Numerical Differentiation and Integration in MATLAB; Function M-files 259 Lecture 16 Numerical Differentiation and Integration in MATLAB; Function M-files

Derivatives and Integrals We can use MATLAB to numerically differentiate or integrate functions! The key is to remember the definitions of derivative and definite integral: 2

Numerical Differentiation For small x, we have the following estimate for f’(x): Using the MATLAB commands “linspace” and “diff”, we can find reasonable approximations to f’(x), provided x is small and f is differentiable!

Numerical Differentiation Example 1: Use MATLAB to find the numerical derivative of y = sin(x) on the interval [0, 2]. Compare this estimate for dy/dx to the actual derivative of y.

Numerical Differentiation Example 1 (cont.): Try each of the following commands with a = 0, b = 2*pi, and n = 50. linspace(a, b, n) a:(b-a)/(n-1):b What do you notice? Try “linspace(a,b)”. What happens in this case?

Numerical Differentiation Example 1 (cont.): Next, try these commands: x = linspace(1, 10, 10) diff(x) What happens? In general, for x = [a, b, c, d], diff(x) = [b-a, c-b, d-c]. Now we are ready to estimate dy/dx!

Numerical Differentiation Example 1 (cont.): Enter the following commands to create an estimate for dy/dx, which we’ll call yprime: x = linspace(0, 2*pi, 1000); y = sin(x); deltax = diff(x); deltay = diff(y); yprime = deltay./deltax;

Numerical Differentiation Example 1 (cont.) To compare our estimate to the actual derivative, let’s look at a table of the first 10 values of yprime and cos(x), via concatenation: [yprime(1:10); cos(x(1:10))]’. Note the use of the colon (:) and transpose (‘) commands!

Numerical Differentiation Example 1 (cont.) Let’s also compare graphically! One way to do this is with the “subplot” command. Try the following commands: subplot(1,2,1) plot(x(1:999), yprime, 'r‘) title(‘yprime = \Deltay/\Deltax’) subplot(1,2,2) plot(x(1:999), cos(x(1:999)),‘b') title(‘dy/dx = cos(x)’) Why can’t we just use “x, yprime”, etc. in our plot commands?

Numerical Integration For integrable function f(x), choosing xi = (b-a)/n and xi* to be the right endpoint of the ith subinterval, i.e. xi*=a+i*(b-a)/n, we get the following estimate for large n: Using the “sum” command, we can find an estimate for definite integrals via Riemann sums!

Numerical Integration Example 2: Use MATLAB to numerically estimate the definite integral Compare this estimate as n gets larger to the actual value for the integral.

Numerical Integration Example 2 (cont.) Enter the following commands to compute Riemann sums for our integral! a = 0; b = 1; n = 50; deltax = (b-a)/n; xstar = a+deltax:deltax:b; Rn = sum((xstar.*xstar) *deltax)

Function M-Files In addition to using M-files to run scripts, we can use them to create functions! Function M-files can accept input and produce output. One example of a function defined by a function M-file is “linspace”. Let’s make a function via an M-file!

Function M-Files Within the M-file script editor, type the following: function y = Sample1(x) %Here is where you put in information about how the function is used. %The syntax and variables can be outlined here as well. y = x + x.^2 – x.^4; Save the file as Sample1.m on the Desktop and make sure the Path is set to see the file!

Function M-Files To use the new function, for example to find the value of Sample1(x) at x = 3, type: Sample1(3) Find the numerical derivative of Sample1(x) and plot both y = Sample1(x) and its numerical derivative on [-1,1].

References Using MATLAB in Calculus by Gary Jenson 16