Difference between revisions of "Numerical Differentiation"
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Lagrange polynomials, which are commonly used for [[Interpolation#Lagrange_Polynomial_Interpolation|interpolation]], can also be used for differentiation. The formula is | Lagrange polynomials, which are commonly used for [[Interpolation#Lagrange_Polynomial_Interpolation|interpolation]], can also be used for differentiation. The formula is | ||
| − | + | :<math>f^{\prime}(x) = \sum_{k=0}^n y_k L_{k}^{\prime}(x),</math> | |
where <math>L_{k}^{\prime}(x)</math> is given as | where <math>L_{k}^{\prime}(x)</math> is given as | ||
| − | + | :<math>L_{k}^{\prime}(x) = \left[ \sum_{{j=0}\atop{j\ne k}}^{n} \left( \prod_{ {i=0}\atop{i \ne j,k} }^n (x-x_i) \right) \right] \left[ \prod_{{i=0}\atop{i\ne k}}^{n} (x_k-x_i) \right]^{-1}. </math> | |
Here <math>n</math> is the order of the polynomial and we require <math>n_p=n+1</math> points to form the Lagrange polynomial. | Here <math>n</math> is the order of the polynomial and we require <math>n_p=n+1</math> points to form the Lagrange polynomial. | ||
Here are the results for <var>n = 2, 3, 4</var> | Here are the results for <var>n = 2, 3, 4</var> | ||
| − | {| border=1 cellpadding=10 cellspacing=1 | + | :{| border=1 cellpadding=10 cellspacing=1 |
! | ! | ||
! <var>n</var>=2 | ! <var>n</var>=2 | ||
Revision as of 15:32, 28 August 2009
Introduction
Often we need to approximate the derivative of a function when we cannot obtain it analytically. Here we discuss several ways to do this numerically.
Taylor Series
Let's consider the situation where we have samples of a function
at discrete points 
seperated by spacing
as depicted in the following figure:
Consider a Taylor series expansions about some
arbitrary point 
. Since 
we can write these as follows:
Approximation location Taylor Series Expansion about
If we subtract the Taylor series expansion at
from the one at , we find
Now we solve this for 
to find
Now if is small, then the second term (with the
in it) is small and we can approximate the derivative
as
We call this a second order approximation to
because when we truncated the series
approximation to the largest term there was
of the order of .
Note that we now have a way to approximate the derivative of a function if we have the function's values at two locations.
On a uniform mesh, we can use this technique to generate a variety of approximations to derivatives, as summarized in the following table:
|
 Error in approximation of  as a function of the size of the interval . Click here to download the matlab file that produced this plot. |
As can be seen in the figure above, higher order approximations result
in significantly lower error for a given spacing . Note
that for 
the fourth-order approximation is
contaminated by roundoff error. The same would happen for the other
derivative approximations, but at smaller .
Lagrange Polynomials
Lagrange polynomials, which are commonly used for interpolation, can also be used for differentiation. The formula is
where 
is given as
![L_{k}^{\prime}(x) = \left[ \sum_{{j=0}\atop{j\ne k}}^{n} \left( \prod_{ {i=0}\atop{i \ne j,k} }^n (x-x_i) \right) \right] \left[ \prod_{{i=0}\atop{i\ne k}}^{n} (x_k-x_i) \right]^{-1}.](/wiki/images/math/3/c/c/3cced1f629b2f0354163c2308a6df53c.png)
Here 
is the order of the polynomial and we require 
points to form the Lagrange polynomial.
Here are the results for n = 2, 3, 4
n=2 n=3 n=4 
