Contents

% DERIVEST demo script

% This script file is designed to be used in cell mode
% from the matlab editor, or best of all, use the publish
% to HTML feature from the matlab editor. Older versions
% of matlab can copy and paste entire blocks of code into
% the Matlab command window.

% DERIVEST is property/value is driven for its arguments.
% Properties can be shortened to the

derivative of exp(x), at x == 0

[deriv,err] = derivest(@(x) exp(x),0)

deriv =

    1.0000


err =

  1.5088e-014

DERIVEST can also use an inline function

[deriv,err] = derivest(inline('exp(x)'),0)

deriv =

    1.0000


err =

  1.5088e-014

Higher order derivatives (second derivative)

Truth: 0

[deriv,err] = derivest(@(x) sin(x),pi,'deriv',2)

deriv =

 -5.5372e-019


err =

  1.8650e-018

Higher order derivatives (third derivative)

Truth: 1

[deriv,err] = derivest(@(x) cos(x),pi/2,'der',3)

deriv =

    1.0000


err =

  2.0718e-012

Higher order derivatives (up to the fourth derivative)

Truth: sqrt(2)/2 = 0.707106781186548

[deriv,err] = derivest(@(x) sin(x),pi/4,'d',4)

deriv =

    0.7071


err =

  1.9122e-005

Evaluate the indicated (default = first) derivative at multiple points

[deriv,err] = derivest(@(x) sin(x),linspace(0,2*pi,13))

deriv =

    1.0000    0.8660    0.5000         0   -0.5000   -0.8660   -1.0000   -0.8660   -0.5000         0    0.5000    0.8660    1.0000


err =

  1.0e-012 *

    0.0033    0.0039    0.0251         0    0.0126    0.0290    0.0031    0.0298    0.0488         0    0.0035    0.1291    0.0043

Specify the step size (default stepsize = 0.1)

deriv = derivest(@(x) polyval(1:5,x),1,'deriv',4,'FixedStep',1)

deriv =

   24.0000

Provide other parameters via an anonymous function

At a minimizer of a function, its derivative should be essentially zero. So, first, find a local minima of a first kind bessel function of order nu.

nu = 0;
fun = @(t) besselj(nu,t);
fplot(fun,[0,10])
x0 = fminbnd(fun,0,10,optimset('TolX',1.e-15))
hold on
plot(x0,fun(x0),'ro')
hold off

deriv = derivest(fun,x0,'d',1)

x0 =

    3.8317


deriv =

 -2.3301e-009

The second derivative should be positive at a minimizer.

deriv = derivest(fun,x0,'d',2)

deriv =

    0.4028

Compute the numerical gradient vector of a 2-d function

Note: the gradient at this point should be [4 6]

fun = @(x,y) x.^2 + y.^2;
xy = [2 3];
gradvec = [derivest(@(x) fun(x,xy(2)),xy(1),'d',1), ...
           derivest(@(y) fun(xy(1),y),xy(2),'d',1)]

gradvec =

    4.0000    6.0000

Compute the numerical Laplacian function of a 2-d function

Note: The Laplacian of this function should be everywhere == 4

fun = @(x,y) x.^2 + y.^2;
xy = [2 3];
lapval = derivest(@(x) fun(x,xy(2)),xy(1),'d',2) + ...
           derivest(@(y) fun(xy(1),y),xy(2),'d',2)

lapval =

     4

Compute the derivative of a function using a central difference scheme

Sometimes you may not want your function to be evaluated above or below a given point. A 'central' difference scheme will look in both directions equally.

[deriv,err] = derivest(@(x) sinh(x),0,'Style','central')

deriv =

     1


err =

  2.0824e-015

Compute the derivative of a function using a forward difference scheme

But a forward scheme will only look above x0.

[deriv,err] = derivest(@(x) sinh(x),0,'Style','forward')

deriv =

    1.0000


err =

  4.3314e-015

Compute the derivative of a function using a backward difference scheme

And a backward scheme will only look below x0.

[deriv,err] = derivest(@(x) sinh(x),0,'Style','backward')

deriv =

    1.0000


err =

  4.3314e-015

Although a central rule may put some samples in the wrong places, it may still succeed

[d,e,del]=derivest(@(x) log(x),.001,'style','central')

d =

  1.0000e+003


e =

  1.7156e-010


del =

  3.0518e-005

But forcing the use of a one-sided rule may be smart anyway

[d,e,del]=derivest(@(x) log(x),.001,'style','forward')

d =

 1000.0000


e =

  6.5546e-008


del =

  1.2207e-004

Control the behavior of DERIVEST - forward 2nd order method, with only 1 Romberg term

Compute the first derivative, also return the final stepsize chosen

[deriv,err,fdelta] = derivest(@(x) tan(x),pi,'deriv',1,'Style','for','MethodOrder',2,'RombergTerms',1)

deriv =

    1.0000


err =

  2.8379e-013


fdelta =

    0.0012

Functions should be vectorized for speed, but its not always easy to do.

[deriv,err] = derivest(@(x) x.^2,0:5,'deriv',1)
[deriv,err] = derivest(@(x) x^2,0:5,'deriv',1,'vectorized','no')

deriv =

         0    2.0000    4.0000    6.0000    8.0000   10.0000


err =

  1.0e-013 *

         0    0.0807    0.1613    0.3226    0.3226    0.4712


deriv =

         0    2.0000    4.0000    6.0000    8.0000   10.0000


err =

  1.0e-013 *

         0    0.0807    0.1613    0.3226    0.3226    0.4712

Published with MATLAB® 7.4