Mapytex/pymath/polynom.py

#!/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, name = "P"):
        """ 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(coefs = coefs, letter = letter, name = name)

    def __init__(self, coefs = [1], letter = "x", name = "P"):
        """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
        :param name: Name of the polynom

        >>> P = Polynom([1, 2, 3])
        >>> P.mainOp
        '+'
        >>> P.name
        'P'
        >>> P._letter
        'x'
        >>> Polynom([1]).mainOp
        '*'
        >>> Polynom([0, 0, 3]).mainOp
        '*'
        >>> Polynom([1, 2, 3])._letter
        'x'
        >>> Polynom([1, 2, 3], "y")._letter
        'y'
        >>> Polynom([1, 2, 3], name = "Q").name
        'Q'
        """
        super(Polynom, self).__init__()
        self.feed_coef(coefs)
        self._letter = letter
        self.name = name

        
        if self.is_monom():
            self.mainOp = op.mul
        else:
            self.mainOp = op.add

        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]>
        >>> print(Q.name)
        P'
        >>> 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])]

        ans = Polynom(derv_coefs[1:]).simplify()
        ans.name = self.name + "'"
        return ans

    @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)


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