import sympy as s
from sympy import init_printing
init_printing()
# Reconstructs a quadratic interpolant from x1...x3, then gets the derivative at x2
# Define the symbols
x1, x2, x3 = s.symbols('x1 x2 x3', real=True)
f1, f2, f3 = s.symbols('f1 f2 f3', real=True)
# hm = x2 - x1
# hp = x3 - x2
hm, hp = s.symbols("hm hp")
[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.
# q2 = s.symbols('q2', real=True) # (x2 - x1) / (x3 - x1)
# Define the Bernstein basis polynomials
c1, c2, c3 = s.symbols('c1 c2 c3', real=True)
# Can solve for c2 and c3 exactly
f = c1 * b1 + c2 * b2 + c3 * b3
[docs]f2_quadratic = f.subs(q, q2)#.simplify()
# factors = [f1, f2, f3, f4]
# Solve for c2 and c3
[docs]sol = s.solve(
[
f2_quadratic - f2,
],
[
c2,
],
)
[docs]c2 = sol[c2].factor(factors).simplify()
[docs]f = c1 * b1 + c2 * b2 + c3 * b3
[docs]dfdq = f.diff(q).simplify()
# dqdx = 1 / (x3 - x1)
dfm, dfp = s.symbols("dfm dfp")
[docs]def simplify(expr):
import copy
original_expr = copy.copy(expr)
expr = expr.subs({
f3 - f2: dfp,
f2 - f1: dfm,
f3 - f1: dfp + dfm,
x3 - x1: hm + hp,
})
expr = expr.subs({
f3 - f2: dfp,
f2 - f1: dfm,
f3 - f1: dfp + dfm,
x3 - x1: hm + hp,
})
expr = expr.factor([
hp,
hm
]).simplify()
if expr != original_expr:
expr = simplify(expr)
return expr
[docs]dfdx_q1 = simplify(dfdx.subs(q, q1))
[docs]dfdx_q2 = simplify(dfdx.subs(q, q2))
[docs]dfdx_q3 = simplify(dfdx.subs(q, q3))
# integral = (c1 + c2 + c3) / 3 # God I love Bernstein polynomials
# integral = s.simplify(integral)
[docs]parsimony = len(str(dfdx_q1))
print(s.pretty(dfdx_q1, num_columns=100))
print(f"Parsimony: {parsimony}")