Chebychev polynomial and spline approximantion of various functions
Contents
Chebychev polynomial and spline approximantion of various functions¶
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: demapp05.m
Running this file requires the Python version of CompEcon. This can be installed with pip by running
!pip install compecon --upgrade
Last updated: 2021-Oct-01
About¶
Demonstrates Chebychev polynomial, cubic spline, and linear spline approximation for the following functions
(4)¶\[\begin{align}
y &= 1 + x + 2x^2 - 3x^3 \\
y &= \exp(-x) \\
y &= \frac{1}{1+25x^2} \\
y &= \sqrt{|x|}
\end{align}\]
Initial tasks¶
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from compecon import BasisChebyshev, BasisSpline, nodeunif
Functions to be approximated¶
funcs = [lambda x: 1 + x + 2 * x ** 2 - 3 * x ** 3,
lambda x: np.exp(-x),
lambda x: 1 / ( 1 + 25 * x ** 2),
lambda x: np.sqrt(np.abs(x))]
fst = ['$y = 1 + x + 2x^2 - 3x^3$', '$y = \exp(-x)$',
'$y = 1/(1+25x^2)$', '$y = \sqrt{|x|}$']
Set degree of approximation and endpoints of approximation interval
n = 7 # degree of approximation
a = -1 # left endpoint
b = 1 # right endpoint
Construct uniform grid for error ploting
x = np.linspace(a, b, 2001)
def subfig(f, title):
# Construct interpolants
C = BasisChebyshev(n, a, b, f=f)
S = BasisSpline(n, a, b, f=f)
L = BasisSpline(n, a, b, k=1, f=f)
data = pd.DataFrame({
'actual': f(x),
'Chebyshev': C(x),
'Cubic Spline': S(x),
'Linear Spline': L(x)},
index = x)
fig1, axs = plt.subplots(2,2, figsize=[12,6], sharex=True, sharey=True)
fig1.suptitle(title)
data.plot(ax=axs, subplots=True)
errors = data[['Chebyshev', 'Cubic Spline']].subtract(data['actual'], axis=0)
fig2, ax = plt.subplots(figsize=[12,3])
fig2.suptitle("Approximation Error")
errors.plot(ax=ax)
Polynomial¶
\(y = 1 + x + 2x^2 - 3x^3\)
subfig(lambda x: 1 + x + 2*x**2 - 3*x**3, '$y = 1 + x + 2x^2 - 3x^3$')
Exponential¶
\(y = \exp(-x)\)
subfig(lambda x: np.exp(-x),'$y = \exp(-x)$')
Rational¶
\(y = 1/(1+25x^2)\)
subfig(lambda x: 1 / ( 1 + 25 * x ** 2),'$y = 1/(1+25x^2)$')
Kinky¶
\(y = \sqrt{|x|}\)
subfig(lambda x: np.sqrt(np.abs(x)), '$y = \sqrt{|x|}$')
