Basis functions and standard nodes for major approximation schemes

Basis functions and standard nodes for major approximation schemes¶

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

Initial tasks¶

import numpy as np
import matplotlib.pyplot as plt
from compecon import BasisChebyshev, BasisSpline

FIGSIZE = [9,6]

Set degree of approximation and approximation interval¶

n, a, b = 12, -1, 1

Routine for plotting basis functions¶

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

Construct plotting grid¶

m = 1001
x = np.linspace(a, b, m)

Plot monomial basis functions¶

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)])

../../_images/03 Basis functions and standard nodes for major approximation schemes_10_0.png

Plot Chebychev basis functions and nodes¶

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)])

../../_images/03 Basis functions and standard nodes for major approximation schemes_12_0.png

Plot linear spline basis functions and nodes¶

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)])

../../_images/03 Basis functions and standard nodes for major approximation schemes_14_0.png

Plot cubic spline basis functions and nodes¶

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)])

../../_images/03 Basis functions and standard nodes for major approximation schemes_16_0.png

Routine for plotting approximation nodes¶

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) 

../../_images/03 Basis functions and standard nodes for major approximation schemes_18_0.png

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);

../../_images/03 Basis functions and standard nodes for major approximation schemes_19_0.png

Approximating functions on \(R^2\)

Approximating Runge’s function

A Python Implementation of CompEcon