Numerical
Derivative
We are going to develop a Matlab function to calculate the numerical
derivative of any unidimensional scalar function fun(x) at a point x0.
The function is going to have the following functionality:
Usage:
D = Deriv(fun, x0)
fun: name of the unidimensional scalar function
(string)
x0: point of interest (scalar)
D: derivative of fun at x0 (scalar)
As you may remember, the very well known way to analytically derivate
any function is calculating:
This roughly means that we have to find the value of the function
in two very close values (one of them is the point of interest), get
the difference of those values and divide it by the displacement of the
independent variable. In other words,
First step:
Find ,
where is a very small value compared to
x.
Second step:
Obtain the difference .
Third
step:
Divide the result above by the small value .
The result is the numerical
derivative of your function.
We can implement this algorithm very easily in Matlab, in
this way:
function
D = Deriv(fun, x0)
%
delta is
relative to x0
delta =
x0 / 1000;
if x0 ==
0
% avoids
delta = 0 (**arbitrary value**)
delta = 1e12;
end
f1 =
feval ( fun, x0 + delta );
f2 =
feval ( fun, x0  delta );
D = (f1
 f2) / (2 * delta);
Note, that we are really
evaluating
which
should be the same as explained above, since the displacement is almost
zero. You can try both options.
Function
'feval'
evaluates a given function (the string in the parameter fun) in a specific
value (number in second
parameter of 'feval').
Now, let's try our derivative function. We create a function in a
separate mfile:
function
y = inverse(x)
y = 1/x;
And we can call it like this:
Deriv('inverse',
1)
The
result is:
Expected:
1
Obtained: 1.0000
Deriv('inverse',
5)
The
result is:
Expected:
0.04
Obtained: 0.0400
We can save another totally arbitrary (unidimensional scalar) function
in a different mfile:
function
y = myfun(x)
y =
3*x^2 + 2*x;
And we can perform the derivative in several ways...
f
= 'myfun'
x = 0
Deriv(f,
x)
The
result is:
Expected:
2 Obtained:
2
x = 5
Deriv(f,
x)
The
result is:
Expected:
32
Obtained: 32.0000
Deriv(f,
3)
The
result is:
Expected:
16 Obtained:
16.0000
From
'Derivative' to home
From
'Derivative' to 'Matlab Cookbook'
