CompEcon Toolbox:
DemApp08
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.

Last updated: 2020-Sep-09

The function is defined implicitly by \begin{equation*} f(x)^{-2} + f(x)^{-5} - 2x = 0 \end{equation*}

In [ ]:
if 'google.colab' in str(get_ipython()):
print("This notebook is running on Google Colab. Installing the compecon package.")
!pip install compecon

In [ ]:
import numpy as np
import matplotlib.pyplot as plt
from compecon import BasisChebyshev, NLP, demo


### Approximation structure¶

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n, a, b = 31, 1, 5
F = BasisChebyshev(n, a, b, y=5*np.ones(31), labels=['f(x)'])  # define basis functions
x = F.nodes                  # compute standard nodes

In [ ]:
F.plot()


### Residual function¶

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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¶

In [ ]:
c0 = np.zeros(n)  # set initial guess for coeffs
c0 = 0.2
problem = NLP(resid)
F.c = problem.broyden(c0)  # compute coeff by Broyden's method


### Plot function inverse¶

In [ ]:
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¶

In [ ]:
fig2, ax = plt.subplots()
ax.set(title='Functional Equation Residual',
xlabel='x',
ylabel='Residual')
ax.hlines(0, a, b, 'k', '--')
ax.plot(x, r);


### Save all figures to disc¶

In [ ]:
#demo.savefig([fig1, fig2], name='demapp08')