def build_orthogonal_basis(n, p):
# Build a basis from total degree monomials
monomial_basis = []
x = [Symbol('x%d' % k) for k in range(1,n+1)]
for alpha in MultiIndex(n, p):
term = 1
for j in range(n):
term *= x[j]**alpha[j]
monomial_basis.append(term)
# Now build the corresponding mass matrix
M = zeros(len(monomial_basis), len(monomial_basis))
for i, psi1 in enumerate(monomial_basis):
for j, psi2 in enumerate(monomial_basis):
if i <= j:
out = psi1*psi2
for k in range(n):
out = integrate(out, (x[k], -1,1))
M[i,j] = out
M[j,i] = out
R = M.cholesky().inv()
# Now build our orthogonal basis
basis_terms = []
basis = []
for i in range(len(monomial_basis)):
term = 0
for j, psi in enumerate(monomial_basis):
term += R[i,j]*psi
basis.append(term)
# Make the sparse version
term = Poly(term, x)
term = [ (alpha, float(term.coeff_monomial(alpha)) ) for alpha in term.monoms()]
basis_terms.append(term)
# Now build the derivatives
basis_terms_der = []
for i in range(n):
basis_terms_der_curr = []
for psi in basis:
# Make the sparse version
term = Poly(diff(psi, x[i]), x)
term = [ (alpha, float(term.coeff_monomial(alpha)) ) for alpha in term.monoms()]
basis_terms_der_curr.append(term)
basis_terms_der.append(basis_terms_der_curr)
return basis_terms, basis_terms_der
def get_matrix(self):
"""
Returns
-------
macaulay_matrix: Matrix
The Macaulay's matrix
"""
rows = []
row_coefficients = self.get_row_coefficients()
for i in range(self.n):
for multiplier in row_coefficients[i]:
coefficients = []
poly = Poly(self.polynomials[i] * multiplier, *self.variables)
for mono in self.monomial_set:
coefficients.append(poly.coeff_monomial(mono))
rows.append(coefficients)
macaulay_matrix = Matrix(rows)
return macaulay_matrix
def get_matrix(self):
"""
Returns
-------
macaulay_matrix: Matrix
The Macaulay's matrix
"""
rows = []
row_coefficients = self.get_row_coefficients()
for i in range(self.n):
for multiplier in row_coefficients[i]:
coefficients = []
poly = Poly(self.polynomials[i] * multiplier,
*self.variables)
for mono in self.monomial_set:
coefficients.append(poly.coeff_monomial(mono))
rows.append(coefficients)
macaulay_matrix = Matrix(rows)
return macaulay_matrix
