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: demapp03.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
import numpy as np
import matplotlib.pyplot as plt
from compecon import BasisChebyshev, BasisSpline
FIGSIZE = [9,6]
n, a, b = 12, -1, 1
def basisplot(x,Phi,figtitle, titles):
fig, axs = plt.subplots(3, 4, figsize=FIGSIZE, sharex=True,sharey=True)
ymin = np.round(Phi.min())
ymax = np.round(Phi.max())
degree = 0
for phi, ax, ttl in zip(Phi, axs.flatten(), titles):
ax.plot(x, phi, lw=4)
ax.set_title(ttl, size=14)
ax.set_xticklabels([a, b], fontsize=11)
ax.set_yticklabels([ymin, ymax], fontsize=11)
degree += 1
ax.set(ylim=[ymin,ymax], xlim=[a,b],xticks=[a, b], yticks=[ymin, ymax])
fig.suptitle(figtitle, size=16)
return fig
m = 1001
x = np.linspace(a, b, m)
Phi = np.array([x ** j for j in np.arange(n)])
figm = basisplot(x,Phi,'Monomial Basis Functions on [-1,1]',['$x^{%d}$' % d for d in range(12)])
B = BasisChebyshev(n,a,b)
figch = basisplot(x, B.Phi(x).T,'Chebychev Polynomial Basis Functions on [-1,1]',['$T_{%d}(z)$' % d for d in range(12)])
L = BasisSpline(n,a,b,k=1)
figl = basisplot(x, L.Phi(x).T.toarray(),'Linear Spline Basis Functions on [-1,1]', [f'Spline {d}' for d in range(12)])
C = BasisSpline(n,a,b,k=3)
figc = basisplot(x, C.Phi(x).T.toarray(),'Cubic Spline Basis Functions on [-1,1]',[f'Spline {d}' for d in range(12)])
fignodos, axs = plt.subplots(7,1,figsize=FIGSIZE, sharex=True, sharey=True)
for i, ax in enumerate(axs.flatten()):
n = i+3
B = BasisChebyshev(n,a,b)
ax.plot(B.nodes, 0, 'bo')
ax.set_xticks([a,0,b])
ax.set_xticklabels([a,0,b], fontsize=11)
ax.set_yticks([])
ax.text(1.07, 0, f'$n={n}$', size=12)
fignodos.suptitle('Chebyshev nodes', size=16)
fig0x, ax = plt.subplots(figsize=[9,3])
B = BasisChebyshev(5,a,b)
ax.plot(x, B.Phi(x)[:,-1])
xextr=np.cos(np.arange(5)*np.pi/4)
ax.plot(xextr, B.Phi(xextr)[:,-1],'r*')
for i, vv in enumerate(xextr):
ax.text(vv,B.Phi(vv)[:,-1],r'$\frac{%d \pi}{8}$' % (2*i), ha='center', va='bottom')
xcero = np.cos((np.arange(4)+0.5)*np.pi/4)
ax.plot(xcero, B.Phi(xcero)[:,-1],'bo')
for i, vv in enumerate(xcero):
ax.text(vv,0,r' $\frac{%d \pi}{8}$' % (2*i+1), ha='left', va='center')
ax.set_xticks([-1,0,1])
ax.set_xticklabels([-1,0,1],fontsize=11)
ax.set_yticks([-1,0,1])
ax.set_yticklabels([-1,0,1],fontsize=11)
ax.set_title('Extrema and zeros of a Chebyshev polynomial\n', size=16);