## Step 2: simplification of coefficients
__dprint("-- Starting step 2", _debug);
- poset, relation = __simplify_coefficients(R, coeffs, l_of_derivatives, _debug);
+ poset, relations = __simplify_coefficients(R, coeffs, l_of_derivatives, _debug);
maximal_elements = poset.maximal_elements();
## Detecting case: all coefficients are already simpler
## Step 3: simplification with the derivatives
__dprint("-- Starting step 3", _debug);
- drelation = __simplify_derivatives(l_of_derivatives, _debug);
+ drelations = __simplify_derivatives(l_of_derivatives, _debug);
## Building the vector of final generators
gens = [];
if(-1 in poset):
gens = [1];
- gens = sum([l_of_derivatives[:drelation[i]] for i in range(len(coeffs))], gens);
+ 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(coeffs, drelation, _debug);
+ C = __build_derivation_matrix(gens, coeffs, drelations, _debug);
## Step 5: build the vector v
__dprint("-- Starting step 5", _debug);
- v = __build_vector(ocoeffs, monomials, poset, relation, drelation, _debug);
+ v = __build_vector(ocoeffs, monomials, poset, relations, drelations, gens, _debug);
## Step 6: build the square matrix M = (v, v',...)
__dprint("-- Starting step 6", _debug);
return (parent, poly, dR);
-def __simplify_coefficients(R, coeffs, derivatives, _debug=False):
+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.
# Checking the simplest cases (coeff[i] in R)
for i in range(n):
- if(coeffs[i] in R):
+ if(coeffs[i] in base):
if((-1) not in E):
E += [-1];
R += [(i,-1)];
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]
'''
- for k in range(len(derivatives)):
+ for k in range(len(df)):
res = __find_linear_relation(df[k], g);
if(not (res is None)):
- __dprint("Found relation:\n\t(%s) == [%s]*(%s) + [%s]" %(repr(g),repr(res[0]), repr(res[1]), repr(df[k]), _debug);
+ __dprint("Found relation:\n\t(%s) == [%s]*(%s) + [%s]" %(repr(g),repr(res[0]), repr(df[k]), repr(res[1])), _debug);
return (k,res);
return None;
raise TypeError("find_linear_relation: The functions are not DDFunctions");
try:
- of = 1; while(f.getSequenceElement(of) == 0): of += 1;
- og = 1; while(g.getSequenceElement(og) == 0): og += 1;
+ 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);
elif(g.is_constant):
return None;
-def __build_derivation_matrix(coeffs, drelations, _debug=False):
- ## TODO: Implement this
- raise NotImplementedError("__build_derivation_matrix: method not implemented");
+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];
+
+ if(_debug):
+ print "Matrix:";
+ for row in rows:
+ print row;
+
+ return Matrix(rows);
+
+def __build_vector(coeffs, monomials, poset, relations, drelations, gens, _debug=False):
+ '''
+ This method builds a vector that represent the polynomial generated by
+ 'coeffs' and 'monomials' using the relations contained in 'poset', 'relations'
+ and 'drelations' w.r.t. the vector 'gens'.
+
+ The output vector v satisfies:
+ sum(coeff[i]*monomials[i]) == sum(v[j]*gens[j])
+ '''
-def __build_vector(coeffs, monomials, poset, relation, drelation, _debug=False):
+ ## Getting the minimal chain for each coefficient up to a maximal coefficient
+ maximal_elements = poset.maximal_elements();
+ chains = {i : [el for el in poset.chains() if (len(el) > 0 and el[0] == i and (el[-1] in maximal_elements)] for i in range(len(coeffs))};
+ min_chains = [];
+ for i in range(len(coeffs)):
+ current = chains[i];
+ shortest = current[0];
+ for j in range(1,len(current)):
+ if(len(shortest) > len(current[j])):
+ shortest = current[j];
+ min_chains += [shortest];
+
+ ## For each coefficient, compute the representation in gens
+ ## We must follow the shortest chain to a maximal element to know where and
+ ## how put the coefficients in the final vector.
## TODO: Implement this
raise NotImplementedError("__build_vector: method not implemented");
####################################################################################################
#### PACKAGE ENVIRONMENT VARIABLES
####################################################################################################
-__all__ = ["is_InfinitePolynomialRing", "get_InfinitePolynomialRingGen", "get_InfinitePolynomialRingVaribale", "infinite_derivative", "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", "get_InfinitePolynomialRingVaribale", "infinite_derivative", "diffalg_reduction", "toDifferentiallyAlgebraic_Below", "diff_to_diffalg", "inverse_DA", "func_inverse_DA", "guess_DA_DDfinite", "guess_homogeneous_DNfinite", "FaaDiBruno_polynomials", "Exponential_polynomials"];
-def simplify_coefficients(R, coeffs, derivatives, _debug=True):
- return __simplify_coefficients(R,coeffs,derivatives,_debug);
+# 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_linear_relation(f,g):
return __find_linear_relation(g,df);
-def build_derivation_matrix(coeffs, drelations, _debug=True):
- return __build_derivation_matrix(coeffs, drelations, _debug);
+def build_derivation_matrix(gens, coeffs, drelations, _debug=True):
+ return __build_derivation_matrix(gens, coeffs, drelations, _debug);
+
+def build_vector(coeffs, monomials, poset, relations, drelations, gens, _debug=True):
+ return __build_vector(coeffs, monomials, poset, relations, drelations, gens, _debug);
-def build_vector(coeffs, monomials, poset, relation, drelation, _debug=True):
- return __build_vector(coeffs, monomials, poset, relation, drelation, _debug);
-# Extra functions for debuggin
__all__ += ["simplify_coefficients", "simplify_derivatives", "find_relation", "find_linear_relation", "build_derivation_matrix", "build_vector"];
projects / ajpastor / diff_defined_functions.git / commitdiff