## Contents

- derivative of exp(x), at x == 0
- DERIVEST can also use an inline function
- Higher order derivatives (second derivative)
- Higher order derivatives (third derivative)
- Higher order derivatives (up to the fourth derivative)
- Evaluate the indicated (default = first) derivative at multiple points
- Specify the step size (default stepsize = 0.1)
- Provide other parameters via an anonymous function
- The second derivative should be positive at a minimizer.
- Compute the numerical gradient vector of a 2-d function
- Compute the numerical Laplacian function of a 2-d function
- Compute the derivative of a function using a central difference scheme
- Compute the derivative of a function using a forward difference scheme
- Compute the derivative of a function using a backward difference scheme
- Although a central rule may put some samples in the wrong places, it may still succeed
- But forcing the use of a one-sided rule may be smart anyway
- Control the behavior of DERIVEST - forward 2nd order method, with only 1 Romberg term
- Functions should be vectorized for speed, but its not always easy to do.

    % 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
    err =
       1.4046e-14

## DERIVEST can also use an inline function

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

    deriv =
                1
    err =
       1.4046e-14

## Higher order derivatives (second derivative)

Truth: 0

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

    deriv =
      -5.5372e-19
    err =
        1.865e-18

## Higher order derivatives (third derivative)

Truth: 1

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

    deriv =
                1
    err =
       4.3657e-12

## 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 =
          0.70711
    err =
       1.9122e-05

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

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

    deriv =
      Columns 1 through 7
                1      0.86603          0.5            0         -0.5     -0.86603           -1
      Columns 8 through 13
         -0.86603         -0.5            0          0.5      0.86603            1
    err =
      Columns 1 through 7
       1.0412e-15   1.4725e-15   2.5102e-14            0   1.3754e-14   2.7429e-14   1.8034e-15
      Columns 8 through 13
       3.0284e-14   4.9044e-14            0   3.2092e-15   1.2987e-13   2.5504e-15

## Specify the step size (default stepsize = 0.1)

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

    deriv =
               24

## 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 =
      -2.3285e-09

## The second derivative should be positive at a minimizer.

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

    deriv =
          0.40276

## 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            6

## 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 =
       1.0412e-15

## 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
    err =
       3.1516e-15

## 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
    err =
       3.1516e-15

## 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 =
             1000
    e =
       1.7072e-10
    del =
       3.0518e-05

## 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
    e =
       6.5547e-08
    del =
       0.00012207

## 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
    err =
       2.8399e-13
    fdelta =
        0.0011984

## 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            4            6            8           10
    err =
                0   4.6563e-15   9.3127e-15    1.178e-14   1.8625e-14   2.3559e-14
    deriv =
                0            2            4            6            8           10
    err =
                0   4.6563e-15   9.3127e-15    1.178e-14   1.8625e-14   2.3559e-14