Verify Numerical Recipes' Nelder-Mead formula

_code/pset_08/py/lagrange_poly.py

+import sympy as sp
+
+# Verifying that the Nelder-Mead reflection scheme in Numerical Methods is equivalent to the
+# standard presentation.
+
+x = sp.symbols("x", real=True)
+x_1, x_2, x_3 = sp.symbols("x_1 x_2 x_3", real=True)
+y_1, y_2, y_3 = sp.symbols("y_1 y_2 y_3", real=True)
+
+term_1 = y_1 * (x - x_2) * (x - x_3) / (x_1 - x_2) / (x_1 - x_3)
+term_2 = y_2 * (x - x_1) * (x - x_3) / (x_2 - x_1) / (x_2 - x_3)
+term_3 = y_3 * (x - x_1) * (x - x_2) / (x_3 - x_1) / (x_3 - x_2)
+poly = term_1 + term_2 + term_3
+dpoly = sp.diff(poly, x)
+sol = sp.solve(dpoly, x)
+assert(len(sol) == 1)
+sol = sol[0]
+
+print("lagrange polynomial")
+print(sp.latex(poly))
+print("derivative")
+print(sp.latex(dpoly))
+print("minimum")
+print(sp.latex(sol))
+
+q = (x_2 - x_3) * (y_2 - y_1)
+r = (x_2 - x_1) * (y_2 - y_3)
+alt = x_2 - ((x_2 - x_3) * q - (x_2 - x_1) * r) / 2 / (q - r)
+
+print(sp.simplify(sol - alt))