import sympy as s
from sympy import init_printing
init_printing()
# Gets the integral from x2 to x3, looking at the cubic spline interpolant from x1...x4
# Define the symbols
x1, x2, x3, x4 = s.symbols('x1 x2 x3 x4', real=True)
f1, f2, f3, f4 = s.symbols('f1 f2 f3 f4', real=True)
[docs]q = s.symbols('q') # Normalized space for a Bernstein basis.
# Mapping from x-space to q-space has x=x2 -> q=0, x=x3 -> q=1.
# q1 = q2 - hm / h
# q4 = q3 + hp / h
q1, q4 = s.symbols('q1 q4', real=True)
# Define the Bernstein basis polynomials
[docs]b2 = 3 * q * (1 - q) ** 2
[docs]b3 = 3 * q ** 2 * (1 - q)
c1, c2, c3, c4 = s.symbols('c1 c2 c3 c4', real=True)
# Can solve for c2 and c3 exactly
[docs]f = c1 * b1 + c2 * b2 + c3 * b3 + c4 * b4
[docs]f1_cubic = f.subs(q, q1)#.simplify()
[docs]f4_cubic = f.subs(q, q4)#.simplify()
# factors = [f1, f2, f3, f4]
# Solve for c2 and c3
[docs]sol = s.solve(
[
f1_cubic - f1,
f4_cubic - f4,
],
[
c2,
c3,
],
)
[docs]c2 = sol[c2].factor(factors).simplify()
[docs]c3 = sol[c3].factor(factors).simplify()
integral = (c1 + c2 + c3 + c4) / 4 # God I love Bernstein polynomials
# integral = s.simplify(integral)
[docs]integral = integral.factor(factors).simplify()
[docs]parsimony = len(str(integral))
print(s.pretty(integral, num_columns=100))
print(f"Parsimony: {parsimony}")