Using Richardson Extrapolation with Finite Differences

Copyright (C) 2020 Andreas Kloeckner

MIT License Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
In [1]:
from math import sin, cos

Here are a function and its derivative. We also choose a "center" about which we carry out our experiments:

In [2]:
f = sin

df = cos



x = 2.3

We then compare the accuracy of:

  • First-order (right) differences

  • First-order (right) differences with half the step size

  • An estimate based on these two using Richardson extrapolation

against true, the actual derivative

In [3]:
for k in range(3, 10):

    h = 2**(-k)



    fd1 = (f(x+2*h) - f(x))/(2*h)

    fd2 = (f(x+h) - f(x))/h

    

    richardson = (-1)*fd1 + 2*fd2

    

    true = df(x)

    

    print("Err FD1: %g\tErr FD: %g\tErr Rich: %g" % (

            abs(true-fd1),

            abs(true-fd2),    

            abs(true-richardson)))

Err FD1: 0.08581	Err FD: 0.0448122	Err Rich: 0.00381441
Err FD1: 0.0448122	Err FD: 0.022862	Err Rich: 0.000911846
Err FD1: 0.022862	Err FD: 0.0115423	Err Rich: 0.000222501
Err FD1: 0.0115423	Err FD: 0.00579859	Err Rich: 5.49282e-05
Err FD1: 0.00579859	Err FD: 0.00290612	Err Rich: 1.3644e-05
Err FD1: 0.00290612	Err FD: 0.00145476	Err Rich: 3.39995e-06
Err FD1: 0.00145476	Err FD: 0.000727804	Err Rich: 8.48602e-07
In [ ]: