From 9f75d3e345c162a1f5d49fb39dbf9d3ba533efa5 Mon Sep 17 00:00:00 2001
From: root <root@ebook.dk-compmath.jku.at>
Date: Tue, 23 Apr 2019 11:11:33 +0200
Subject: [PATCH] Updating to last version

---
 ajpastor/dd_functions/toDiffAlgebraic.py           |   99 +-
 ajpastor/dd_functions/toDiffAlgebraic.py.Zb1982    | 1180 +++++++++++++++++++
 ajpastor/dd_functions/toDiffAlgebraic.py.jd2153    | 1187 ++++++++++++++++++++
 releases/diff_defined_functions.zip                |  Bin 144641 -> 169347 bytes
 releases/diff_defined_functions__0.6.zip           |  Bin 144641 -> 169347 bytes
 ...f_defined_functions__0.6__19.04.23_11:11:32.zip |  Bin 0 -> 169347 bytes
 6 files changed, 2465 insertions(+), 1 deletion(-)
 create mode 100644 ajpastor/dd_functions/toDiffAlgebraic.py.Zb1982
 create mode 100644 ajpastor/dd_functions/toDiffAlgebraic.py.jd2153
 create mode 100644 releases/old/diff_defined_functions__0.6__19.04.23_11:11:32.zip

diff --git a/ajpastor/dd_functions/toDiffAlgebraic.py b/ajpastor/dd_functions/toDiffAlgebraic.py
index 7ecfdbf..7eddad5 100644
--- a/ajpastor/dd_functions/toDiffAlgebraic.py
+++ b/ajpastor/dd_functions/toDiffAlgebraic.py
@@ -70,6 +70,13 @@ def infinite_derivative(element, times=1, var=x, _infinite=True):
     except AttributeError:
         return 0;
     
+    ##Simple call: parent is a Fraction Field
+    if(is_FractionField(parent)):
+        parent = parent.base();
+        n = parent(element.numerator()); dn = infinite_derivative(n);
+        d = parent(element.denominator()); dd = infinite_derivative(d);
+        return (dn*d - n*dd)/d**2;
+    
     ## Simple call: not an InfinitePolynomialRing
     if(not is_InfinitePolynomialRing(parent)):
         try:
@@ -1114,6 +1121,96 @@ def Exponential_polynomials(n, parent):
         prev = Exponential_polynomials(n-1,parent);
         return infinite_derivative(prev) + y[0]*prev;
     
+@cached_function
+def Logarithmic_polynomials(n, parent):
+    if(n < 0):
+        raise ValueError("No Logarithmic polynomial can be computed for non-positive index");
+    elif(not(is_InfinitePolynomialRing(parent))):
+        return Logarithmic_polynomials(n, InfinitePolynomialRing(parent, "y"));
+    
+    y = parent.gens()[0];
+    
+    if(n == 0):
+        return y[1]/y[0];
+    else:
+        return infinite_derivative(Logarithmic_polynomials(n-1, parent));
+    
+def change_variable(poly, new_var):
+    parent = poly.parent();
+    if(not (new_var in parent.fraction_field())):
+        raise TypeError("The new value must be in the same InfinitePolynomialRing");
+    elif(not is_InfinitePolynomialRing(parent)):
+        raise TypeError("The parent must be an InfinitePolynomialRing");
+    
+    gens = poly.polynomial().parent().gens();
+        
+    index = [get_InfinitePolynomialRingGen(parent, v, True)[1] for v in gens]; m = max(index);
+    derivatives = [new_var];
+    for i in range(m):
+        derivatives += [infinite_derivative(derivatives[-1])];
+    to_plug = [derivatives[i] for i in index];
+    
+    return parent.fraction_field()(poly.polynomial()(**{str(gens[i]) : to_plug[i] for i in range(len(gens))}));
+    
+def is_Riccati(poly):
+    '''
+        Method that checks if a non-linear differential equation is of Riccatti type 
+        and returns the change of coordinates and the linear differential equation in case
+        we can.
+    '''
+    poly = fromFinitePolynomial_toInfinitePolynomial(poly);
+    parent = poly.parent(); y = parent.gens()[0];
+    var = poly.polynomial().parent().gens(); index = [get_InfinitePolynomialRingGen(parent, v, True)[1] for v in var];
+    order = max(index);
+    
+    ## Checking the degree of y[0] is appropriate in the therm of y[order-1]
+    try:
+        if(not poly.degrees()[var.index(y[order-1])] == 1):
+            raise ValueError("The degree of the function in the equation is not valid");
+    except:
+        raise TypeError("Error getting the degree of the function in the equation");
+    
+    C = poly.coefficient(y[order-1]*y[0]);
+    ## Simple case: no element inside or is not in the base field
+    if(C == 0 or (not C in parent.base())):
+        return False, poly;
+    
+    ## The coefficient is in the appropriate place
+    try:
+        if(C.derivative(x) != 0):
+            new_eq = change_variable(poly, y[0]/C)*C;
+            print "Recursion in Raccati, new equation: %s" %new_eq; 
+            return is_Riccati(new_eq);
+    except:
+        # Impossible to compute the derivative -- Assuming zero
+        pass;
+    
+    ## Computing the constant
+    C = C/(order+1);
+    
+    to_plug = [Logarithmic_polynomials(i, parent)/C for i in index];
+    
+    new_poly = poly.polynomial()(**{str(var[i]): to_plug[i] for i in range(len(var))});
+    
+    if(is_FractionField(new_poly.parent())):
+        new_poly = new_poly.numerator();
+    if(is_InfinitePolynomialRing(new_poly.parent())):
+        new_poly = new_poly.polynomial();
+    
+    var = new_poly.parent().gens(); min_degree = {v:new_poly.degree() for v in var};
+    for m in new_poly.monomials():
+        for v in var:
+            d = m.degree(v);
+            if(d < min_degree[v]):
+                min_degree[v] = d;
+    gcd = prod([v**min_degree[v] for v in var], new_poly.parent().one());
+    new_poly = new_poly.parent()(new_poly/gcd);
+    
+    if(new_poly.degree() == 1):
+        return True, new_poly;
+    
+    return False, new_poly;
+    
 ###################################################################################################
 ### Private functions
 ###################################################################################################
@@ -1123,7 +1220,7 @@ def __dprint(obj, _debug):
 ####################################################################################################
 #### PACKAGE ENVIRONMENT VARIABLES
 ####################################################################################################
-__all__ = ["is_InfinitePolynomialRing", "get_InfinitePolynomialRingGen", "infinite_derivative", "diffalg_reduction", "toDifferentiallyAlgebraic_Below", "diff_to_diffalg", "inverse_DA", "func_inverse_DA", "guess_DA_DDfinite", "guess_homogeneous_DNfinite", "FaaDiBruno_polynomials", "Exponential_polynomials"];
+__all__ = ["is_InfinitePolynomialRing", "get_InfinitePolynomialRingGen", "infinite_derivative", "diffalg_reduction", "toDifferentiallyAlgebraic_Below", "diff_to_diffalg", "inverse_DA", "func_inverse_DA", "guess_DA_DDfinite", "guess_homogeneous_DNfinite", "FaaDiBruno_polynomials", "Exponential_polynomials", "Logarithmic_polynomials", "is_Riccati", "change_variable"];
 
 # Extra functions for debugging
 def simplify_coefficients(base, coeffs, derivatives, _debug=True):
diff --git a/ajpastor/dd_functions/toDiffAlgebraic.py.Zb1982 b/ajpastor/dd_functions/toDiffAlgebraic.py.Zb1982
new file mode 100644
index 0000000..a22b666
--- /dev/null
+++ b/ajpastor/dd_functions/toDiffAlgebraic.py.Zb1982
@@ -0,0 +1,1180 @@
+   
+from sage.all_cmdline import *   # import sage library
+
+from sage.graphs.digraph import DiGraph;
+
+from sage.rings.polynomial.polynomial_ring import is_PolynomialRing; 
+from sage.rings.polynomial.multi_polynomial_ring import is_MPolynomialRing;
+from sage.rings.fraction_field import is_FractionField;
+
+from ajpastor.dd_functions.ddFunction import *;
+from ajpastor.dd_functions.ddExamples import *;
+from sage.rings.polynomial.infinite_polynomial_ring import InfinitePolynomialRing;
+
+from ajpastor.misc.matrix import matrix_of_dMovement;
+from ajpastor.misc.matrix import differential_movement;
+
+################################################################################
+###
+### Utilities functions for InfinitePolynomialRings
+###
+################################################################################
+def is_InfinitePolynomialRing(parent):
+    from sage.rings.polynomial.infinite_polynomial_ring import InfinitePolynomialRing_dense as dense;                                                                 
+    from sage.rings.polynomial.infinite_polynomial_ring import InfinitePolynomialRing_sparse as sparse;                                
+    
+    return isinstance(parent, dense) or isinstance(parent, sparse);
+    
+def get_InfinitePolynomialRingGen(parent, var, with_number = False):
+    for gen in parent.gens():
+        spl = str(var).split(repr(gen)[:-1]);
+        if(len(spl) == 2):
+            if(with_number):
+                return (gen, int(spl[1]));
+            else:
+                return gen;
+    return None;
+    
+#### TODO: Review this function
+def infinite_derivative(element, times=1, var=x, _infinite=True):
+    '''
+        Method that takes an element and compute its times-th derivative
+        with respect to the variable given by var.
+        
+        If the element is not in a InfinitePolynomialRing then the method
+        'derivative' of the object will be called. Otherwise this method
+        returns the derivative following the rules:
+            - The coefficients of the polynomial ring are differentiated using
+            a recursive call to this method (usually this leads to a call of the
+            method '.derivative' of that coefficient).
+            - Any variable "#_n" appearing in the InfinitePolynomialRing has
+            as derivative the variable "#_{n+1}", i.e., the same name but with
+            one higher index.
+    '''
+    ## Checking the input _infinite
+    if(_infinite):
+        r_method = fromFinitePolynomial_toInfinitePolynomial;
+    else:
+        r_method = fromInfinityPolynomial_toFinitePolynomial;
+    
+    if(not ((times in ZZ) and times >=0)):
+        raise ValueError("The argument 'times' must be a non-negative integer");
+    elif(times > 1): # Recursive call
+        return infinite_derivative(infinite_derivative(element, times-1))
+    elif(times == 0): # Empty call
+        return element;
+        
+    ## Call with times = 1
+    try:
+        parent = element.parent();
+    except AttributeError:
+        return 0;
+    
+    ## Simple call: not an InfinitePolynomialRing
+    if(not is_InfinitePolynomialRing(parent)):
+        try:
+            try:
+                return element.derivative(var);
+            except Exception:
+                return element.derivative();
+        except AttributeError:
+            return parent.zero();
+            
+    ## IPR call
+    ## Symbolic case?
+    if(sage.symbolic.ring.is_SymbolicExpressionRing(parent.base())):
+        raise TypeError("Base ring for the InfinitePolynomialRing not valid (found SymbolicRing)");
+    ## Zero case
+    if(element == 0):
+        return 0;
+    ### Monomial case
+    if(element.is_monomial()):
+        ## Case of degree 1 (add one to the index of the variable)
+        if(len(element.variables()) == 1 and element.degree() == 1):
+            g,n = get_InfinitePolynomialRingGen(parent, element, True);
+            return r_method(g[n+1]);
+        ## Case of higher degree
+        else:
+            # Computing the variables in the monomial and their degrees (in order)
+            variables = element.variables();
+            degrees = [element.degree(v) for v in variables];
+            variables = [parent(v) for v in variables]
+            ## Computing the common factor
+            com_factor = prod([variables[i]**(degrees[i]-1) for i in range(len(degrees))]);
+            ## Computing the derivative of each variable
+            der_variables = [infinite_derivative(parent(v)) for v in variables];
+            
+            ## Computing the particular part for each summand
+            each_sum = [];
+            for i in range(len(variables)):
+                if(degrees[i] > 0):
+                    part_prod = prod([variables[j] for j in range(len(variables)) if j != i]);
+                    each_sum += [degrees[i]*infinite_derivative(parent(variables[i]))*part_prod];
+            
+            ## Computing the final result
+            return r_method(com_factor*sum(each_sum));
+    ### Non-monomial case
+    else:
+        coefficients = element.coefficients();
+        monomials = [parent(el) for el in element.monomials()];
+        return r_method(sum([infinite_derivative(coefficients[i])*monomials[i] + coefficients[i]*infinite_derivative(monomials[i]) for i in range(len(monomials))]));
+    
+################################################################################
+###
+### Changes from and to InfinitePolynomialRings and MultivariatePolynomialRings
+###
+################################################################################
+def fromInfinityPolynomial_toFinitePolynomial(poly):
+    if(not is_InfinitePolynomialRing(poly.parent())):
+        return poly;
+    
+    gens = poly.variables();
+    base = poly.parent().base();
+    
+    parent = PolynomialRing(base, gens);
+    return parent(str(poly));
+
+def fromFinitePolynomial_toInfinitePolynomial(poly):
+    parent = poly.parent();
+    if(is_InfinitePolynomialRing(poly) and (not is_MPolynomialRing(parent))):
+        return poly;
+    
+    
+    names = [str(el) for el in poly.parent().gens()];
+    pos = names[0].rfind("_");
+    if(pos == -1):
+        return poly;
+    prefix = names[0][:pos];
+    if(not all(el.find(prefix) == 0 for el in names)):
+        return poly;
+    R = InfinitePolynomialRing(parent.base(), prefix);
+    return R(poly);
+    
+################################################################################
+###
+### Methods for computing Diff. Algebraic equations with simpler coefficients
+###
+################################################################################
+def diffalg_reduction(poly, _infinite=False, _debug=False):
+    '''
+    Method that recieves a polynomial 'poly' with variables y_0,...,y_m with coefficients in some ring DD(R) and reduce it to a polynomial
+    'result' with coefficients in R where, for any function f(x) such that poly(f) == 0, result(f) == 0.
+    
+    The algorithm works as follows:
+        1 - Collect all the coefficients and their derivatives
+        2 - Perform the simplification to each pair of coefficients
+        3 - Perform the simplification for the last derivatives of the remaining coefficients --> gens
+        4 - Build the final derivation matrix for 'gens'
+        5 - Create a vector v for represent 'poly' w.r.t. 'gens'
+        6 - Build a square matrix --> M = (v | v' | ... | v^(n))
+        7 - Return det(M)
+    
+    INPUT:
+        - poly: a polynomial with variables y_0,...,y_m. Also an infinite polynomial with variable y_* is valid.
+        - _infinite: if True the result polynomial will be an infinite polynomial. Otherwise, a finite variable polynomial will be returned.
+        - _debug: if True, the steps of the algorithm will be printed.
+    '''
+    ## Processing the input _infinite
+    if(_infinite):
+        r_method = fromFinitePolynomial_toInfinitePolynomial;
+    else:
+        r_method = fromInfinityPolynomial_toFinitePolynomial;
+        
+    ### Preprocessing the input poly
+    parent, poly, dR = __check_input(poly, _debug);
+    
+    if(not is_DDRing(dR)):
+        raise TypeError("diffalg_reduction: polynomial is not over a DDRing");
+    
+    R = dR.base();
+    
+    ## Step 1: collecting function and derivatives
+    __dprint("---- DEBUGGING PRINTING FOR diffalg_reduction", _debug);
+    __dprint("R: %s" %R, _debug);
+    __dprint("-- Starting step 1", _debug);
+    coeffs = poly.coefficients(); # List of original coefficients
+    __dprint("Coefficients: %s" %("["+reduce(lambda p,q : p+", "+q, [repr(coeff) for coeff in coeffs])+"]"), _debug);            
+        
+    ocoeffs = coeffs; # Extra variable with the same list
+    monomials = poly.monomials(); # List of monomials
+    l_of_derivatives = [[f.derivative(times=i) for i in range(f.getOrder())] for f in coeffs]; # List of derivatives for each coefficient
+    
+    ## Step 2: simplification of coefficients
+    __dprint("-- Starting step 2", _debug);
+    graph = __simplify_coefficients(R, coeffs, l_of_derivatives, _debug);
+    maximal_elements = __digraph_roots(graph, _debug);
+    
+    ## Detecting case: all coefficients are already simpler
+    if(len(maximal_elements) == 1 and maximal_elements[0] == -1 and len(graph.outgoing_edges(-1)) == len(graph.edges())):
+        __dprint("Detected simplicity: returning the same polynomial over the basic ring", _debug);
+        return r_method(InfinitePolynomialRing(R, names=["y"])(poly));
+    
+    # Reducing the considered coefficients
+    coeffs = [coeffs[i] for i in maximal_elements if i != (-1)]; 
+    l_of_derivatives = [l_of_derivatives[i] for i in maximal_elements if i != (-1)]; 
+                
+    ## Step 3: simplification with the derivatives
+    __dprint("-- Starting step 3", _debug);
+    drelations = __simplify_derivatives(l_of_derivatives, _debug);
+    
+    ## Building the vector of final generators
+    gens = [];
+    if(-1 in maximal_elements):
+        gens = [1];
+    gens = sum([l_of_derivatives[i][:drelations[i][0]] for i in range(len(coeffs))], gens);
+    S = len(gens);
+    __dprint("Resulting vector of generators: %s" %gens, _debug);
+    
+    ## Step 4: build the derivation matrix
+    __dprint("-- Starting step 4", _debug);
+    C = __build_derivation_matrix(gens, coeffs, drelations, _debug);
+    cR = InfinitePolynomialRing(C.base_ring(), names=["y"]);
+    
+    ## Step 5: build the vector v
+    __dprint("-- Starting step 5", _debug);
+    v = __build_vector(ocoeffs, monomials, graph, drelations, cR, _debug);
+    
+    ## Step 6: build the square matrix M = (v, v',...)
+    __dprint("-- Starting step 6", _debug);
+    __dprint("Computing derivatives from the vector (step 5) using the matrix (step 4)", _debug);
+    derivative = infinite_derivative;
+    M = matrix_of_dMovement(C, v, derivative, S);
+    
+    __dprint("Vector of generators:\n\t%s" %gens, _debug);
+    __dprint("Final matrix (vector in rows):\n\t%s" %(str(M.transpose()).replace('\n', '\n\t')), _debug);
+    __dprint("Returning the determinant", _debug);
+    __dprint("---- DEBUGGING PRINTING FOR diffalg_reduction (finished)", _debug);
+    return r_method(M.determinant());
+
+################################################################################
+################################################################################
+### Some private methods for diffalg_reduction
+################################################################################
+################################################################################
+
+def __check_input(poly, _debug=False):
+    '''
+    Method that check and cast the polynomial input accordingly to our standards.
+    
+    The element 'poly' must be a polynomial with variables y_0,...,y_m with coefficients in some ring
+    '''
+    parent = poly.parent();
+    if(not is_InfinitePolynomialRing(parent)):
+        if(not is_MPolynomialRing(parent)):
+            if(not is_PolynomialRing(parent)):
+                raise TypeError("__check_input: the input is not a valid polynomial. Obtained something in %s" %parent);
+            if(not str(parent.gens()[0]).startswith("y_")):
+                raise TypeError("The input is not a valid polynomial. Obtained %s but wanted something with variables y_*" %poly);
+            parent = InfinitePolynomialRing(parent.base(), "y");
+        else:
+            gens = list(parent.gens());
+            to_delete = [];
+            for gen in gens:
+                if(str(gen).startswith("y_")):
+                    to_delete += [gen];
+            if(len(to_delete) == 0):
+                raise TypeError("__check_input: the input is not a valid polynomial. Obtained %s but wanted something with variables y_*" %poly);
+            for gen in to_delete:
+                gens.remove(gen);
+            parent = InfinitePolynomialRing(PolynomialRing(parent.base(), gens), "y");
+    else:
+        if(parent.ngens() > 1 or repr(parent.gen()) != "y_*"):
+            raise TypeError("__check_input: the input is not a valid polynomial. Obtained %s but wanted something with variables y_*" %poly);
+    
+    poly = parent(poly);
+    dR = parent.base();
+    
+    return (parent, poly, dR);
+
+def __simplify_coefficients(base, coeffs, derivatives, _debug=False):
+    '''
+    Method that get the relations for the coefficients within their derivatives. The list 'coeffs' is the list to simplify
+    and ;derivatives' is a list of lists with the derivatives of each element of 'coeffs' that we will consider.
+    
+    It returns a pair (graph, relations) where:
+        - graph: a forest of the points of minimal depth with all the relations
+        - relations: a dictionary which tells for each i < j, how the ith coefficient is related with the jth coefficient
+            (see __find_relation to know how that relation is expressed).
+    '''
+    # Checking the input
+    n = len(coeffs);
+    if(len(derivatives) < n):
+        raise ValueError("__simplify_coefficients: error in size of the input 'derivatives'");
+    
+    E = range(n);
+    R = [];
+    
+    # Checking the simplest cases (coeff[i] in R)
+    simpler = [];
+    for i in range(n):
+        if(coeffs[i] in base):
+            if((-1) not in E):
+                E += [-1];
+            R += [(-1,i,(0,0,coeffs[i]))];
+            simpler += [i];
+    
+    __dprint("Simpler coefficiets (related with -1): %s" %simpler, _debug);
+    # Starting the comparison
+    for i in range(n):
+        for j in range(i+1, n):
+            # Checking (i,j)
+            if(not(j in simpler)):
+                rel = __find_relation(coeffs[j], derivatives[i], _debug);
+                if(not(rel is None)):
+                    R += [(i,j,rel)];
+                    continue;
+            # Checking (j,i)
+            if(not(i in simpler)):
+                rel = __find_relation(coeffs[i], derivatives[j], _debug);
+                if(not(rel is None)):
+                    R += [(j,i,rel)];
+    
+    # Checking if the basic case will be used because of the relations
+    if((-1) not in E and any(e[2][1][1] != 0 for e in R)):
+        E += [-1];
+    
+    # Building the graph and returning
+    __dprint("All relations: %s" %R, _debug);
+    graph = DiGraph([E,R]);
+    __dprint("Edges: %s" %graph.edges(), _debug);
+    
+    # Deleting spurious relations and returning
+    return __min_span_tree(graph, _debug);
+
+def __simplify_derivatives(derivatives, _debug=False):
+    '''
+    Method to get the relations for the derivatives of the coefficients. The argument 'derivatives' contains a list of lists for which
+    one we will see if the last elements have relations with the firsts.
+    
+    It returns a list of tuples 'res' where 'res[i][0] = k' if the kth fucntion in 'derivatives[i]' is the first element on the list
+    with relations with the previous elements and 'res[i][1]' is the relation found.
+    If no relation is found, a tuple (None,None) is added to the list.
+    '''
+    res = [];
+    for i in range(len(derivatives)):
+        to_add = (None,None);
+        for k in range(len(derivatives[i])-1,0,-1):
+            relation = __find_relation(derivatives[i][k], derivatives[i][:k-1], _debug);
+            if(not (relation is None)):
+                to_add = (k,relation);
+                break;
+        res += [to_add];
+            
+    __dprint("Found relations with derivatives: %s" %res, _debug);
+    return res;
+    
+def __find_relation(g,df, _debug=False):
+    '''
+    Method that get the relation between g and the list df. 
+    
+    It returns a tuple (k,res) where:
+        - k: the first index which we found a relation
+        - res: the relation (see __find_linear_relation to see how such relation is expressed).
+        
+    Then the following statement is always True:
+        g == df[k]*res[0] + res[1]
+    '''
+    __dprint("** Checking relations between %s and %s" %(repr(g), df), _debug);
+    for k in range(len(df)):
+        res = __find_linear_relation(g,df[k]);
+        if(not (res is None)):
+            __dprint("Found relation:\n\t(%s) == [%s]*(%s) + [%s]" %(repr(g),repr(res[0]), repr(df[k]), repr(res[1])), _debug);
+            __dprint("*************************", _debug);
+            return (k,res);
+    __dprint("Nothing found\n*************************", _debug);
+    
+    return None;
+    
+def __find_linear_relation(f,g):
+    '''
+    This method receives two DDFunctions and return two constants (c,d) such that f = cg+d. 
+    None is return if those constants do not exist.
+    '''
+    if(not (is_DDFunction(f) and is_DDFunction(g))):
+        raise TypeError("find_linear_relation: The functions are not DDFunctions");
+    
+    ## Simplest case: some of the functions are constants is a constant
+    if(f.is_constant):
+        return (f.parent().zero(), f(0));
+    elif(g.is_constant):
+        return None;
+    
+    try:
+        of = 1; og = 1;
+        while(f.getSequenceElement(of) == 0): of += 1;
+        while(g.getSequenceElement(og) == 0): og += 1;
+        
+        if(of == og):
+            c = f.getSequenceElement(of)/g.getSequenceElement(og);
+            d = f(0) - c*g(0);
+            
+            if(f == c*g + d):
+                return (c,d);
+    except Exception:
+        pass;
+    
+    return None;
+    
+def __min_span_tree(digraph, _debug=False):
+    '''
+    Method that computes a forest from a directed graph containing all vertices and having the minimal depth.
+    
+    It is computed with a breadth first search.
+    '''
+    to_search = __digraph_roots(digraph, _debug);
+    i = 0;
+    done = [];
+    edges = [];
+    while(i < len(to_search)):
+        v = to_search[i];
+        if(not (v in done)):
+            for e in digraph.outgoing_edges(v):
+                if(not(e[1] in to_search)):
+                    to_search += [e[1]];
+                    edges += [e];
+            done += [v];
+        i += 1;
+    return DiGraph([digraph.vertices(),edges]);
+
+def __digraph_roots(digraph, _debug=False):
+    '''
+    Method that computes the roots of a directed graph. We consider a vertex v to be a root if there
+    are not edges (v,w) in the graph.
+    '''
+    return [v for v in digraph.vertices() if digraph.in_degree(v) == 0];
+    
+def __build_derivation_matrix(gens, coeffs, drelations, _debug=False):
+    '''
+        Method to build a derivation matrix 'C' for the vector of generators 'gen'.
+        
+        All the information about the coefficients and their relations are given
+        by the arguments 'coeffs' (list of basic elements for 'gens') and 
+        'drelations' (relations between elements in 'coeffs' and the last derivative
+        on 'gens').
+        
+        It could be that gens[0] == 1. This element must be treated in a specific
+        way collecting all the relations on 'drelations'. 
+    '''
+    __dprint("Building derivation matrix for %s" %gens, _debug);
+    
+    rows = []; # Variable for the rows of the matrix M
+    
+    constant = (gens[0] == 1); # Flag variable for gens[0] == 1
+    coeff_order = []; # Collecting the real order of each coefficient in the vector space
+    for i in range(len(coeffs)):
+        if(drelations[i][0] is None):
+            coeff_order += [coeffs[i].getOrder()];
+        else:
+            coeff_order += [drelations[i][0]];
+    coeff_index = [0]; # Variable for the index of coeff[i] in gens
+    if(constant): coeff_index[0] = 1;
+    for i in range(1, len(coeffs)): coeff_index += [coeff_index[-1]+coeff_order[i-1]];
+    
+    __dprint("Is there constant: %s" %constant, _debug);
+    __dprint("Orders: %s\nIndices: %s" %(coeff_order, coeff_index), _debug);
+    
+    ## Considering the case that gens[0] == -1
+    if(constant):
+        new_row = [0];
+        for i in range(len(coeffs)):
+            new_row += [0 for j in range(coeff_order[i])];
+            if(not (drelations[i][1] is None)):
+                new_row[-1] = drelations[i][1][1][1];
+        rows += [new_row];
+        
+    ## Now we loop through all the coefficients in coeffs
+    for i in range(len(coeffs)):
+        com = coeffs[i].equation.companion();
+        for j in range(coeff_order[i]):
+            ## Building the jth row for coeff[i]
+            new_row = [0 for k in range(coeff_index[i])]; # First columns for the row
+            
+            ## Middle columns for the row
+            new_row += [com[j][k] for k in range(coeff_order[i]-1)];
+            if((not (drelations[i][0] is None)) and (drelations[i][1][0] == j)):
+                new_row += [drelations[i][1][1][0]];
+            else:
+                new_row += [com[j][coeff_order[i]-1]];
+            
+            new_row += [0 for k in range(coeff_index[i]+coeff_order[i], len(gens))]; # Last columns for the row
+            
+            ## Adding the resulting row
+            rows += [new_row];
+            
+    rows = Matrix(rows);
+    
+    __dprint("** Matrix:\n%s\n*******" %rows, _debug);
+    
+    return rows;
+
+def __build_vector(coeffs, monomials, graph, drelations, cR, _debug=False):
+    '''
+        This method builds a vector that represent the polynomial generated by
+        'coeffs' and 'monomials' using the relations contained in 'graph'
+        and 'drelations' w.r.t. the vector 'gens'.
+        
+        The output vector v satisfies:
+            sum(coeffs[i]*monomials[i]) == sum(v[j]*gens[j])
+    '''
+    # Computing important data for each coefficient
+    maximal_elements = [el for el in __digraph_roots(graph) if el != -1];
+    coeff_order = [];
+    for i in range(len(coeffs)):
+        if(i in maximal_elements and (not (drelations[maximal_elements.index(i)][0] is None))):
+            coeff_order += [drelations[maximal_elements.index(i)][0]];
+        else:
+            coeff_order += [coeffs[i].getOrder()];
+    constant = (-1 in graph.vertices());
+    
+    __dprint("Is there constant: %s" %constant, _debug);
+    
+    # Computing vectors and companion matrices for each element
+    vectors = [vector(cR, [monomials[i]] + [0 for i in range(coeff_order[i]-1)]) for i in range(len(coeffs))];
+    trans = [];
+    for i in range(len(coeffs)):
+        if(i in maximal_elements and (not (drelations[maximal_elements.index(i)][0] is None))):
+            relation = drelations[maximal_elements.index(i)];
+            C = [];
+            for j in range(relation[0]):
+                new_row = [el for el in coeffs[i].equation.companion()[j][:relation[0]-1]] + [0];
+                if(j == relation[1][0]):
+                    new_row[-1] = relation[1][1][0];
+                C += [new_row];
+            C = Matrix(C);
+            trans += [(drelations[maximal_elements.index(i)][1][1][1], C)];
+        else:
+            trans += [(0, coeffs[i].equation.companion())];
+            
+    if(_debug):
+        print "--- Transitions to nodes:";
+        for i in range(len(trans)):
+            print "+++ %s --> Cosntant: %s; Matrix:\n%s" %(i, trans[i][0], trans[i][1]);
+        print "-------------------------";
+    
+    const_val = 0;    
+    # Doing a Tree Transversal in POstorder in the graph to pull up the vectors
+    # Case for -1
+    if(constant):
+        maximal_elements = [-1] + maximal_elements;
+    #    const_val = sum(cR(coeffs[e[1]])*monomials[coeffs[e[1]]] for e in graph.outgoing_edges(-1));
+    stack = copy(maximal_elements); stack.reverse();
+        
+    # Rest of the graph
+    ready = [];
+    while(len(stack) > 0):
+        current = stack[-1];
+        if(current in ready):
+            __dprint("Visiting node %s" %current, _debug);
+            if(not (current in maximal_elements)):
+                # Visit the node: pull-up the vector
+                edge = graph.incoming_edges(current)[0];
+                cu_vector = vectors[current];
+                
+                ## Constant case; reducing to the constant term
+                if(edge[0] == -1):
+                    __dprint("** Reducing node %d to constant" %(current), _debug);
+                    __dprint("\t- Current vector: %s" %(cu_vector), _debug);
+                    __dprint("\t- Prev. constant: %s" %const_val, _debug);
+                    const_val += sum(cu_vector[i]*cR(coeffs[current].derivative(times=i)) for i in range(len(cu_vector)));
+                    __dprint("\t- New constant: %s" %const_val, _debug);
+                
+                ## General case
+                else:                
+                    de_vector = vectors[edge[0]];
+                    relation = edge[2];
+                    C = trans[edge[0]][1];
+                    
+                    __dprint("** Reducing node %d to %d" %(current, edge[0]), _debug);
+                    __dprint("\t- Current vector: %s" %(cu_vector), _debug);
+                    __dprint("\t- Destiny vector: %s" %(de_vector), _debug);
+                    __dprint("\t- Relation: %s" %(str(relation)), _debug);
+                    __dprint("\t- Matrix:\n\t\t%s" %(str(C).replace('\n','\n\t\t')), _debug); 
+                    __dprint("\t- Prev. constant: %s" %const_val, _debug);
+                                
+                    # Building vectors for all the required derivatives
+                    ivectors = [vector(cR, [0 for i in range(relation[0])] + [1] + [0 for i in range(relation[0]+1,coeff_order[edge[0]])])];
+                    extra_cons = [relation[1][1]];
+                    
+                    for i in range(len(cu_vector)):
+                        extra_cons += [ivectors[-1][-1]*trans[edge[0]][0]];
+                        ivectors += [differential_movement(C, ivectors[-1], infinite_derivative)];
+                    
+                    vectors[edge[0]] = sum([vector(cR,[cu_vector[i]*ivectors[i][j] for j in range(len(ivectors[i]))]) for i in range(len(cu_vector))], de_vector);
+                    const_val += sum([cu_vector[i]*extra_cons[i] for i in range(len(cu_vector))]);
+                    
+                    __dprint("\t- New vector: %s" %vectors[edge[0]], _debug);
+                    __dprint("\t- New constant: %s" %const_val, _debug);
+                    __dprint("*************************", _debug);
+                
+            # Getting out the element of the stack
+            stack.pop();
+        else:
+            # Add the children and visit them
+            ready += [current];
+            for e in graph.outgoing_edges(current):
+                stack.append(e[1]);
+        
+    final_vector = [];
+    for i in maximal_elements:
+        if(i == -1):
+            final_vector += [const_val];
+        else:
+            final_vector += [el for el in vectors[i]];
+            
+    final_vector = vector(cR, final_vector);
+    __dprint("Vector: %s" %final_vector, _debug);
+    
+    return vector(final_vector);
+    
+    
+################################################################################
+################################################################################
+################################################################################
+
+def toDifferentiallyAlgebraic_Below(poly, _infinite=False, _debug=False):
+    '''
+    Method that receives a polynomial with variables y_0,...,y_m with coefficients in some ring DD(R) and reduce it to a polynomial
+    with coefficients in R.
+    
+    The optional input '_debug' allow the user to print extra information as the matrix that the determinant is computed.
+    '''
+    ## Processing the input _infinite
+    if(_infinite):
+        r_method = fromFinitePolynomial_toInfinitePolynomial;
+    else:
+        r_method = fromInfinityPolynomial_toFinitePolynomial;
+    
+    ### Preprocessing the input
+    parent = poly.parent();
+    if(not is_InfinitePolynomialRing(parent)):
+        if(not is_MPolynomialRing(parent)):
+            if(not is_PolynomialRing(parent)):
+                raise TypeError("The input is not a valid polynomial. Obtained something in %s" %parent);
+            if(not str(parent.gens()[0]).startswith("y_")):
+                raise TypeError("The input is not a valid polynomial. Obtained %s but wanted something with variables y_*" %poly);
+            parent = InfinitePolynomialRing(parent.base(), "y");
+        else:
+            gens = list(parent.gens());
+            to_delete = [];
+            for gen in gens:
+                if(str(gen).startswith("y_")):
+                    to_delete += [gen];
+            if(len(to_delete) == 0):
+                raise TypeError("The input is not a valid polynomial. Obtained %s but wanted something with variables y_*" %poly);
+            for gen in to_delete:
+                gens.remove(gen);
+            parent = InfinitePolynomialRing(PolynomialRing(parent.base(), gens), "y");
+    else:
+        if(parent.ngens() > 1 or repr(parent.gen()) != "y_*"):
+            raise TypeError("The input is not a valid polynomial. Obtained %s but wanted something with variables y_*" %poly);
+    
+    poly = parent(poly);
+    
+    ### Now poly is in a InfinitePolynomialRing with one generator "y_*" and some base ring.
+    if(not isinstance(parent.base(), DDRing)):
+        print "The base ring is not a DDRing. Returning the original polynomial (reached the bottom)";
+        return r_method(poly);
+    
+    up_ddring = parent.base()
+    dw_ddring = up_ddring.base();
+    goal_ring = InfinitePolynomialRing(dw_ddring, "y");
+    
+    coefficients = poly.coefficients();
+    monomials = poly.monomials();
+    
+    ## Arraging the coefficients and organize the vector-space notation
+    dict_to_derivatives = {};
+    dict_to_vectors = {};
+    S = 0;
+    for i in range(len(coefficients)):
+        coeff = coefficients[i]; monomial = goal_ring(str(monomials[i]));
+        if(coeff in dw_ddring):
+            if(1 not in dict_to_vectors):
+                S += 1;
+                dict_to_vectors[1] = goal_ring.zero();
+            dict_to_vectors[1] += dw_ddring(coeff)*monomial;
+        else:
+            used = False;
+            for el in dict_to_derivatives:
+                try:
+                    index = dict_to_derivatives[el].index(coeff);
+                    dict_to_vectors[el][index] += monomial;
+                    used = True;
+                    break;
+                except ValueError:
+                    pass;                  
+            if(not used):
+                list_of_derivatives = [coeff]; 
+                for j in range(coeff.getOrder()-1):
+                    list_of_derivatives += [list_of_derivatives[-1].derivative()];
+                dict_to_derivatives[coeff] = list_of_derivatives;
+                dict_to_vectors[coeff] = [monomial] + [0 for j in range(coeff.getOrder()-1)];
+                S += coeff.getOrder();
+    
+    rows = [];
+    for el in dict_to_vectors:
+        if(el == 1):
+            row = [dict_to_vectors[el]];
+            for i in range(S-1):
+                row += [infinite_derivative(row[-1])];
+            rows += [row];
+        else:
+            C = el.equation.companion();
+            C = Matrix(goal_ring.base(),[[goal_ring.base()(row_el) for row_el in row] for row in C]);
+            rows += [row for row in matrix_of_dMovement(C, vector(goal_ring, dict_to_vectors[el]), infinite_derivative, S)];
+        
+    M = Matrix(rows);
+    if(_debug): print M;
+    return r_method(M.determinant().numerator());
+
+def diff_to_diffalg(func, _infinite=False, _debug=False):
+    '''
+        Method that compute an differentially algebraic equation for the obect func.
+        This objet may be any element in QQ(x) or an element in some DDRing. In the first case
+        the equation returned is the simple "y_0 - func". In the latter, we compute the differential
+        equation using the linear differential representation of the obect.
+        
+        The result is always a polynomial with variables "y_*" where the number in the index
+        represent the derivative.
+    
+        The optional argument _debug allows the user to see during execution more data of the computation.
+    '''
+    ## Processing the input _infinite
+    if(_infinite):
+        r_method = fromFinitePolynomial_toInfinitePolynomial;
+    else:
+        r_method = fromInfinityPolynomial_toFinitePolynomial;
+    
+    ## Computations
+    try:
+        parent = func.parent();
+    except AttributeError:
+        return r_method(PolynomialRing(QQ,"y_0")("y_0 + %s" %func));
+    
+    if(isinstance(parent, DDRing)):
+        R = InfinitePolynomialRing(parent.base(), "y");
+        p = sum([R("y_%d" %i)*func[i] for i in range(func.getOrder()+1)], R.zero());
+        for i in range(parent.depth()-1):
+            p = toDifferentiallyAlgebraic_Below(p, _infinite=True, _debug=_debug);
+        return r_method(p);
+    else:
+        R = PolynomialRing(PolynomialRing(QQ,x).fraction_field, "y_0");
+        return r_method(R.gens()[0] - func);
+    
+################################################################################
+###
+### Methods for computing Diff. Algebraic equation for inverses
+### 
+###     - Multiplicative inverse from DA equation
+###     - Functional inverse from Dn-finite equation
+###
+################################################################################
+def inverse_DA(poly, vars=None, _infinite=False):
+    '''
+        Method that computes the DA equation for the multiplicative inverse 
+        of the solutions of a DA equation.
+        
+        We assume that the variables given by vars are the variables representing
+        the solution, namely vars[i+1] = vars[i].derivative().
+        If vars is not provided, we take all the variables from the polynomial
+        that is given.
+    '''
+    ## Processing the input _infinite
+    if(_infinite):
+        r_method = fromFinitePolynomial_toInfinitePolynomial;
+    else:
+        r_method = fromInfinityPolynomial_toFinitePolynomial;
+        
+    ## Checking that poly is a polynomial
+    
+    parent = poly.parent();
+    if(is_FractionField(parent)):
+        parent = parent.base();
+    if(is_InfinitePolynomialRing(parent)):
+        poly = fromInfinityPolynomial_toFinitePolynomial(poly);
+        return inverse_DA(poly, vars, _infinite=_infinite);
+    if(not (is_PolynomialRing(parent) or is_MPolynomialRing(parent))):
+        raise TypeError("No polynomial is given");
+    poly = parent(poly);
+    
+    ## Getting the list of variables
+    if(vars is None):
+        g = list(poly.parent().gens()); g.reverse();
+    else:
+        if(any(v not in poly.parent().gens())):
+            raise TypeError("The variables given are not in the polynomial ring");
+        g = vars;
+        
+    ## Computing the derivative of the inverse using Faa di Bruno's formula
+    derivatives = [1/g[0]] + [sum((falling_factorial(-1, k)/g[0]**(k+1))*bell_polynomial(n,k)(*g[1:n-k+2]) for k in range(1,n+1)) for n in range(1,len(g
+))];
+    
+    ## Computing the final equation
+    return r_method(poly(**{str(g[i]): derivatives[i] for i in range(len(g))}).numerator());
+
+def func_inverse_DA(func, _inifnite=False, _debug=False):
+    '''
+        TODO: Add documentation
+    '''
+    ## Processing the input _infinite
+    if(_infinite):
+        r_method = fromFinitePolynomial_toInfinitePolynomial;
+    else:
+        r_method = fromInfinityPolynomial_toFinitePolynomial;
+        
+    ## Computations
+    equation = fromFinitePolynomial_toInfinitePolynomial(diff_to_diffalg(func, _debug=_debug));
+    
+    if(_debug): print equation;
+    
+    parent = equation.parent();
+    y = parent.gens()[0];
+    
+    n = len(equation.variables());
+        
+    quot = [FaaDiBruno_polynomials(i, parent)[0]/FaaDiBruno_polynomials(i,parent)[1] for i in range(n)];
+    coeffs = [el(**{str(parent.base().gens()[0]) : y[0]}) for el in equation.coefficients()];
+    monomials = [el(**{str(y[j]) : quot[j] for j in range(n)}) for el in equation.monomials()];
+    
+    if(_debug):
+        print monomials;
+        print coeffs;
+    
+    sol = sum(monomials[i]*coeffs[i] for i in range(len(monomials)));
+    
+    if(_debug): print sol;
+    if(hasattr(sol, "numerator")):
+        return parent(sol.numerator());
+    return r_method(parent(sol));
+
+
+################################################################################
+###
+### Methods for compute Dn-finite equations from DA-equations
+###
+################################################################################
+def guess_Dnfinite(poly, init):
+    ## TODO: Method not public
+    res = guess_homogeneous_DNfinite(poly, init);
+    if(is_DDFunction(res)):
+        return res;
+    return guess_DA_DDfinite(poly, init);
+
+def guess_DA_DDfinite(poly, init=[], bS=None, bd = None, all=True):
+    '''
+        Method that tries to compute a DD-finite differential equation
+        for a DA-function with constant coefficients.
+        
+        It just tries all the possibilities. It uses the Composition class
+        to get the possibilities of the orders for the coefficients.
+        
+        INPUT:
+            - poly: polynomial with the differential equation we want to mimic with DD-finite elements
+            - init: initial values of the solution of poly
+            - bS: bound to the sum of orders in the DD-finite equation
+            - db: bound to the order of the DD-finite equation
+            - all: compute all the posibles equations
+    '''
+    poly = fromInfinityPolynomial_toFinitePolynomial(poly);
+    
+    if(not poly.is_homogeneous()):
+        raise TypeError("We require a homogeneous polynomial");
+    
+    if(bS is None):
+        S = poly.degree(); 
+    else:
+        S = bS;
+    if(bd is None):
+        d = S - poly.parent().ngens() + 2;
+    else:
+        d = bd;
+    
+    compositions = Compositions(S, length = d+1);
+    coeffs = [["a_%d_%d" %(i,j) for j in range(S)] for i in range(d+1)];
+    cInit = [["i_%d_%d" %(i,j) for j in range(S-1)] for i in range(d+1)];
+    R = ParametrizedDDRing(DFinite, sum(coeffs, [])+sum(cInit,[]));
+    R2 = R.to_depth(2);
+    coeffs = [[R.parameter(coeffs[i][j]) for j in range(S)] for i in range(d+1)];
+    cInit = [[R.parameter(cInit[i][j]) for j in range(S-1)] for i in range(d+1)];
+    sols = [];
+    
+    for c in compositions:
+        ## Building the parametrized guess
+        e = [R.element([coeffs[i][j] for j in range(c[i])] + [1],cInit[i]) for i in range(d+1)];
+        f = R2.element(e);
+        if(len(init) > 0):
+            try:
+                f = f.change_init_values([init[i] for i in range(min(len(init), f.equation.jp_value+1))]);
+            except ValueError:
+                continue;
+        
+        ## Computing the DA equation
+        poly2 = diff_to_diffalg(f);
+        
+        ###############################################
+        ### Comparing the algebraic equations
+        ## Getting a possible constant difference
+        m = poly.monomials();
+        m2 = poly2.monomials();
+        eqs = [];
+        C = None;
+        for mon in m2:
+            c2 = poly2.coefficient(mon);
+            if(poly2.parent().base()(poly2.coefficient(mon)).is_constant()):
+                C = poly.coefficient(poly.parent()(mon))/c2;
+                break;
+        if(C is None):
+            C = 1;
+        elif(C == 0):
+            continue;
+        
+        ### Building all equations from the algebraic equation
+        for mon in m2:
+            c2 = poly2.coefficient(mon); c = poly.coefficient(poly.parent()(mon));
+            eqs += [C*c2 - c];
+        
+        ##################################################
+        ### Comparing the initial values
+        for i in range(min(len(init), f.equation.jp_value+1),len(init)):
+            try:
+                eqs += [f.getInitialValue(i) - init[i]];
+            except TypeError:
+                break; ## Case when no initial value can be computed
+            
+        ##################################################
+        ### Solving the system (groebner basis)
+        P = R.base_field.base(); 
+        fEqs = [];
+        for el in eqs:
+            if(el.parent().is_field()):
+                fEqs += [P(str(el.numerator()))];
+            else:
+                fEqs += [P(str(el))];
+                
+        gb = ideal(fEqs).groebner_basis();
+        if(1 not in gb):
+            sols += [(f,gb)];
+            if(not all):
+                break;
+        
+        
+    if(len(sols) > 0 and not all):
+        return sols[0];
+    
+    return sols;
+
+def guess_homogeneous_DNfinite(poly, init):
+    '''
+        Method that tries to compute a Dn-finite differential equation
+        for a DA-function defined from an homogeneous equation.
+        
+        This method simplifies the equation supposing that the solution
+        is of the form y(x) = exp(int(u(x))), getting a differential equation
+        for u(x) of less order.
+        
+        If we end up with a linear equation, then we can return a Dn-finite function
+        where n depends on how many iterations we needed.
+        
+        If the final algebraic equation is not linear, then we return this
+        last equation together with the number of iterations done.
+        
+        INPUT:
+            - poly: polynomial with the differential equation we want to mimic with DD-finite elements
+            - init: initial values of the solution of poly (must be enough).
+    '''
+    new_poly, depth = simplify_homogeneous(poly);
+    parent = new_poly.parent(); base = parent.base(); y = parent.gens()[0];
+    
+    order = max(get_InfinitePolynomialRingGen(parent, v, True)[1] for v in new_poly.variables());
+    
+    if(new_poly.degree() == 1 and new_poly.is_homogeneous()): ## We can build something
+        coeffs = [];
+        for i in range(order+1):
+            if(y[i] in new_poly.variables()):
+                coeffs += new_poly.coefficients()[new_poly.monomials().index(y[i])];
+            else:
+                coeffs += [0];
+        
+        inhom = new_poly.constant_coefficient();
+        if(is_DDRing(base)):
+            base_field = base.base_field;
+            base = base.to_depth(base.depth()+1);
+        else:
+            base_field = base.fraction_field();
+            base = DDRing(PolynomialRing(base_field, 'x'));
+        
+        ## We can build a D(depth+1)-finite function
+        deep_init = build_initial_from_homogeneous(tuple(init), order, depth, base_field);
+                
+        res = base.element(coeffs, deep_init, inhomogeneous=inhom);
+        for i in range(depth):
+            res = Exp(res.integrate(0));
+        return res;
+    
+    else:
+        return (new_poly, depth);
+    
+
+@cached_function
+def build_initial_from_homogeneous(init, n, depth, base):
+    ## Checking we have enough data
+    if(len(init) < n+depth):
+        raise TypeError("Not enought initial data was provided (required %d)" %(n+depth));
+    elif(len(init) > n+depth):
+        return build_initial_from_homogeneous(tuple(list(init)[:n+depth]), n, depth, base);
+    
+    ## Base case with depth = 0
+    if(depth == 0):
+        return init[:n];
+    
+    ## Casting the input to the desired ring
+    init = [base(el) for el in init];
+    
+    ## Checking the initial condition init[0]
+    if(init[0] == 0):
+        raise ValueError("Initial condition is zero. Impossible to compute a solution of the form e(int(u(x)))");
+    
+    ## Computing the initial conditions for depth "depth-1"
+    new_init = [init[1]/init[0]];
+    
+    parent = Exponential_polynomials(1,base).parent(); y = parent.gens()[0];
+    
+    for i in range(1,n+depth-1):
+        poly = fromInfinityPolynomial_toFinitePolynomial(-(Exponential_polynomials(i+1, parent) - y[i]));
+        new_init += [base(poly(**{str(y[j]) : new_init[j] for j in range(i)})) + init[i+1]/init[0]];
+        
+    ## Returning the recurive call with one less depth
+    return build_initial_from_homogeneous(tuple(new_init), n, depth-1, base);
+
+@cached_function
+def simplify_homogeneous(poly, _stop_linear=True):
+    '''
+        Given an homogeneous differential polynomial 'poly', we reduce the order of the equation by 1
+        using the change of variables y(x) = exp(int(u(x))).
+        
+        This leads to a differentially algebraic equation of order 1 less than 'poly'. If we obtain a 
+        new homogeneous equation, we can iterate. The return of this function is this final result toether
+        with the number of steps performed.
+    '''
+    poly = fromFinitePolynomial_toInfinitePolynomial(poly);
+    parent = poly.parent(); y = parent.gens()[0];
+    
+    if(not(poly.is_homogeneous()) or (_stop_linear and poly.degree() == 1)):
+        return (poly,0);
+    
+    d = {str(y[0]) : parent.one()};
+    for v in poly.variables():
+        if(v != y[0]):
+            d[str(v)] = Exponential_polynomials(get_InfinitePolynomialRingGen(parent, v, True)[1], parent);
+                
+    new_poly = poly(**d);
+    result,n = simplify_homogeneous(new_poly);
+    
+    return (result, n+1);
+    
+###################################################################################################
+### Polynomial functions
+###################################################################################################
+@cached_function
+def FaaDiBruno_polynomials(n, parent):
+    if(n < 0):
+        raise ValueError("No Faa Di Bruno polynomial can be computed for negative index");
+    elif(not(is_InfinitePolynomialRing(parent))):
+        if((not(is_PolynomialRing(parent))) and (not(is_MPolynomialRing(parent)))):
+            raise TypeError("The parent ring is not valid: needed polynomial rings or InfinitePolynomialRing");
+        return FaaDiBruno_polynomials(n, InfinitePolynomialRing(parent, "y"));        
+    
+    if(parent.base().ngens() == 0 or parent.base().gens()[0] == 1):
+        raise TypeError("Needed a inner variable in the coefficient ring");
+        
+        
+    x = parent.base().gens()[0];
+    y = parent.gens()[0];
+    if(n == 0):
+        return (parent(parent.base().gens()[0]), parent.one());
+    if(n == 1):
+        return (parent.one(), y[1]);
+    else:
+        prev = [FaaDiBruno_polynomials(k, parent) for k in range(n)];
+        ele = -sum(prev[k][0]*bell_polynomial(n,k)(**{"x%d" %i : y[i+1] for i in range(n-k+1)})/prev[k][1] for k in range(1,n))/(y[1]**n);
+        return (ele.numerator(), ele.denominator());
+
+@cached_function
+def Exponential_polynomials(n, parent):
+    if(n <= 0):
+        raise ValueError("No Exponential polynomial can be computed for non-positive index");
+    elif(not(is_InfinitePolynomialRing(parent))):
+        return Exponential_polynomials(n, InfinitePolynomialRing(parent, "y"));
+    
+    y = parent.gens()[0];
+    
+    if(n == 1):
+        return y[0];
+    else:
+        prev = Exponential_polynomials(n-1,parent);
+        return infinite_derivative(prev) + y[0]*prev;
+    
+@cached_function
+def Logarithmic_polynomials(n, parent):
+    if(n < 0):
+        raise ValueError("No Logarithmic polynomial can be computed for non-positive index");
+    elif(not(is_InfinitePolynomialRing(parent))):
+        return Logarithmic_polynomials(n, InfinitePolynomialRing(parent, "y"));
+    
+    y = parent.gens()[0];
+    
+    if(n == 0):
+        return y[1]/y[0];
+    else:
+        return infinite_derivative(Logarithmic_polynomials(n-1, parent));
+    
+def is_Riccati(poly):
+    '''
+        Method that checks if a non-linear differential equation is of Riccatti type 
+        and returns the change of coordinates and the linear differential equation in case
+        we can.
+    '''
+    poly = fromFinitePolynomial_toInfinitePolynomial(poly);
+    parent = poly.parent(); y = parent.gens()[0];
+    
+    var = poly.variables(); index = [get_InfinitePolynomialRingGen(parent, v, True) for v in var];
+    to_plug = [Logarithmic_polynomials(i, parent) for i in index];
+    
+    new_poly = poly(**{str(var[i]): to_plug[i] for i in range(len(var))});
+    
+    if(new_poly.degree() <= 1):
+        return new_poly, y[1]/y[0];
+    
+    return None;
+    
+###################################################################################################
+### Private functions
+###################################################################################################
+def __dprint(obj, _debug):
+    if(_debug): print obj;
+    
+####################################################################################################
+#### PACKAGE ENVIRONMENT VARIABLES
+####################################################################################################
+__all__ = ["is_InfinitePolynomialRing", "get_InfinitePolynomialRingGen", "infinite_derivative", "diffalg_reduction", "toDifferentiallyAlgebraic_Below", "diff_to_diffalg", "inverse_DA", "func_inverse_DA", "guess_DA_DDfinite", "guess_homogeneous_DNfinite", "FaaDiBruno_polynomials", "Exponential_polynomials", "Logarithmic_polynomials", "is_Riccati"];
+
+# Extra functions for debugging
+def simplify_coefficients(base, coeffs, derivatives, _debug=True):
+    return __simplify_coefficients(base,coeffs,derivatives,_debug);
+
+def simplify_derivatives(derivatives, _debug=True):
+    return __simplify_derivatives(derivatives, _debug);
+    
+def find_relation(g,df, _debug=True):
+    return __find_relation(g,df,_debug);
+    
+def find_linear_relation(f,g):
+    return __find_linear_relation(g,df);
+    
+def build_derivation_matrix(gens, coeffs, drelations, _debug=True):
+    return __build_derivation_matrix(gens, coeffs, drelations, _debug);
+
+def build_vector(coeffs, monomials, graph, drelations, cR, _debug=True):
+    return __build_vector(coeffs, monomials, graph, drelations, cR, _debug);
+
+__all__ += ["simplify_coefficients", "simplify_derivatives", "find_relation", "find_linear_relation", "build_derivation_matrix", "build_vector"];
diff --git a/ajpastor/dd_functions/toDiffAlgebraic.py.jd2153 b/ajpastor/dd_functions/toDiffAlgebraic.py.jd2153
new file mode 100644
index 0000000..ec1c85d
--- /dev/null
+++ b/ajpastor/dd_functions/toDiffAlgebraic.py.jd2153
@@ -0,0 +1,1187 @@
+   
+from sage.all_cmdline import *   # import sage library
+
+from sage.graphs.digraph import DiGraph;
+
+from sage.rings.polynomial.polynomial_ring import is_PolynomialRing; 
+from sage.rings.polynomial.multi_polynomial_ring import is_MPolynomialRing;
+from sage.rings.fraction_field import is_FractionField;
+
+from ajpastor.dd_functions.ddFunction import *;
+from ajpastor.dd_functions.ddExamples import *;
+from sage.rings.polynomial.infinite_polynomial_ring import InfinitePolynomialRing;
+
+from ajpastor.misc.matrix import matrix_of_dMovement;
+from ajpastor.misc.matrix import differential_movement;
+
+################################################################################
+###
+### Utilities functions for InfinitePolynomialRings
+###
+################################################################################
+def is_InfinitePolynomialRing(parent):
+    from sage.rings.polynomial.infinite_polynomial_ring import InfinitePolynomialRing_dense as dense;                                                                 
+    from sage.rings.polynomial.infinite_polynomial_ring import InfinitePolynomialRing_sparse as sparse;                                
+    
+    return isinstance(parent, dense) or isinstance(parent, sparse);
+    
+def get_InfinitePolynomialRingGen(parent, var, with_number = False):
+    for gen in parent.gens():
+        spl = str(var).split(repr(gen)[:-1]);
+        if(len(spl) == 2):
+            if(with_number):
+                return (gen, int(spl[1]));
+            else:
+                return gen;
+    return None;
+    
+#### TODO: Review this function
+def infinite_derivative(element, times=1, var=x, _infinite=True):
+    '''
+        Method that takes an element and compute its times-th derivative
+        with respect to the variable given by var.
+        
+        If the element is not in a InfinitePolynomialRing then the method
+        'derivative' of the object will be called. Otherwise this method
+        returns the derivative following the rules:
+            - The coefficients of the polynomial ring are differentiated using
+            a recursive call to this method (usually this leads to a call of the
+            method '.derivative' of that coefficient).
+            - Any variable "#_n" appearing in the InfinitePolynomialRing has
+            as derivative the variable "#_{n+1}", i.e., the same name but with
+            one higher index.
+    '''
+    ## Checking the input _infinite
+    if(_infinite):
+        r_method = fromFinitePolynomial_toInfinitePolynomial;
+    else:
+        r_method = fromInfinityPolynomial_toFinitePolynomial;
+    
+    if(not ((times in ZZ) and times >=0)):
+        raise ValueError("The argument 'times' must be a non-negative integer");
+    elif(times > 1): # Recursive call
+        return infinite_derivative(infinite_derivative(element, times-1))
+    elif(times == 0): # Empty call
+        return element;
+        
+    ## Call with times = 1
+    try:
+        parent = element.parent();
+    except AttributeError:
+        return 0;
+    
+    ##Simple call: parent is a Fraction Field
+    if(is_FractionField(parent)):
+        parent = parent.base();
+        n = parent(element.numerator()); dn = infinite_derivative(n);
+        d = parent(element.denominator()); dd = infinite_derivative(d);
+        return (n*dd - dn*d)/d**2;
+    
+    ## Simple call: not an InfinitePolynomialRing
+    if(not is_InfinitePolynomialRing(parent)):
+        try:
+            try:
+                return element.derivative(var);
+            except Exception:
+                return element.derivative();
+        except AttributeError:
+            return parent.zero();
+            
+    ## IPR call
+    ## Symbolic case?
+    if(sage.symbolic.ring.is_SymbolicExpressionRing(parent.base())):
+        raise TypeError("Base ring for the InfinitePolynomialRing not valid (found SymbolicRing)");
+    ## Zero case
+    if(element == 0):
+        return 0;
+    ### Monomial case
+    if(element.is_monomial()):
+        ## Case of degree 1 (add one to the index of the variable)
+        if(len(element.variables()) == 1 and element.degree() == 1):
+            g,n = get_InfinitePolynomialRingGen(parent, element, True);
+            return r_method(g[n+1]);
+        ## Case of higher degree
+        else:
+            # Computing the variables in the monomial and their degrees (in order)
+            variables = element.variables();
+            degrees = [element.degree(v) for v in variables];
+            variables = [parent(v) for v in variables]
+            ## Computing the common factor
+            com_factor = prod([variables[i]**(degrees[i]-1) for i in range(len(degrees))]);
+            ## Computing the derivative of each variable
+            der_variables = [infinite_derivative(parent(v)) for v in variables];
+            
+            ## Computing the particular part for each summand
+            each_sum = [];
+            for i in range(len(variables)):
+                if(degrees[i] > 0):
+                    part_prod = prod([variables[j] for j in range(len(variables)) if j != i]);
+                    each_sum += [degrees[i]*infinite_derivative(parent(variables[i]))*part_prod];
+            
+            ## Computing the final result
+            return r_method(com_factor*sum(each_sum));
+    ### Non-monomial case
+    else:
+        coefficients = element.coefficients();
+        monomials = [parent(el) for el in element.monomials()];
+        return r_method(sum([infinite_derivative(coefficients[i])*monomials[i] + coefficients[i]*infinite_derivative(monomials[i]) for i in range(len(monomials))]));
+    
+################################################################################
+###
+### Changes from and to InfinitePolynomialRings and MultivariatePolynomialRings
+###
+################################################################################
+def fromInfinityPolynomial_toFinitePolynomial(poly):
+    if(not is_InfinitePolynomialRing(poly.parent())):
+        return poly;
+    
+    gens = poly.variables();
+    base = poly.parent().base();
+    
+    parent = PolynomialRing(base, gens);
+    return parent(str(poly));
+
+def fromFinitePolynomial_toInfinitePolynomial(poly):
+    parent = poly.parent();
+    if(is_InfinitePolynomialRing(poly) and (not is_MPolynomialRing(parent))):
+        return poly;
+    
+    
+    names = [str(el) for el in poly.parent().gens()];
+    pos = names[0].rfind("_");
+    if(pos == -1):
+        return poly;
+    prefix = names[0][:pos];
+    if(not all(el.find(prefix) == 0 for el in names)):
+        return poly;
+    R = InfinitePolynomialRing(parent.base(), prefix);
+    return R(poly);
+    
+################################################################################
+###
+### Methods for computing Diff. Algebraic equations with simpler coefficients
+###
+################################################################################
+def diffalg_reduction(poly, _infinite=False, _debug=False):
+    '''
+    Method that recieves a polynomial 'poly' with variables y_0,...,y_m with coefficients in some ring DD(R) and reduce it to a polynomial
+    'result' with coefficients in R where, for any function f(x) such that poly(f) == 0, result(f) == 0.
+    
+    The algorithm works as follows:
+        1 - Collect all the coefficients and their derivatives
+        2 - Perform the simplification to each pair of coefficients
+        3 - Perform the simplification for the last derivatives of the remaining coefficients --> gens
+        4 - Build the final derivation matrix for 'gens'
+        5 - Create a vector v for represent 'poly' w.r.t. 'gens'
+        6 - Build a square matrix --> M = (v | v' | ... | v^(n))
+        7 - Return det(M)
+    
+    INPUT:
+        - poly: a polynomial with variables y_0,...,y_m. Also an infinite polynomial with variable y_* is valid.
+        - _infinite: if True the result polynomial will be an infinite polynomial. Otherwise, a finite variable polynomial will be returned.
+        - _debug: if True, the steps of the algorithm will be printed.
+    '''
+    ## Processing the input _infinite
+    if(_infinite):
+        r_method = fromFinitePolynomial_toInfinitePolynomial;
+    else:
+        r_method = fromInfinityPolynomial_toFinitePolynomial;
+        
+    ### Preprocessing the input poly
+    parent, poly, dR = __check_input(poly, _debug);
+    
+    if(not is_DDRing(dR)):
+        raise TypeError("diffalg_reduction: polynomial is not over a DDRing");
+    
+    R = dR.base();
+    
+    ## Step 1: collecting function and derivatives
+    __dprint("---- DEBUGGING PRINTING FOR diffalg_reduction", _debug);
+    __dprint("R: %s" %R, _debug);
+    __dprint("-- Starting step 1", _debug);
+    coeffs = poly.coefficients(); # List of original coefficients
+    __dprint("Coefficients: %s" %("["+reduce(lambda p,q : p+", "+q, [repr(coeff) for coeff in coeffs])+"]"), _debug);            
+        
+    ocoeffs = coeffs; # Extra variable with the same list
+    monomials = poly.monomials(); # List of monomials
+    l_of_derivatives = [[f.derivative(times=i) for i in range(f.getOrder())] for f in coeffs]; # List of derivatives for each coefficient
+    
+    ## Step 2: simplification of coefficients
+    __dprint("-- Starting step 2", _debug);
+    graph = __simplify_coefficients(R, coeffs, l_of_derivatives, _debug);
+    maximal_elements = __digraph_roots(graph, _debug);
+    
+    ## Detecting case: all coefficients are already simpler
+    if(len(maximal_elements) == 1 and maximal_elements[0] == -1 and len(graph.outgoing_edges(-1)) == len(graph.edges())):
+        __dprint("Detected simplicity: returning the same polynomial over the basic ring", _debug);
+        return r_method(InfinitePolynomialRing(R, names=["y"])(poly));
+    
+    # Reducing the considered coefficients
+    coeffs = [coeffs[i] for i in maximal_elements if i != (-1)]; 
+    l_of_derivatives = [l_of_derivatives[i] for i in maximal_elements if i != (-1)]; 
+                
+    ## Step 3: simplification with the derivatives
+    __dprint("-- Starting step 3", _debug);
+    drelations = __simplify_derivatives(l_of_derivatives, _debug);
+    
+    ## Building the vector of final generators
+    gens = [];
+    if(-1 in maximal_elements):
+        gens = [1];
+    gens = sum([l_of_derivatives[i][:drelations[i][0]] for i in range(len(coeffs))], gens);
+    S = len(gens);
+    __dprint("Resulting vector of generators: %s" %gens, _debug);
+    
+    ## Step 4: build the derivation matrix
+    __dprint("-- Starting step 4", _debug);
+    C = __build_derivation_matrix(gens, coeffs, drelations, _debug);
+    cR = InfinitePolynomialRing(C.base_ring(), names=["y"]);
+    
+    ## Step 5: build the vector v
+    __dprint("-- Starting step 5", _debug);
+    v = __build_vector(ocoeffs, monomials, graph, drelations, cR, _debug);
+    
+    ## Step 6: build the square matrix M = (v, v',...)
+    __dprint("-- Starting step 6", _debug);
+    __dprint("Computing derivatives from the vector (step 5) using the matrix (step 4)", _debug);
+    derivative = infinite_derivative;
+    M = matrix_of_dMovement(C, v, derivative, S);
+    
+    __dprint("Vector of generators:\n\t%s" %gens, _debug);
+    __dprint("Final matrix (vector in rows):\n\t%s" %(str(M.transpose()).replace('\n', '\n\t')), _debug);
+    __dprint("Returning the determinant", _debug);
+    __dprint("---- DEBUGGING PRINTING FOR diffalg_reduction (finished)", _debug);
+    return r_method(M.determinant());
+
+################################################################################
+################################################################################
+### Some private methods for diffalg_reduction
+################################################################################
+################################################################################
+
+def __check_input(poly, _debug=False):
+    '''
+    Method that check and cast the polynomial input accordingly to our standards.
+    
+    The element 'poly' must be a polynomial with variables y_0,...,y_m with coefficients in some ring
+    '''
+    parent = poly.parent();
+    if(not is_InfinitePolynomialRing(parent)):
+        if(not is_MPolynomialRing(parent)):
+            if(not is_PolynomialRing(parent)):
+                raise TypeError("__check_input: the input is not a valid polynomial. Obtained something in %s" %parent);
+            if(not str(parent.gens()[0]).startswith("y_")):
+                raise TypeError("The input is not a valid polynomial. Obtained %s but wanted something with variables y_*" %poly);
+            parent = InfinitePolynomialRing(parent.base(), "y");
+        else:
+            gens = list(parent.gens());
+            to_delete = [];
+            for gen in gens:
+                if(str(gen).startswith("y_")):
+                    to_delete += [gen];
+            if(len(to_delete) == 0):
+                raise TypeError("__check_input: the input is not a valid polynomial. Obtained %s but wanted something with variables y_*" %poly);
+            for gen in to_delete:
+                gens.remove(gen);
+            parent = InfinitePolynomialRing(PolynomialRing(parent.base(), gens), "y");
+    else:
+        if(parent.ngens() > 1 or repr(parent.gen()) != "y_*"):
+            raise TypeError("__check_input: the input is not a valid polynomial. Obtained %s but wanted something with variables y_*" %poly);
+    
+    poly = parent(poly);
+    dR = parent.base();
+    
+    return (parent, poly, dR);
+
+def __simplify_coefficients(base, coeffs, derivatives, _debug=False):
+    '''
+    Method that get the relations for the coefficients within their derivatives. The list 'coeffs' is the list to simplify
+    and ;derivatives' is a list of lists with the derivatives of each element of 'coeffs' that we will consider.
+    
+    It returns a pair (graph, relations) where:
+        - graph: a forest of the points of minimal depth with all the relations
+        - relations: a dictionary which tells for each i < j, how the ith coefficient is related with the jth coefficient
+            (see __find_relation to know how that relation is expressed).
+    '''
+    # Checking the input
+    n = len(coeffs);
+    if(len(derivatives) < n):
+        raise ValueError("__simplify_coefficients: error in size of the input 'derivatives'");
+    
+    E = range(n);
+    R = [];
+    
+    # Checking the simplest cases (coeff[i] in R)
+    simpler = [];
+    for i in range(n):
+        if(coeffs[i] in base):
+            if((-1) not in E):
+                E += [-1];
+            R += [(-1,i,(0,0,coeffs[i]))];
+            simpler += [i];
+    
+    __dprint("Simpler coefficiets (related with -1): %s" %simpler, _debug);
+    # Starting the comparison
+    for i in range(n):
+        for j in range(i+1, n):
+            # Checking (i,j)
+            if(not(j in simpler)):
+                rel = __find_relation(coeffs[j], derivatives[i], _debug);
+                if(not(rel is None)):
+                    R += [(i,j,rel)];
+                    continue;
+            # Checking (j,i)
+            if(not(i in simpler)):
+                rel = __find_relation(coeffs[i], derivatives[j], _debug);
+                if(not(rel is None)):
+                    R += [(j,i,rel)];
+    
+    # Checking if the basic case will be used because of the relations
+    if((-1) not in E and any(e[2][1][1] != 0 for e in R)):
+        E += [-1];
+    
+    # Building the graph and returning
+    __dprint("All relations: %s" %R, _debug);
+    graph = DiGraph([E,R]);
+    __dprint("Edges: %s" %graph.edges(), _debug);
+    
+    # Deleting spurious relations and returning
+    return __min_span_tree(graph, _debug);
+
+def __simplify_derivatives(derivatives, _debug=False):
+    '''
+    Method to get the relations for the derivatives of the coefficients. The argument 'derivatives' contains a list of lists for which
+    one we will see if the last elements have relations with the firsts.
+    
+    It returns a list of tuples 'res' where 'res[i][0] = k' if the kth fucntion in 'derivatives[i]' is the first element on the list
+    with relations with the previous elements and 'res[i][1]' is the relation found.
+    If no relation is found, a tuple (None,None) is added to the list.
+    '''
+    res = [];
+    for i in range(len(derivatives)):
+        to_add = (None,None);
+        for k in range(len(derivatives[i])-1,0,-1):
+            relation = __find_relation(derivatives[i][k], derivatives[i][:k-1], _debug);
+            if(not (relation is None)):
+                to_add = (k,relation);
+                break;
+        res += [to_add];
+            
+    __dprint("Found relations with derivatives: %s" %res, _debug);
+    return res;
+    
+def __find_relation(g,df, _debug=False):
+    '''
+    Method that get the relation between g and the list df. 
+    
+    It returns a tuple (k,res) where:
+        - k: the first index which we found a relation
+        - res: the relation (see __find_linear_relation to see how such relation is expressed).
+        
+    Then the following statement is always True:
+        g == df[k]*res[0] + res[1]
+    '''
+    __dprint("** Checking relations between %s and %s" %(repr(g), df), _debug);
+    for k in range(len(df)):
+        res = __find_linear_relation(g,df[k]);
+        if(not (res is None)):
+            __dprint("Found relation:\n\t(%s) == [%s]*(%s) + [%s]" %(repr(g),repr(res[0]), repr(df[k]), repr(res[1])), _debug);
+            __dprint("*************************", _debug);
+            return (k,res);
+    __dprint("Nothing found\n*************************", _debug);
+    
+    return None;
+    
+def __find_linear_relation(f,g):
+    '''
+    This method receives two DDFunctions and return two constants (c,d) such that f = cg+d. 
+    None is return if those constants do not exist.
+    '''
+    if(not (is_DDFunction(f) and is_DDFunction(g))):
+        raise TypeError("find_linear_relation: The functions are not DDFunctions");
+    
+    ## Simplest case: some of the functions are constants is a constant
+    if(f.is_constant):
+        return (f.parent().zero(), f(0));
+    elif(g.is_constant):
+        return None;
+    
+    try:
+        of = 1; og = 1;
+        while(f.getSequenceElement(of) == 0): of += 1;
+        while(g.getSequenceElement(og) == 0): og += 1;
+        
+        if(of == og):
+            c = f.getSequenceElement(of)/g.getSequenceElement(og);
+            d = f(0) - c*g(0);
+            
+            if(f == c*g + d):
+                return (c,d);
+    except Exception:
+        pass;
+    
+    return None;
+    
+def __min_span_tree(digraph, _debug=False):
+    '''
+    Method that computes a forest from a directed graph containing all vertices and having the minimal depth.
+    
+    It is computed with a breadth first search.
+    '''
+    to_search = __digraph_roots(digraph, _debug);
+    i = 0;
+    done = [];
+    edges = [];
+    while(i < len(to_search)):
+        v = to_search[i];
+        if(not (v in done)):
+            for e in digraph.outgoing_edges(v):
+                if(not(e[1] in to_search)):
+                    to_search += [e[1]];
+                    edges += [e];
+            done += [v];
+        i += 1;
+    return DiGraph([digraph.vertices(),edges]);
+
+def __digraph_roots(digraph, _debug=False):
+    '''
+    Method that computes the roots of a directed graph. We consider a vertex v to be a root if there
+    are not edges (v,w) in the graph.
+    '''
+    return [v for v in digraph.vertices() if digraph.in_degree(v) == 0];
+    
+def __build_derivation_matrix(gens, coeffs, drelations, _debug=False):
+    '''
+        Method to build a derivation matrix 'C' for the vector of generators 'gen'.
+        
+        All the information about the coefficients and their relations are given
+        by the arguments 'coeffs' (list of basic elements for 'gens') and 
+        'drelations' (relations between elements in 'coeffs' and the last derivative
+        on 'gens').
+        
+        It could be that gens[0] == 1. This element must be treated in a specific
+        way collecting all the relations on 'drelations'. 
+    '''
+    __dprint("Building derivation matrix for %s" %gens, _debug);
+    
+    rows = []; # Variable for the rows of the matrix M
+    
+    constant = (gens[0] == 1); # Flag variable for gens[0] == 1
+    coeff_order = []; # Collecting the real order of each coefficient in the vector space
+    for i in range(len(coeffs)):
+        if(drelations[i][0] is None):
+            coeff_order += [coeffs[i].getOrder()];
+        else:
+            coeff_order += [drelations[i][0]];
+    coeff_index = [0]; # Variable for the index of coeff[i] in gens
+    if(constant): coeff_index[0] = 1;
+    for i in range(1, len(coeffs)): coeff_index += [coeff_index[-1]+coeff_order[i-1]];
+    
+    __dprint("Is there constant: %s" %constant, _debug);
+    __dprint("Orders: %s\nIndices: %s" %(coeff_order, coeff_index), _debug);
+    
+    ## Considering the case that gens[0] == -1
+    if(constant):
+        new_row = [0];
+        for i in range(len(coeffs)):
+            new_row += [0 for j in range(coeff_order[i])];
+            if(not (drelations[i][1] is None)):
+                new_row[-1] = drelations[i][1][1][1];
+        rows += [new_row];
+        
+    ## Now we loop through all the coefficients in coeffs
+    for i in range(len(coeffs)):
+        com = coeffs[i].equation.companion();
+        for j in range(coeff_order[i]):
+            ## Building the jth row for coeff[i]
+            new_row = [0 for k in range(coeff_index[i])]; # First columns for the row
+            
+            ## Middle columns for the row
+            new_row += [com[j][k] for k in range(coeff_order[i]-1)];
+            if((not (drelations[i][0] is None)) and (drelations[i][1][0] == j)):
+                new_row += [drelations[i][1][1][0]];
+            else:
+                new_row += [com[j][coeff_order[i]-1]];
+            
+            new_row += [0 for k in range(coeff_index[i]+coeff_order[i], len(gens))]; # Last columns for the row
+            
+            ## Adding the resulting row
+            rows += [new_row];
+            
+    rows = Matrix(rows);
+    
+    __dprint("** Matrix:\n%s\n*******" %rows, _debug);
+    
+    return rows;
+
+def __build_vector(coeffs, monomials, graph, drelations, cR, _debug=False):
+    '''
+        This method builds a vector that represent the polynomial generated by
+        'coeffs' and 'monomials' using the relations contained in 'graph'
+        and 'drelations' w.r.t. the vector 'gens'.
+        
+        The output vector v satisfies:
+            sum(coeffs[i]*monomials[i]) == sum(v[j]*gens[j])
+    '''
+    # Computing important data for each coefficient
+    maximal_elements = [el for el in __digraph_roots(graph) if el != -1];
+    coeff_order = [];
+    for i in range(len(coeffs)):
+        if(i in maximal_elements and (not (drelations[maximal_elements.index(i)][0] is None))):
+            coeff_order += [drelations[maximal_elements.index(i)][0]];
+        else:
+            coeff_order += [coeffs[i].getOrder()];
+    constant = (-1 in graph.vertices());
+    
+    __dprint("Is there constant: %s" %constant, _debug);
+    
+    # Computing vectors and companion matrices for each element
+    vectors = [vector(cR, [monomials[i]] + [0 for i in range(coeff_order[i]-1)]) for i in range(len(coeffs))];
+    trans = [];
+    for i in range(len(coeffs)):
+        if(i in maximal_elements and (not (drelations[maximal_elements.index(i)][0] is None))):
+            relation = drelations[maximal_elements.index(i)];
+            C = [];
+            for j in range(relation[0]):
+                new_row = [el for el in coeffs[i].equation.companion()[j][:relation[0]-1]] + [0];
+                if(j == relation[1][0]):
+                    new_row[-1] = relation[1][1][0];
+                C += [new_row];
+            C = Matrix(C);
+            trans += [(drelations[maximal_elements.index(i)][1][1][1], C)];
+        else:
+            trans += [(0, coeffs[i].equation.companion())];
+            
+    if(_debug):
+        print "--- Transitions to nodes:";
+        for i in range(len(trans)):
+            print "+++ %s --> Cosntant: %s; Matrix:\n%s" %(i, trans[i][0], trans[i][1]);
+        print "-------------------------";
+    
+    const_val = 0;    
+    # Doing a Tree Transversal in POstorder in the graph to pull up the vectors
+    # Case for -1
+    if(constant):
+        maximal_elements = [-1] + maximal_elements;
+    #    const_val = sum(cR(coeffs[e[1]])*monomials[coeffs[e[1]]] for e in graph.outgoing_edges(-1));
+    stack = copy(maximal_elements); stack.reverse();
+        
+    # Rest of the graph
+    ready = [];
+    while(len(stack) > 0):
+        current = stack[-1];
+        if(current in ready):
+            __dprint("Visiting node %s" %current, _debug);
+            if(not (current in maximal_elements)):
+                # Visit the node: pull-up the vector
+                edge = graph.incoming_edges(current)[0];
+                cu_vector = vectors[current];
+                
+                ## Constant case; reducing to the constant term
+                if(edge[0] == -1):
+                    __dprint("** Reducing node %d to constant" %(current), _debug);
+                    __dprint("\t- Current vector: %s" %(cu_vector), _debug);
+                    __dprint("\t- Prev. constant: %s" %const_val, _debug);
+                    const_val += sum(cu_vector[i]*cR(coeffs[current].derivative(times=i)) for i in range(len(cu_vector)));
+                    __dprint("\t- New constant: %s" %const_val, _debug);
+                
+                ## General case
+                else:                
+                    de_vector = vectors[edge[0]];
+                    relation = edge[2];
+                    C = trans[edge[0]][1];
+                    
+                    __dprint("** Reducing node %d to %d" %(current, edge[0]), _debug);
+                    __dprint("\t- Current vector: %s" %(cu_vector), _debug);
+                    __dprint("\t- Destiny vector: %s" %(de_vector), _debug);
+                    __dprint("\t- Relation: %s" %(str(relation)), _debug);
+                    __dprint("\t- Matrix:\n\t\t%s" %(str(C).replace('\n','\n\t\t')), _debug); 
+                    __dprint("\t- Prev. constant: %s" %const_val, _debug);
+                                
+                    # Building vectors for all the required derivatives
+                    ivectors = [vector(cR, [0 for i in range(relation[0])] + [1] + [0 for i in range(relation[0]+1,coeff_order[edge[0]])])];
+                    extra_cons = [relation[1][1]];
+                    
+                    for i in range(len(cu_vector)):
+                        extra_cons += [ivectors[-1][-1]*trans[edge[0]][0]];
+                        ivectors += [differential_movement(C, ivectors[-1], infinite_derivative)];
+                    
+                    vectors[edge[0]] = sum([vector(cR,[cu_vector[i]*ivectors[i][j] for j in range(len(ivectors[i]))]) for i in range(len(cu_vector))], de_vector);
+                    const_val += sum([cu_vector[i]*extra_cons[i] for i in range(len(cu_vector))]);
+                    
+                    __dprint("\t- New vector: %s" %vectors[edge[0]], _debug);
+                    __dprint("\t- New constant: %s" %const_val, _debug);
+                    __dprint("*************************", _debug);
+                
+            # Getting out the element of the stack
+            stack.pop();
+        else:
+            # Add the children and visit them
+            ready += [current];
+            for e in graph.outgoing_edges(current):
+                stack.append(e[1]);
+        
+    final_vector = [];
+    for i in maximal_elements:
+        if(i == -1):
+            final_vector += [const_val];
+        else:
+            final_vector += [el for el in vectors[i]];
+            
+    final_vector = vector(cR, final_vector);
+    __dprint("Vector: %s" %final_vector, _debug);
+    
+    return vector(final_vector);
+    
+    
+################################################################################
+################################################################################
+################################################################################
+
+def toDifferentiallyAlgebraic_Below(poly, _infinite=False, _debug=False):
+    '''
+    Method that receives a polynomial with variables y_0,...,y_m with coefficients in some ring DD(R) and reduce it to a polynomial
+    with coefficients in R.
+    
+    The optional input '_debug' allow the user to print extra information as the matrix that the determinant is computed.
+    '''
+    ## Processing the input _infinite
+    if(_infinite):
+        r_method = fromFinitePolynomial_toInfinitePolynomial;
+    else:
+        r_method = fromInfinityPolynomial_toFinitePolynomial;
+    
+    ### Preprocessing the input
+    parent = poly.parent();
+    if(not is_InfinitePolynomialRing(parent)):
+        if(not is_MPolynomialRing(parent)):
+            if(not is_PolynomialRing(parent)):
+                raise TypeError("The input is not a valid polynomial. Obtained something in %s" %parent);
+            if(not str(parent.gens()[0]).startswith("y_")):
+                raise TypeError("The input is not a valid polynomial. Obtained %s but wanted something with variables y_*" %poly);
+            parent = InfinitePolynomialRing(parent.base(), "y");
+        else:
+            gens = list(parent.gens());
+            to_delete = [];
+            for gen in gens:
+                if(str(gen).startswith("y_")):
+                    to_delete += [gen];
+            if(len(to_delete) == 0):
+                raise TypeError("The input is not a valid polynomial. Obtained %s but wanted something with variables y_*" %poly);
+            for gen in to_delete:
+                gens.remove(gen);
+            parent = InfinitePolynomialRing(PolynomialRing(parent.base(), gens), "y");
+    else:
+        if(parent.ngens() > 1 or repr(parent.gen()) != "y_*"):
+            raise TypeError("The input is not a valid polynomial. Obtained %s but wanted something with variables y_*" %poly);
+    
+    poly = parent(poly);
+    
+    ### Now poly is in a InfinitePolynomialRing with one generator "y_*" and some base ring.
+    if(not isinstance(parent.base(), DDRing)):
+        print "The base ring is not a DDRing. Returning the original polynomial (reached the bottom)";
+        return r_method(poly);
+    
+    up_ddring = parent.base()
+    dw_ddring = up_ddring.base();
+    goal_ring = InfinitePolynomialRing(dw_ddring, "y");
+    
+    coefficients = poly.coefficients();
+    monomials = poly.monomials();
+    
+    ## Arraging the coefficients and organize the vector-space notation
+    dict_to_derivatives = {};
+    dict_to_vectors = {};
+    S = 0;
+    for i in range(len(coefficients)):
+        coeff = coefficients[i]; monomial = goal_ring(str(monomials[i]));
+        if(coeff in dw_ddring):
+            if(1 not in dict_to_vectors):
+                S += 1;
+                dict_to_vectors[1] = goal_ring.zero();
+            dict_to_vectors[1] += dw_ddring(coeff)*monomial;
+        else:
+            used = False;
+            for el in dict_to_derivatives:
+                try:
+                    index = dict_to_derivatives[el].index(coeff);
+                    dict_to_vectors[el][index] += monomial;
+                    used = True;
+                    break;
+                except ValueError:
+                    pass;                  
+            if(not used):
+                list_of_derivatives = [coeff]; 
+                for j in range(coeff.getOrder()-1):
+                    list_of_derivatives += [list_of_derivatives[-1].derivative()];
+                dict_to_derivatives[coeff] = list_of_derivatives;
+                dict_to_vectors[coeff] = [monomial] + [0 for j in range(coeff.getOrder()-1)];
+                S += coeff.getOrder();
+    
+    rows = [];
+    for el in dict_to_vectors:
+        if(el == 1):
+            row = [dict_to_vectors[el]];
+            for i in range(S-1):
+                row += [infinite_derivative(row[-1])];
+            rows += [row];
+        else:
+            C = el.equation.companion();
+            C = Matrix(goal_ring.base(),[[goal_ring.base()(row_el) for row_el in row] for row in C]);
+            rows += [row for row in matrix_of_dMovement(C, vector(goal_ring, dict_to_vectors[el]), infinite_derivative, S)];
+        
+    M = Matrix(rows);
+    if(_debug): print M;
+    return r_method(M.determinant().numerator());
+
+def diff_to_diffalg(func, _infinite=False, _debug=False):
+    '''
+        Method that compute an differentially algebraic equation for the obect func.
+        This objet may be any element in QQ(x) or an element in some DDRing. In the first case
+        the equation returned is the simple "y_0 - func". In the latter, we compute the differential
+        equation using the linear differential representation of the obect.
+        
+        The result is always a polynomial with variables "y_*" where the number in the index
+        represent the derivative.
+    
+        The optional argument _debug allows the user to see during execution more data of the computation.
+    '''
+    ## Processing the input _infinite
+    if(_infinite):
+        r_method = fromFinitePolynomial_toInfinitePolynomial;
+    else:
+        r_method = fromInfinityPolynomial_toFinitePolynomial;
+    
+    ## Computations
+    try:
+        parent = func.parent();
+    except AttributeError:
+        return r_method(PolynomialRing(QQ,"y_0")("y_0 + %s" %func));
+    
+    if(isinstance(parent, DDRing)):
+        R = InfinitePolynomialRing(parent.base(), "y");
+        p = sum([R("y_%d" %i)*func[i] for i in range(func.getOrder()+1)], R.zero());
+        for i in range(parent.depth()-1):
+            p = toDifferentiallyAlgebraic_Below(p, _infinite=True, _debug=_debug);
+        return r_method(p);
+    else:
+        R = PolynomialRing(PolynomialRing(QQ,x).fraction_field, "y_0");
+        return r_method(R.gens()[0] - func);
+    
+################################################################################
+###
+### Methods for computing Diff. Algebraic equation for inverses
+### 
+###     - Multiplicative inverse from DA equation
+###     - Functional inverse from Dn-finite equation
+###
+################################################################################
+def inverse_DA(poly, vars=None, _infinite=False):
+    '''
+        Method that computes the DA equation for the multiplicative inverse 
+        of the solutions of a DA equation.
+        
+        We assume that the variables given by vars are the variables representing
+        the solution, namely vars[i+1] = vars[i].derivative().
+        If vars is not provided, we take all the variables from the polynomial
+        that is given.
+    '''
+    ## Processing the input _infinite
+    if(_infinite):
+        r_method = fromFinitePolynomial_toInfinitePolynomial;
+    else:
+        r_method = fromInfinityPolynomial_toFinitePolynomial;
+        
+    ## Checking that poly is a polynomial
+    
+    parent = poly.parent();
+    if(is_FractionField(parent)):
+        parent = parent.base();
+    if(is_InfinitePolynomialRing(parent)):
+        poly = fromInfinityPolynomial_toFinitePolynomial(poly);
+        return inverse_DA(poly, vars, _infinite=_infinite);
+    if(not (is_PolynomialRing(parent) or is_MPolynomialRing(parent))):
+        raise TypeError("No polynomial is given");
+    poly = parent(poly);
+    
+    ## Getting the list of variables
+    if(vars is None):
+        g = list(poly.parent().gens()); g.reverse();
+    else:
+        if(any(v not in poly.parent().gens())):
+            raise TypeError("The variables given are not in the polynomial ring");
+        g = vars;
+        
+    ## Computing the derivative of the inverse using Faa di Bruno's formula
+    derivatives = [1/g[0]] + [sum((falling_factorial(-1, k)/g[0]**(k+1))*bell_polynomial(n,k)(*g[1:n-k+2]) for k in range(1,n+1)) for n in range(1,len(g
+))];
+    
+    ## Computing the final equation
+    return r_method(poly(**{str(g[i]): derivatives[i] for i in range(len(g))}).numerator());
+
+def func_inverse_DA(func, _inifnite=False, _debug=False):
+    '''
+        TODO: Add documentation
+    '''
+    ## Processing the input _infinite
+    if(_infinite):
+        r_method = fromFinitePolynomial_toInfinitePolynomial;
+    else:
+        r_method = fromInfinityPolynomial_toFinitePolynomial;
+        
+    ## Computations
+    equation = fromFinitePolynomial_toInfinitePolynomial(diff_to_diffalg(func, _debug=_debug));
+    
+    if(_debug): print equation;
+    
+    parent = equation.parent();
+    y = parent.gens()[0];
+    
+    n = len(equation.variables());
+        
+    quot = [FaaDiBruno_polynomials(i, parent)[0]/FaaDiBruno_polynomials(i,parent)[1] for i in range(n)];
+    coeffs = [el(**{str(parent.base().gens()[0]) : y[0]}) for el in equation.coefficients()];
+    monomials = [el(**{str(y[j]) : quot[j] for j in range(n)}) for el in equation.monomials()];
+    
+    if(_debug):
+        print monomials;
+        print coeffs;
+    
+    sol = sum(monomials[i]*coeffs[i] for i in range(len(monomials)));
+    
+    if(_debug): print sol;
+    if(hasattr(sol, "numerator")):
+        return parent(sol.numerator());
+    return r_method(parent(sol));
+
+
+################################################################################
+###
+### Methods for compute Dn-finite equations from DA-equations
+###
+################################################################################
+def guess_Dnfinite(poly, init):
+    ## TODO: Method not public
+    res = guess_homogeneous_DNfinite(poly, init);
+    if(is_DDFunction(res)):
+        return res;
+    return guess_DA_DDfinite(poly, init);
+
+def guess_DA_DDfinite(poly, init=[], bS=None, bd = None, all=True):
+    '''
+        Method that tries to compute a DD-finite differential equation
+        for a DA-function with constant coefficients.
+        
+        It just tries all the possibilities. It uses the Composition class
+        to get the possibilities of the orders for the coefficients.
+        
+        INPUT:
+            - poly: polynomial with the differential equation we want to mimic with DD-finite elements
+            - init: initial values of the solution of poly
+            - bS: bound to the sum of orders in the DD-finite equation
+            - db: bound to the order of the DD-finite equation
+            - all: compute all the posibles equations
+    '''
+    poly = fromInfinityPolynomial_toFinitePolynomial(poly);
+    
+    if(not poly.is_homogeneous()):
+        raise TypeError("We require a homogeneous polynomial");
+    
+    if(bS is None):
+        S = poly.degree(); 
+    else:
+        S = bS;
+    if(bd is None):
+        d = S - poly.parent().ngens() + 2;
+    else:
+        d = bd;
+    
+    compositions = Compositions(S, length = d+1);
+    coeffs = [["a_%d_%d" %(i,j) for j in range(S)] for i in range(d+1)];
+    cInit = [["i_%d_%d" %(i,j) for j in range(S-1)] for i in range(d+1)];
+    R = ParametrizedDDRing(DFinite, sum(coeffs, [])+sum(cInit,[]));
+    R2 = R.to_depth(2);
+    coeffs = [[R.parameter(coeffs[i][j]) for j in range(S)] for i in range(d+1)];
+    cInit = [[R.parameter(cInit[i][j]) for j in range(S-1)] for i in range(d+1)];
+    sols = [];
+    
+    for c in compositions:
+        ## Building the parametrized guess
+        e = [R.element([coeffs[i][j] for j in range(c[i])] + [1],cInit[i]) for i in range(d+1)];
+        f = R2.element(e);
+        if(len(init) > 0):
+            try:
+                f = f.change_init_values([init[i] for i in range(min(len(init), f.equation.jp_value+1))]);
+            except ValueError:
+                continue;
+        
+        ## Computing the DA equation
+        poly2 = diff_to_diffalg(f);
+        
+        ###############################################
+        ### Comparing the algebraic equations
+        ## Getting a possible constant difference
+        m = poly.monomials();
+        m2 = poly2.monomials();
+        eqs = [];
+        C = None;
+        for mon in m2:
+            c2 = poly2.coefficient(mon);
+            if(poly2.parent().base()(poly2.coefficient(mon)).is_constant()):
+                C = poly.coefficient(poly.parent()(mon))/c2;
+                break;
+        if(C is None):
+            C = 1;
+        elif(C == 0):
+            continue;
+        
+        ### Building all equations from the algebraic equation
+        for mon in m2:
+            c2 = poly2.coefficient(mon); c = poly.coefficient(poly.parent()(mon));
+            eqs += [C*c2 - c];
+        
+        ##################################################
+        ### Comparing the initial values
+        for i in range(min(len(init), f.equation.jp_value+1),len(init)):
+            try:
+                eqs += [f.getInitialValue(i) - init[i]];
+            except TypeError:
+                break; ## Case when no initial value can be computed
+            
+        ##################################################
+        ### Solving the system (groebner basis)
+        P = R.base_field.base(); 
+        fEqs = [];
+        for el in eqs:
+            if(el.parent().is_field()):
+                fEqs += [P(str(el.numerator()))];
+            else:
+                fEqs += [P(str(el))];
+                
+        gb = ideal(fEqs).groebner_basis();
+        if(1 not in gb):
+            sols += [(f,gb)];
+            if(not all):
+                break;
+        
+        
+    if(len(sols) > 0 and not all):
+        return sols[0];
+    
+    return sols;
+
+def guess_homogeneous_DNfinite(poly, init):
+    '''
+        Method that tries to compute a Dn-finite differential equation
+        for a DA-function defined from an homogeneous equation.
+        
+        This method simplifies the equation supposing that the solution
+        is of the form y(x) = exp(int(u(x))), getting a differential equation
+        for u(x) of less order.
+        
+        If we end up with a linear equation, then we can return a Dn-finite function
+        where n depends on how many iterations we needed.
+        
+        If the final algebraic equation is not linear, then we return this
+        last equation together with the number of iterations done.
+        
+        INPUT:
+            - poly: polynomial with the differential equation we want to mimic with DD-finite elements
+            - init: initial values of the solution of poly (must be enough).
+    '''
+    new_poly, depth = simplify_homogeneous(poly);
+    parent = new_poly.parent(); base = parent.base(); y = parent.gens()[0];
+    
+    order = max(get_InfinitePolynomialRingGen(parent, v, True)[1] for v in new_poly.variables());
+    
+    if(new_poly.degree() == 1 and new_poly.is_homogeneous()): ## We can build something
+        coeffs = [];
+        for i in range(order+1):
+            if(y[i] in new_poly.variables()):
+                coeffs += new_poly.coefficients()[new_poly.monomials().index(y[i])];
+            else:
+                coeffs += [0];
+        
+        inhom = new_poly.constant_coefficient();
+        if(is_DDRing(base)):
+            base_field = base.base_field;
+            base = base.to_depth(base.depth()+1);
+        else:
+            base_field = base.fraction_field();
+            base = DDRing(PolynomialRing(base_field, 'x'));
+        
+        ## We can build a D(depth+1)-finite function
+        deep_init = build_initial_from_homogeneous(tuple(init), order, depth, base_field);
+                
+        res = base.element(coeffs, deep_init, inhomogeneous=inhom);
+        for i in range(depth):
+            res = Exp(res.integrate(0));
+        return res;
+    
+    else:
+        return (new_poly, depth);
+    
+
+@cached_function
+def build_initial_from_homogeneous(init, n, depth, base):
+    ## Checking we have enough data
+    if(len(init) < n+depth):
+        raise TypeError("Not enought initial data was provided (required %d)" %(n+depth));
+    elif(len(init) > n+depth):
+        return build_initial_from_homogeneous(tuple(list(init)[:n+depth]), n, depth, base);
+    
+    ## Base case with depth = 0
+    if(depth == 0):
+        return init[:n];
+    
+    ## Casting the input to the desired ring
+    init = [base(el) for el in init];
+    
+    ## Checking the initial condition init[0]
+    if(init[0] == 0):
+        raise ValueError("Initial condition is zero. Impossible to compute a solution of the form e(int(u(x)))");
+    
+    ## Computing the initial conditions for depth "depth-1"
+    new_init = [init[1]/init[0]];
+    
+    parent = Exponential_polynomials(1,base).parent(); y = parent.gens()[0];
+    
+    for i in range(1,n+depth-1):
+        poly = fromInfinityPolynomial_toFinitePolynomial(-(Exponential_polynomials(i+1, parent) - y[i]));
+        new_init += [base(poly(**{str(y[j]) : new_init[j] for j in range(i)})) + init[i+1]/init[0]];
+        
+    ## Returning the recurive call with one less depth
+    return build_initial_from_homogeneous(tuple(new_init), n, depth-1, base);
+
+@cached_function
+def simplify_homogeneous(poly, _stop_linear=True):
+    '''
+        Given an homogeneous differential polynomial 'poly', we reduce the order of the equation by 1
+        using the change of variables y(x) = exp(int(u(x))).
+        
+        This leads to a differentially algebraic equation of order 1 less than 'poly'. If we obtain a 
+        new homogeneous equation, we can iterate. The return of this function is this final result toether
+        with the number of steps performed.
+    '''
+    poly = fromFinitePolynomial_toInfinitePolynomial(poly);
+    parent = poly.parent(); y = parent.gens()[0];
+    
+    if(not(poly.is_homogeneous()) or (_stop_linear and poly.degree() == 1)):
+        return (poly,0);
+    
+    d = {str(y[0]) : parent.one()};
+    for v in poly.variables():
+        if(v != y[0]):
+            d[str(v)] = Exponential_polynomials(get_InfinitePolynomialRingGen(parent, v, True)[1], parent);
+                
+    new_poly = poly(**d);
+    result,n = simplify_homogeneous(new_poly);
+    
+    return (result, n+1);
+    
+###################################################################################################
+### Polynomial functions
+###################################################################################################
+@cached_function
+def FaaDiBruno_polynomials(n, parent):
+    if(n < 0):
+        raise ValueError("No Faa Di Bruno polynomial can be computed for negative index");
+    elif(not(is_InfinitePolynomialRing(parent))):
+        if((not(is_PolynomialRing(parent))) and (not(is_MPolynomialRing(parent)))):
+            raise TypeError("The parent ring is not valid: needed polynomial rings or InfinitePolynomialRing");
+        return FaaDiBruno_polynomials(n, InfinitePolynomialRing(parent, "y"));        
+    
+    if(parent.base().ngens() == 0 or parent.base().gens()[0] == 1):
+        raise TypeError("Needed a inner variable in the coefficient ring");
+        
+        
+    x = parent.base().gens()[0];
+    y = parent.gens()[0];
+    if(n == 0):
+        return (parent(parent.base().gens()[0]), parent.one());
+    if(n == 1):
+        return (parent.one(), y[1]);
+    else:
+        prev = [FaaDiBruno_polynomials(k, parent) for k in range(n)];
+        ele = -sum(prev[k][0]*bell_polynomial(n,k)(**{"x%d" %i : y[i+1] for i in range(n-k+1)})/prev[k][1] for k in range(1,n))/(y[1]**n);
+        return (ele.numerator(), ele.denominator());
+
+@cached_function
+def Exponential_polynomials(n, parent):
+    if(n <= 0):
+        raise ValueError("No Exponential polynomial can be computed for non-positive index");
+    elif(not(is_InfinitePolynomialRing(parent))):
+        return Exponential_polynomials(n, InfinitePolynomialRing(parent, "y"));
+    
+    y = parent.gens()[0];
+    
+    if(n == 1):
+        return y[0];
+    else:
+        prev = Exponential_polynomials(n-1,parent);
+        return infinite_derivative(prev) + y[0]*prev;
+    
+@cached_function
+def Logarithmic_polynomials(n, parent):
+    if(n < 0):
+        raise ValueError("No Logarithmic polynomial can be computed for non-positive index");
+    elif(not(is_InfinitePolynomialRing(parent))):
+        return Logarithmic_polynomials(n, InfinitePolynomialRing(parent, "y"));
+    
+    y = parent.gens()[0];
+    
+    if(n == 0):
+        return y[1]/y[0];
+    else:
+        return infinite_derivative(Logarithmic_polynomials(n-1, parent));
+    
+def is_Riccati(poly):
+    '''
+        Method that checks if a non-linear differential equation is of Riccatti type 
+        and returns the change of coordinates and the linear differential equation in case
+        we can.
+    '''
+    poly = fromFinitePolynomial_toInfinitePolynomial(poly);
+    parent = poly.parent(); y = parent.gens()[0];
+    
+    var = poly.variables(); index = [get_InfinitePolynomialRingGen(parent, v, True) for v in var];
+    to_plug = [Logarithmic_polynomials(i, parent) for i in index];
+    
+    new_poly = poly(**{str(var[i]): to_plug[i] for i in range(len(var))});
+    
+    if(new_poly.degree() <= 1):
+        return new_poly, y[1]/y[0];
+    
+    return None;
+    
+###################################################################################################
+### Private functions
+###################################################################################################
+def __dprint(obj, _debug):
+    if(_debug): print obj;
+    
+####################################################################################################
+#### PACKAGE ENVIRONMENT VARIABLES
+####################################################################################################
+__all__ = ["is_InfinitePolynomialRing", "get_InfinitePolynomialRingGen", "infinite_derivative", "diffalg_reduction", "toDifferentiallyAlgebraic_Below", "diff_to_diffalg", "inverse_DA", "func_inverse_DA", "guess_DA_DDfinite", "guess_homogeneous_DNfinite", "FaaDiBruno_polynomials", "Exponential_polynomials", "Logarithmic_polynomials", "is_Riccati"];
+
+# Extra functions for debugging
+def simplify_coefficients(base, coeffs, derivatives, _debug=True):
+    return __simplify_coefficients(base,coeffs,derivatives,_debug);
+
+def simplify_derivatives(derivatives, _debug=True):
+    return __simplify_derivatives(derivatives, _debug);
+    
+def find_relation(g,df, _debug=True):
+    return __find_relation(g,df,_debug);
+    
+def find_linear_relation(f,g):
+    return __find_linear_relation(g,df);
+    
+def build_derivation_matrix(gens, coeffs, drelations, _debug=True):
+    return __build_derivation_matrix(gens, coeffs, drelations, _debug);
+
+def build_vector(coeffs, monomials, graph, drelations, cR, _debug=True):
+    return __build_vector(coeffs, monomials, graph, drelations, cR, _debug);
+
+__all__ += ["simplify_coefficients", "simplify_derivatives", "find_relation", "find_linear_relation", "build_derivation_matrix", "build_vector"];
diff --git a/releases/diff_defined_functions.zip b/releases/diff_defined_functions.zip
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z_;&B%H^%VuuESkmRUq&`dk{dh+dYWipxx^G{=4V9XqjH1KfH<lmEaJ-_dWms2k;jj
N9RSer2ELU6{2yP>mdpSE

literal 0
HcmV?d00001

-- 
2.1.4