def x_minus_y() -> Polynomial:
x = Symbol('x')
y = Symbol('y')
term_x = PolynomialTerm(Monomial({MonomialFactor(x, 1)}), 1.0)
term_minus_y = PolynomialTerm(Monomial({MonomialFactor(y, 1)}), -1.0)
return Polynomial({term_x, term_minus_y})
def x_sq_minus_y_sq() -> Polynomial:
x = Symbol('x')
y = Symbol('y')
term_x_sq = PolynomialTerm(Monomial({MonomialFactor(x, 2)}), 1.0)
term_minus_y_sq = PolynomialTerm(Monomial({MonomialFactor(y, 2)}), -1.0)
return Polynomial({term_x_sq, term_minus_y_sq})
def two_x_plus_two_y() -> Polynomial:
x = Symbol('x')
y = Symbol('y')
term_two_x = PolynomialTerm(Monomial({MonomialFactor(x, 1)}), 2.0)
term_two_y = PolynomialTerm(Monomial({MonomialFactor(y, 1)}), 2.0)
return Polynomial({term_two_x, term_two_y})
def x_sq_plus_two_xy_plus_y_sq() -> Polynomial:
x = Symbol('x')
y = Symbol('y')
term_2xy = PolynomialTerm(Monomial({MonomialFactor(x, 1), MonomialFactor(y, 1)}), 2.0)
term_x_sq = PolynomialTerm(Monomial({MonomialFactor(x, 2)}), 1.0)
term_y_sq = PolynomialTerm(Monomial({MonomialFactor(y, 2)}), 1.0)
return Polynomial({term_x_sq, term_y_sq, term_2xy})
def first_lotke_volterra_ode() -> ODE:
# return dx/dt = x - x*y
x = Symbol('x')
y = Symbol('y')
rhs = Polynomial({
PolynomialTerm(Monomial({MonomialFactor(x, 1)}), 1),
PolynomialTerm(Monomial({MonomialFactor(x, 1),
MonomialFactor(y, 1)}), -1)
})
return ODE(x, rhs)
def second_lotke_volterra_ode() -> ODE:
# return dy/dt = x*y - y
x = Symbol('x')
y = Symbol('y')
rhs = Polynomial({
PolynomialTerm(Monomial({MonomialFactor(y, 1)}), -1),
PolynomialTerm(Monomial({MonomialFactor(x, 1),
MonomialFactor(y, 1)}), 1)
})
return ODE(y, rhs)
def test_monomial_multiplication_positive_powers() -> None:
x = Symbol('x')
y = Symbol('y')
z = Symbol('z')
first_monomial = Monomial({MonomialFactor(x, 2), MonomialFactor(y, 3), MonomialFactor(z, 4)})
second_monomial = Monomial({MonomialFactor(x, 1), MonomialFactor(y, 1), MonomialFactor(z, 1)})
assert first_monomial * second_monomial == \
Monomial({MonomialFactor(x, 3), MonomialFactor(y, 4), MonomialFactor(z, 5)})
def test_monomial_symbols() -> None:
x = Symbol('x')
y = Symbol('y')
z = Symbol('z')
Monomial({MonomialFactor(x, 2), MonomialFactor(y, 3), MonomialFactor(z, 4)})
assert Monomial({MonomialFactor(x, 2), MonomialFactor(y, 3), MonomialFactor(z, 4)}).symbols == \
{x, y, z}
assert Monomial({MonomialFactor(x, 2), MonomialFactor(y, 3)}).symbols == {x, y}
def test_monomial_multiplication_to_trivial() -> None:
x = Symbol('x')
y = Symbol('y')
z = Symbol('z')
first_monomial = Monomial({MonomialFactor(x, 2), MonomialFactor(y, -3), MonomialFactor(z, 4)})
second_monomial = Monomial({MonomialFactor(x, -2), MonomialFactor(z, -4), MonomialFactor(y, 3)})
assert first_monomial * second_monomial == Monomial({MonomialFactor(TrivialMulSymbol(), 0)})
assert second_monomial * first_monomial == TrivialMonomial()
def one_x() -> Polynomial:
x = Symbol('x')
one_x = PolynomialTerm(Monomial({MonomialFactor(x, 1)}), 1.0)
return Polynomial({one_x})
def const_one() -> Polynomial:
x = TrivialMulSymbol()
one = PolynomialTerm(Monomial({MonomialFactor(x, 0)}), 1.0)
return Polynomial({one})
def two_x() -> Polynomial:
x = Symbol('x')
term_two_x = PolynomialTerm(Monomial({MonomialFactor(x, 1)}), 2.0)
return Polynomial({term_two_x})
def one_y() -> Polynomial:
y = Symbol('y')
one_y = PolynomialTerm(Monomial({MonomialFactor(y, 1)}), 1.0)
return Polynomial({one_y})