Difference between revisions of "Numerical Differentiation"

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== Introduction ==
 
== 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 ==
 
== Taylor Series ==
  
 +
Let's consider the situation where we have samples of a function
 +
<math>f(x)</math> at discrete points <math>\begin{array}{cccc}x_1 &
 +
x_2 & \cdots & x_n\end{array}</math> seperated by spacing
 +
<math>h</math> as depicted in the following figure:
 +
[[Image:uniform_grid_1D.png|thumb|400px|center]]
  
 +
Consider a [[Taylor_Series|Taylor series]] expansions about some
 +
arbitrary point <math>x_i</math>. Since <math>x_{i+1}-x_i =
 +
x_i-x_{i-1} = h</math> we can write these as follows:
 +
:{| border=1 cellpadding=5 cellspacing=0 style="text-align:left"
 +
! Approximation location
 +
! Taylor Series Expansion about <math>x_i</math>
 +
|-
 +
| <math>x_{i+1}</math>
 +
||<math>
 +
f(x_{i+1}) = f(x_i)
 +
  + f^\prime(x_i) h
 +
  + \tfrac{1}{2}f^{\prime\prime}(x_i) h^2
 +
  + \tfrac{1}{6}f^{\prime\prime\prime}(x_i) h^3
 +
  + \tfrac{1}{24}f^{(4)}(x_i) h^4
 +
  + \cdots
 +
</math>
 +
|-
 +
| <math>x_{i-1}</math>
 +
||<math>
 +
f(x_{i-1}) = f(x_i)
 +
  - f^\prime(x_i)h
 +
  + \tfrac{1}{2}f^{\prime\prime}(x_i)h^2
 +
  - \tfrac{1}{6}f^{\prime\prime\prime}(x_i)h^3
 +
  + \tfrac{1}{24}f^{(4)}(x_i)h^4
 +
  - \cdots
 +
</math>
 +
|}
  
 +
If we subtract the Taylor series expansion at <math>x_{i-1}</math>
 +
from the one at <math>x_{i+1}</math>, we find
 +
:<math>
 +
f(x_{i+1})-f(x_{i-1}) = 2f^{\prime}(x_i)h + \tfrac{1}{3} f^{\prime\prime\prime}(x_i) h^3 + \cdots
 +
</math>
 +
Now we solve this for <math>f^\prime(x_i)</math> to find
 +
:<math>
 +
  f^{\prime}(x_i) = \frac{f(x_{i+1})-f(x_{i-1})}{2h} - \tfrac{1}{6} f^{\prime\prime\prime}(x_i)h^2 - \cdots
 +
</math>
  
 +
Now if <math>h</math> is small, then the second term (with the
 +
<math>h^2</math> in it) is small and we can approximate the derivative
 +
as
 +
:{|border=2
 +
|-
 +
|<math>
 +
f^\prime(x_i) = \frac{f(x_{i+1})-f(x_{i-1})}{2h}
 +
</math>
 +
|}
 +
We call this a '''second order approximation''' to
 +
<math>f^\prime(x)</math> because when we truncated the series
 +
approximation to <math>f^\prime(x_i)</math> the largest term there was
 +
of the order of <math>h^2</math>.
  
−
== Lagrange Polynomials ==
+
Note that we now have a way to approximate the derivative of a function if we have the function's values at two locations.  
−
 
 
−
Lagrange polynomials, which are commonly used for [[interpolation]], can also be used for differentiation.
 
−
 
 
−
 
 
−
 
 
  
−
== Tables for Derivatives on Uniform Grids ==
+
On a uniform mesh, we can use this technique to generate a variety of
 +
approximations to derivatives, as summarized in the following table:
  
−
<center>
+
{| align=center
−
{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"
+
|
−
|+ '''Some First Derivative Expressions for a Uniform Mesh'''
+
{| border=1 cellpadding=5 cellspacing=0 style="text-align:center"
−
|-
+
! Formula for <math> \left. \frac{\mathrm{d} f}{\mathrm{d}x} \right|_{i}</math>
−
! Derivative at point <math>i</math>
 
−
! Discrete Representation (uniform mesh)
 
 
! Order
 
! Order
 
|-
 
|-
−
| Forward Difference
+
|<math>
−
| <math> \left. \frac{\mathrm{d}\phi}{\mathrm{d}x} \right|_{i} \approx \frac{\phi_{i+1}-\phi_{i}}{\Delta x}</math>
+
  \frac{f(x_{i+1})-f(x_{i})}{h}</math>
−
| <math>\mathcal{O}\left(\Delta x \right) </math>
+
||<math>\mathcal{O}\left( h \right) </math>
 
|-
 
|-
−
| Backward Difference
+
|<math>
−
| <math> \left. \frac{\mathrm{d}\phi}{\mathrm{d}x} \right|_{i} \approx \frac{\phi_{i}-\phi_{i-1}}{\Delta x}</math>
+
  \frac{f(x_{i})-f(x_{i-1})}{h}</math>
−
| <math>\mathcal{O}\left(\Delta x \right) </math>
+
||<math>\mathcal{O}\left( h \right) </math>
 
|-
 
|-
−
| Central Difference
+
|<math>
−
| | <math> \left. \frac{\mathrm{d\phi}}{\mathrm{d}x} \right|_{i} \approx \frac{\phi_{i+1}-\phi_{i-1}}{2\Delta x}</math>
+
  \frac{f(x_{i+1})-f(x_{i-1})}{2 h}</math>
−
| <math>\mathcal{O}\left(\Delta x^2 \right) </math>
+
| <math>\mathcal{O}\left(h^2 \right) </math>
 
|-
 
|-
 +
|<math>
 +
  \frac{f(x_{i-2}) - 8 f(x_{i-1}) + 8 f(x_{i+1}) - f(x_{i+2})}{12 h}
 +
</math>
 +
||<math>\mathcal{O}\left( h^4 \right) </math>
 +
|}
 +
||
 +
[[Image:derConverge_uniform_mesh.png|thumb|450px|Error in approximation
 +
of <math>\frac{df}{dx}</math> as a function of the size of the
 +
interval <math>h</math>. Click [[media:der_demo.m|here]] to download
 +
the matlab file that produced this plot.]]
 
|}
 
|}
−
</center>
 
  
 +
As can be seen in the figure above, higher order approximations result
 +
in significantly lower error for a given spacing <math>h</math>. Note
 +
that for <math>h<10^{-3}</math> the fourth-order approximation is
 +
contaminated by roundoff error. The same would happen for the other
 +
derivative approximations, but at smaller <math>h</math>.
  
−
<center>
+
<!-- jcs need to fill this in
−
{| border="1" cellpadding="5" cellspacing="0" align="center" style="text-align:center"
+
{| border=1 cellpadding=5 cellspacing=0 align=center style="text-align:center"
−
|+ '''Some Second Derivative Expressions for a Uniform Mesh'''
+
|+ Some Second Derivative Expressions for a Uniform Mesh
 
|-
 
|-
 
! Derivative at point <math>i</math>
 
! Derivative at point <math>i</math>
 
! Discrete Representation (uniform mesh)
 
! Discrete Representation (uniform mesh)
 
! Order
 
! Order
−
</center>
+
|}
 +
-->
 +
 
 +
== Lagrange Polynomials ==
 +
 
 +
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
 +
:<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 are the results for <var>n = 2, 3, 4</var>
 +
 
 +
:{| border=1 cellpadding=10 cellspacing=1
 +
!
 +
! <var>n</var>=2
 +
! <var>n=3</var>
 +
! <var>n=4</var>
 +
|-
 +
| <math>L_{0}^{\prime}(x)</math>
 +
| <math>\frac{1}{x_{0}-x_{1}}</math>
 +
| <math>\frac{2x-x_{1}-x_{2}}{(x_{0}-x_{1})(x_{0}-x_{2})}</math>
 +
| <math>\frac{3x^{2}-2x\left(x_{1}+x_{2}+x_{3}\right)+x_{1}\left(x_{2}+x_{3}\right)+x_{2}x_{3}}{\left(x_{0}-x_{1}\right)\left(x_{0}-x_{2}\right)\left(x_{0}-x_{3}\right)}</math>
 +
|-
 +
| <math>L_{1}^{\prime}(x)</math>
 +
| <math>\frac{1}{x_{1}-x_{0}}</math>
 +
| <math>\frac{2x-x_{0}-x_{2}}{(x_{1}-x_{0})(x_{1}-x_{2})}</math>
 +
| <math>-\frac{3x^{2}-2x(x_{0}+x_{2}+x_{3})+x_{0}\left(x_{2}+x_{3}\right)+x_{2}x_{3}}{\left(x_{0}-x_{1}\right)\left(x_{1}-x_{2}\right)\left(x_{1}-x_{3}\right)}</math>
 +
|-
 +
| <math>L_{2}^{\prime}(x)</math>
 +
|
 +
| <math>\frac{2x-x_{0}-x_{1}}{(x_{2}-x_{0})(x_{2}-x_{1})}</math>
 +
| <math>\frac{3x^{2}-2x(x_{0}+x_{1}+x_{3})+x_{0}\left(x_{1}+x_{3}\right)+x_{1}x_{3}}{\left(x_{0}-x_{2}\right)\left(x_{1}-x_{2}\right)\left(x_{2}-x_{3}\right)}</math>
 +
|-
 +
| <math>L_{3}^{\prime}(x)</math>
 +
|
 +
|
 +
|<math>-\frac{3x^{2}-2x(x_{0}+x_{1}+x_{2})+x_{0}\left(x_{1}+x_{2}\right)+x_{1}x_{2}}{\left(x_{0}-x_{3}\right)\left(x_{1}-x_{3}\right)\left(x_{2}-x_{3}\right)}</math>
 +
|}

Latest revision as of 15:56, 26 August 2010

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 f(x) at discrete points \begin{array}{cccc}x_1 & x_2 & \cdots & x_n\end{array} seperated by spacing h as depicted in the following figure:

Uniform grid 1D.png

Consider a Taylor series expansions about some arbitrary point x_i. Since x_{i+1}-x_i = x_i-x_{i-1} = h we can write these as follows:

Approximation location Taylor Series Expansion about x_i
x_{i+1} f(x_{i+1}) = f(x_i) + f^\prime(x_i) h + \tfrac{1}{2}f^{\prime\prime}(x_i) h^2 + \tfrac{1}{6}f^{\prime\prime\prime}(x_i) h^3 + \tfrac{1}{24}f^{(4)}(x_i) h^4 + \cdots
x_{i-1} f(x_{i-1}) = f(x_i) - f^\prime(x_i)h + \tfrac{1}{2}f^{\prime\prime}(x_i)h^2 - \tfrac{1}{6}f^{\prime\prime\prime}(x_i)h^3 + \tfrac{1}{24}f^{(4)}(x_i)h^4 - \cdots

If we subtract the Taylor series expansion at x_{i-1} from the one at x_{i+1}, we find

f(x_{i+1})-f(x_{i-1}) = 2f^{\prime}(x_i)h + \tfrac{1}{3} f^{\prime\prime\prime}(x_i) h^3 + \cdots

Now we solve this for f^\prime(x_i) to find

f^{\prime}(x_i) = \frac{f(x_{i+1})-f(x_{i-1})}{2h} - \tfrac{1}{6} f^{\prime\prime\prime}(x_i)h^2 - \cdots

Now if h is small, then the second term (with the h^2 in it) is small and we can approximate the derivative as

f^\prime(x_i) = \frac{f(x_{i+1})-f(x_{i-1})}{2h}

We call this a second order approximation to f^\prime(x) because when we truncated the series approximation to f^\prime(x_i) the largest term there was of the order of h^2.

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:

Formula for \left. \frac{\mathrm{d} f}{\mathrm{d}x} \right|_{i} Order
\frac{f(x_{i+1})-f(x_{i})}{h} \mathcal{O}\left( h \right)
\frac{f(x_{i})-f(x_{i-1})}{h} \mathcal{O}\left( h \right)
\frac{f(x_{i+1})-f(x_{i-1})}{2 h} \mathcal{O}\left(h^2 \right)
\frac{f(x_{i-2}) - 8 f(x_{i-1}) + 8 f(x_{i+1}) - f(x_{i+2})}{12 h} \mathcal{O}\left( h^4 \right)
Error in approximation of \frac{df}{dx} as a function of the size of the interval h. 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 h. Note that for h<10^{-3} the fourth-order approximation is contaminated by roundoff error. The same would happen for the other derivative approximations, but at smaller h.


Lagrange Polynomials

Lagrange polynomials, which are commonly used for interpolation, can also be used for differentiation. The formula is

f^{\prime}(x) = \sum_{k=0}^n y_k L_{k}^{\prime}(x),

where L_{k}^{\prime}(x) 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}.

Here n is the order of the polynomial and we require n_p=n+1 points to form the Lagrange polynomial.

Here are the results for n = 2, 3, 4

n=2 n=3 n=4
L_{0}^{\prime}(x) \frac{1}{x_{0}-x_{1}} \frac{2x-x_{1}-x_{2}}{(x_{0}-x_{1})(x_{0}-x_{2})} \frac{3x^{2}-2x\left(x_{1}+x_{2}+x_{3}\right)+x_{1}\left(x_{2}+x_{3}\right)+x_{2}x_{3}}{\left(x_{0}-x_{1}\right)\left(x_{0}-x_{2}\right)\left(x_{0}-x_{3}\right)}
L_{1}^{\prime}(x) \frac{1}{x_{1}-x_{0}} \frac{2x-x_{0}-x_{2}}{(x_{1}-x_{0})(x_{1}-x_{2})} -\frac{3x^{2}-2x(x_{0}+x_{2}+x_{3})+x_{0}\left(x_{2}+x_{3}\right)+x_{2}x_{3}}{\left(x_{0}-x_{1}\right)\left(x_{1}-x_{2}\right)\left(x_{1}-x_{3}\right)}
L_{2}^{\prime}(x) \frac{2x-x_{0}-x_{1}}{(x_{2}-x_{0})(x_{2}-x_{1})} \frac{3x^{2}-2x(x_{0}+x_{1}+x_{3})+x_{0}\left(x_{1}+x_{3}\right)+x_{1}x_{3}}{\left(x_{0}-x_{2}\right)\left(x_{1}-x_{2}\right)\left(x_{2}-x_{3}\right)}
L_{3}^{\prime}(x) -\frac{3x^{2}-2x(x_{0}+x_{1}+x_{2})+x_{0}\left(x_{1}+x_{2}\right)+x_{1}x_{2}}{\left(x_{0}-x_{3}\right)\left(x_{1}-x_{3}\right)\left(x_{2}-x_{3}\right)}
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