Compute function inverse via collocation
Contents
Compute function inverse via collocation¶
Randall Romero Aguilar, PhD
This demo is based on the original Matlab demo accompanying the Computational Economics and Finance 2001 textbook by Mario Miranda and Paul Fackler.
Original (Matlab) CompEcon file: demapp08.m
Running this file requires the Python version of CompEcon. This can be installed with pip by running
!pip install compecon --upgrade
Last updated: 2022-Oct-22
About¶
The function is defined implicitly by
\[\begin{equation*}
f(x)^{-2} + f(x)^{-5} - 2x = 0
\end{equation*}\]
Initial tasks¶
import numpy as np
import matplotlib.pyplot as plt
from compecon import BasisChebyshev, NLP
Approximation structure¶
n, a, b = 31, 1, 5
F = BasisChebyshev(n, a, b, y=5*np.ones(n), labels=['f(x)']) # define basis functions
x = F.nodes # compute standard nodes
F.plot()
Residual function¶
def resid(c):
F.c = c # update basis coefficients
y = F(x) # interpolate at basis nodes x
return y ** -2 + y ** -5 - 2 * x
Compute function inverse¶
c0 = np.zeros(n) # set initial guess for coeffs
c0[0] = 0.2
problem = NLP(resid)
F.c = problem.broyden(c0) # compute coeff by Broyden's method
Plot function inverse¶
n = 1000
x = np.linspace(a, b, n)
r = resid(F.c)
fig1, ax = plt.subplots()
ax.set(title='Implicit Function',
xlabel='x',
ylabel='f(x)')
ax.plot(x, F(x));
Plot residual¶
fig2, ax = plt.subplots()
ax.set(title='Functional Equation Residual',
xlabel='x',
ylabel='Residual')
ax.hlines(0, a, b, 'k', '--')
ax.plot(x, r);
