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Mapytex/pymath/polynom.py

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