## 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