Chebychev and cubic spline derivative approximation errors
Contents
Chebychev and cubic spline derivative approximation errors¶
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: demapp06.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-23
Initial tasks¶
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from compecon import BasisChebyshev, BasisSpline, nodeunif
Function to be approximated¶
f = lambda x: np.exp(-x)
df = lambda x: -np.exp(-x)
d2f = lambda x: np.exp(-x)
Set degree of approximation and endpoints of approximation interval
a = -1 # left endpoint
b = 1 # right endpoint
n = 10 # order of interpolatioin
Construct refined uniform grid for error ploting
x = np.linspace(a,b, 1001)
Construct Chebychev interpolant
C = BasisChebyshev(n, a, b, f=f)
Construct cubic spline interpolant
S = BasisSpline(n, a, b, f=f)
Plot function approximation error¶
y = f(x)
fig1, axs = plt.subplots(2,1,sharex=True)
fig1.suptitle('Function Approximation Error')
(pd.DataFrame({
'Chebychev': y - C(x),
'Cubic Spline': y - S(x)},
index=x)
.plot(subplots=True, ax=axs)
);
Plot first derivative approximation error¶
dy = df(x)
fig1, axs = plt.subplots(2,1,sharex=True)
fig1.suptitle('First Derivative Approximation Error')
(pd.DataFrame({
'Chebychev': dy - C(x, 1),
'Cubic Spline': dy - S(x, 1)},
index=x)
.plot(subplots=True, ax=axs)
);
Plot second derivative approximation error¶
d2y = d2f(x)
fig1, axs = plt.subplots(2,1,sharex=True)
fig1.suptitle('Second Derivative Approximation Error')
(pd.DataFrame({
'Chebychev': d2y - C(x, 2),
'Cubic Spline': d2y - S(x, 2)},
index=x)
.plot(subplots=True, ax=axs)
);
