]
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else:
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ans = [[a, op.add] if i!=0 else [a] for (i,a) in enumerate(numbers)]
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return list(chain.from_iterable(ans))
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def simplify(self):
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"""Same as reduce """
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return self.reduce()
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def __eq__(self, other):
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try:
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o_poly = self.conv2poly(other)
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return self._coef == o_poly._coef
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except TypeError:
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return 0
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def __add__(self, other):
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""" Overload +
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>>> P = Polynom([1,2,3])
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>>> Q = Polynom([4,5])
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>>> R = P+Q
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>>> R
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< Polynom [5, 7, 3]>
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>>> R.steps
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[< <class 'pymath.expression.Expression'> [3, 'x', 2, '^', '*', 2, 'x', '*', '+', 1, '+', 5, 'x', '*', 4, '+', '+'] >, < Polynom [< <class 'pymath.expression.Expression'> [1, 4, '+'] >, < <class 'pymath.expression.Expression'> [2, 5, '+'] >, 3]>]
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"""
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o_poly = self.conv2poly(other)
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n_coef = spe_zip(self._coef, o_poly._coef)
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p = Polynom(n_coef, letter = self._letter)
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ini_step = [Expression(self.postfix_tokens + o_poly.postfix_tokens + [op.add])]
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ans = p.simplify()
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ans.steps = ini_step + ans.steps
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return ans
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def __radd__(self, other):
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o_poly = self.conv2poly(other)
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return o_poly.__add__(self)
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def __neg__(self):
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""" overload - (as arity 1 operator)
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>>> P = Polynom([1,2,3])
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>>> Q = -P
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>>> Q
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< Polynom [-1, -2, -3]>
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>>> Q.steps
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[< <class 'pymath.expression.Expression'> [3, 'x', 2, '^', '*', 2, 'x', '*', '+', 1, '+', '-'] >]
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"""
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ini_step = [Expression(self.postfix_tokens + [op.sub1])]
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ans = Polynom([-i for i in self._coef], letter = self._letter).simplify()
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ans.steps = ini_step + ans.steps
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return ans
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def __sub__(self, other):
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""" overload -
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>>> P = Polynom([1,2,3])
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>>> Q = Polynom([4,5,6])
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>>> R = P - Q
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>>> R
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< Polynom [-3, -3, -3]>
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>>> R.steps
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|
[< <class 'pymath.expression.Expression'> [3, 'x', 2, '^', '*', 2, 'x', '*', '+', 1, '+', 6, 'x', 2, '^', '*', 5, 'x', '*', '+', 4, '+', '-'] >, < <class 'pymath.expression.Expression'> [3, 'x', 2, '^', '*', 2, 'x', '*', '+', 1, '+', 6, 'x', 2, '^', '*', '-', 5, 'x', '*', '-', 4, '-', '+'] >, < Polynom [< <class 'pymath.expression.Expression'> [1, -4, '+'] >, < <class 'pymath.expression.Expression'> [2, -5, '+'] >, < <class 'pymath.expression.Expression'> [3, -6, '+'] >]>]
|
|
"""
|
|
o_poly = self.conv2poly(other)
|
|
ini_step = [Expression(self.postfix_tokens + o_poly.postfix_tokens + [op.sub])]
|
|
o_poly = -o_poly
|
|
#ini_step += o_poly.steps
|
|
|
|
ans = self + o_poly
|
|
ans.steps = ini_step + ans.steps
|
|
|
|
return ans
|
|
|
|
def __rsub__(self, other):
|
|
o_poly = self.conv2poly(other)
|
|
|
|
return o_poly.__sub__(self)
|
|
|
|
def __mul__(self, other):
|
|
""" Overload *
|
|
|
|
>>> p = Polynom([1,2])
|
|
>>> p*3
|
|
< Polynom [3, 6]>
|
|
>>> (p*3).steps
|
|
[[< <class 'pymath.expression.Expression'> [2, 'x', '*', 1, '+', 3, '*'] >], < Polynom [3, < <class 'pymath.expression.Expression'> [2, 3, '*'] >]>]
|
|
>>> q = Polynom([0,0,4])
|
|
>>> q*3
|
|
< Polynom [0, 0, 12]>
|
|
>>> (q*3).steps
|
|
[[< <class 'pymath.expression.Expression'> [4, 'x', 2, '^', '*', 3, '*'] >], < Polynom [0, 0, < <class 'pymath.expression.Expression'> [4, 3, '*'] >]>]
|
|
>>> r = Polynom([0,1])
|
|
>>> r*3
|
|
< Polynom [0, 3]>
|
|
>>> (r*3).steps
|
|
[[< <class 'pymath.expression.Expression'> ['x', 3, '*'] >]]
|
|
>>> p*q
|
|
< Polynom [0, 0, 4, 8]>
|
|
>>> (p*q).steps
|
|
[[< <class 'pymath.expression.Expression'> [2, 'x', '*', 1, '+', 4, 'x', 2, '^', '*', '*'] >], < Polynom [0, 0, 4, < <class 'pymath.expression.Expression'> [2, 4, '*'] >]>]
|
|
>>> p*r
|
|
< Polynom [0, 1, 2]>
|
|
>>> P = Polynom([1,2,3])
|
|
>>> Q = Polynom([4,5,6])
|
|
>>> P*Q
|
|
< Polynom [4, 13, 28, 27, 18]>
|
|
"""
|
|
# TODO: Je trouve qu'elle grille trop d'étapes... |ven. févr. 27 19:08:44 CET 2015
|
|
o_poly = self.conv2poly(other)
|
|
|
|
coefs = [0]*(self.degree + o_poly.degree + 1)
|
|
for (i,a) in enumerate(self._coef):
|
|
for (j,b) in enumerate(o_poly._coef):
|
|
if a == 0 or b == 0:
|
|
elem = 0
|
|
elif a==1:
|
|
elem = b
|
|
elif b==1:
|
|
elem = a
|
|
else:
|
|
elem = Expression([a, b, op.mul])
|
|
|
|
if coefs[i+j]==0:
|
|
coefs[i+j] = elem
|
|
elif elem != 0:
|
|
if type(coefs[i+j]) == list:
|
|
coefs[i+j] += [elem]
|
|
else:
|
|
coefs[i+j] = [coefs[i+j] , elem]
|
|
|
|
p = Polynom(coefs, letter = self._letter)
|
|
ini_step = [Expression(self.postfix_tokens + o_poly.postfix_tokens + [op.mul])]
|
|
ans = p.simplify()
|
|
|
|
ans.steps = [ini_step] + ans.steps
|
|
return ans
|
|
|
|
def __rmul__(self, other):
|
|
o_poly = self.conv2poly(other)
|
|
|
|
return o_poly.__mul__(self)
|
|
|
|
@power_cache
|
|
def __pow__(self, power):
|
|
""" Overload **
|
|
|
|
>>> p = Polynom([0,0,3])
|
|
>>> p**2
|
|
< Polynom [0, 0, 0, 0, 9]>
|
|
>>> (p**2).steps
|
|
[< <class 'pymath.expression.Expression'> [3, 'x', 2, '^', '*', 2, '^'] >, < Polynom [0, 0, 0, 0, < <class 'pymath.expression.Expression'> [3, 2, '^'] >]>]
|
|
>>> p = Polynom([1,2])
|
|
>>> p**2
|
|
< Polynom [1, 4, 4]>
|
|
>>> (p**2).steps
|
|
[< <class 'pymath.expression.Expression'> [2, 'x', '*', 1, '+', 2, '^'] >, [< <class 'pymath.expression.Expression'> [2, 'x', '*', 1, '+', 2, 'x', '*', 1, '+', '*'] >], < Polynom [1, < <class 'pymath.expression.Expression'> [2, 2, '+'] >, < <class 'pymath.expression.Expression'> [2, 2, '*'] >]>]
|
|
>>> p = Polynom([0,0,1])
|
|
>>> p**3
|
|
< Polynom [0, 0, 0, 0, 0, 0, 1]>
|
|
>>> p = Polynom([1,2,3])
|
|
>>> p**2
|
|
< Polynom [1, 4, 10, 12, 9]>
|
|
|
|
"""
|
|
if not type(power):
|
|
raise ValueError("Can't raise Polynom to {} power".format(str(power)))
|
|
|
|
ini_step = [Expression(self.postfix_tokens + [power, op.pw])]
|
|
|
|
if self.is_monom():
|
|
if self._coef[self.degree] == 1:
|
|
coefs = [0]*self.degree*power + [1]
|
|
p = Polynom(coefs, letter = self._letter)
|
|
ans = p
|
|
else:
|
|
coefs = [0]*self.degree*power + [Expression([self._coef[self.degree] , power, op.pw])]
|
|
p = Polynom(coefs, letter = self._letter)
|
|
ans = p.simplify()
|
|
else:
|
|
if power == 2:
|
|
ans = self * self
|
|
else:
|
|
# TODO: faudrait changer ça c'est pas très sérieux |ven. févr. 27 22:08:00 CET 2015
|
|
raise AttributeError("__pw__ not implemented yet when power is greatter than 2")
|
|
|
|
ans.steps = ini_step + ans.steps
|
|
return ans
|
|
|
|
def __xor__(self, power):
|
|
return self.__pow__(power)
|
|
|
|
|
|
|
|
|
|
def test(p,q):
|
|
print("---------------------")
|
|
print("---------------------")
|
|
print("p : ",p)
|
|
print("q : ",q)
|
|
|
|
print("\n Plus ------")
|
|
print(p, "+", q)
|
|
for i in (p + q):
|
|
#print(repr(i))
|
|
#print("\t", str(i.postfix_tokens))
|
|
print(i)
|
|
|
|
print("\n Moins ------")
|
|
for i in (p - q):
|
|
#print(repr(i))
|
|
#print("\t", str(i.postfix_tokens))
|
|
print(i)
|
|
|
|
print("\n Multiplier ------")
|
|
for i in (p * q):
|
|
#print(repr(i))
|
|
#print("\t", str(i.postfix_tokens))
|
|
print(i)
|
|
|
|
print("\n Evaluer p ------")
|
|
for i in p(3).simplify():
|
|
print(i)
|
|
|
|
print("\n Evaluer q ------")
|
|
for i in q(3).simplify():
|
|
print(i)
|
|
|
|
|
|
if __name__ == '__main__':
|
|
#from .fraction import Fraction
|
|
#with Expression.tmp_render(txt):
|
|
# p = Polynom([1, 2, 3])
|
|
# q = Polynom([4, 5, 6])
|
|
# for i in (p*q).explain():
|
|
# print(i)
|
|
# r = Polynom([0,1])
|
|
# for i in (r*3).explain():
|
|
# print(i)
|
|
# print("q = ", q)
|
|
# r = q.reduce()
|
|
# print("r = ", r)
|
|
# for i in r.explain():
|
|
# print("q = ", i)
|
|
# print(p-q)
|
|
# for i in p-q:
|
|
# print(i)
|
|
#Polynom.random(degree = 2, conditions=["{b**2-4*a*c}>0"]) # Polynom deg 2 with positive Delta (ax^2 + bx + c)
|
|
|
|
|
|
import doctest
|
|
doctest.testmod(optionflags=doctest.ELLIPSIS)
|
|
|
|
|
|
# -----------------------------
|
|
# Reglages pour 'vim'
|
|
# vim:set autoindent expandtab tabstop=4 shiftwidth=4:
|
|
# cursor: 16 del
|