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 _simpsol(soleq):
lhs = soleq.lhs
sol = soleq.rhs
sol = powsimp(sol)
gens = list(sol.atoms(exp))
p = Poly(sol, *gens, expand=False)
gens = [factor_terms(g) for g in gens]
if not gens:
gens = p.gens
syms = [Symbol('C1'), Symbol('C2')]
terms = []
for coeff, monom in zip(p.coeffs(), p.monoms()):
coeff = piecewise_fold(coeff)
if type(coeff) is Piecewise:
coeff = Piecewise(*((ratsimp(coef).collect(syms), cond)
for coef, cond in coeff.args))
else:
coeff = ratsimp(coeff).collect(syms)
monom = Mul(*(g**i for g, i in zip(gens, monom)))
terms.append(coeff * monom)
return Eq(lhs, Add(*terms))
