def in_chebyshev_basis(self):
res = poly.Polynomial({poly.ChebyshevVector({}): 1})
for v, p in self:
c = poly2cheb([0] * p + [1])
basis = poly.ChebyshevVector.construct_basis([v], p)
res *= poly.Polynomial(dict(zip(basis, c)))
return res
def in_monomial_basis(self):
res = poly.Polynomial({poly.MonomialVector({}): 1})
for v, p in self:
c = cheb2poly([0] * p + [1])
basis = poly.MonomialVector.construct_basis([v], p)
res *= poly.Polynomial(dict(zip(basis, c)))
return res
def make_polynomial(cls, other):
if isinstance(other, Number):
return poly.Polynomial({cls({}): other})
elif isinstance(other, poly.Variable):
return poly.Polynomial({cls({other: 1}): 1})
else:
raise TypeError(
f'cannot make a polynomial out of a {type(other).__name__}.')
def __mul__(self, cheb):
self._verify_multiplicand(cheb)
variables = set(self.variables() + cheb.variables())
prod_powers = ((self[v] + cheb[v], abs(self[v] - cheb[v]))
for v in variables)
coef = .5**len(variables)
multiplication = poly.Polynomial({})
for powers in product(*prod_powers):
cheb = ChebyshevVector(dict(zip(variables, powers)))
multiplication += poly.Polynomial({cheb: coef})
return multiplication
def derivative(self, variable):
power = self[variable]
if power == 0:
monomial = MonomialVector({})
else:
monomial = deepcopy(self)
monomial[variable] -= 1
return poly.Polynomial({monomial: power})
def substitute(self, evaluation_dict):
coefficient = prod([
self._call_univariate(self[var], val)
for var, val in evaluation_dict.items()
])
vector = type(self)(
{v: p
for v, p in self if not v in evaluation_dict})
return poly.Polynomial({vector: coefficient})
def integral(self, variable):
power = self[variable]
integral = poly.Polynomial({})
cheb = deepcopy(self)
cheb[variable] = power + 1
integral[cheb] = .5 / (1 + power)
cheb = deepcopy(self)
cheb[variable] = abs(power - 1)
integral[cheb] += .25 if power == 1 else .5 / (1 - power)
return integral
def derivative(self, variable):
'''
Uses the formula for the derivative in terms of the polynomials of the
second kind, then express the polynomial of the second kind as a sum of
polynomials of the first.
'''
power = self[variable]
derivative = poly.Polynomial({})
for q in range(power):
if power % 2 ^ q % 2:
cheb = deepcopy(self)
cheb[variable] = q
derivative[cheb] = power if q == 0 else power * 2
return derivative
def integral(self, variable):
monomial = deepcopy(self)
monomial[variable] += 1
return poly.Polynomial({monomial: 1 / (self[variable] + 1)})
def __mul__(self, monomial):
self._verify_multiplicand(monomial)
variables = set(self.variables() + monomial.variables())
monomial = MonomialVector({v: self[v] + monomial[v] for v in variables})
return poly.Polynomial({monomial: 1})