Matlab Section. November 8, 2005

Transcription

1 Matlab Section November 8,

2 1 General commands Clear all variables from memory : clear all Close all figure windows : close all Save a variable in.mat format : save filename name of variable Load a variable in.mat format : load filename Calculate the time it takes for executing a block of code : tic; block of code toc; Getting help : help Cancel code execution at any point : ctr+c Built-in functions : help matlab \ elfun (and many more...) 2 Vectors Defining a vector : x = (x 1, x 2,..., x n ) Row vector : x = [x 1, x 2,..., x n ] or x = [x 1 x 2... x 4 ] Column vector : x = [x 1 ; x 2 ;...; x n ] 2

3 Other ways : x = min value : step : max value, or x = linspace(min value, max value, number of elements) Special vectors Zero : (x = 0): x = zeros(1, n) (row vector), or x = zeros(1, n) (column vector) One : (x = (1, 1,..., 1)): x = ones(1, n) (row vector), or x = ones(n, 1) (column vector) Vector manipulation Finding the ith element of vector x : x(i) Complex conjugate (x ) : x Transpose (x T ) : x. Sum (a + b = (a 1 + b 1, a 2 + b 2,..., a n + b n )) : a + b (the same for subtraction - ) Product (a b = (a 1 b 1, a 2 b 2,..., a n b n )) : a. b (The same for division / ) Inner product (a T b = a 1 b 1 + a 2 b a n b n ) : a b Raising to the nth power : x.ˆ n Minimum value : min(x), or Minimum value and index of minimum element : [in, inn] = min(x) (the same for maximum (max)) Absolute value : abs(x) Sum of the elements of a vector ( n i=1 x i ) : sum(x) 3

4 3 Matrices Defining a matrix : a 11 a a 1n a 21 a a 2n A = (1) a n1 a n2... a nn A = [a 11, a 12,..., a 1n ; a 21, a 22,..., a 2n ;...; a n1, a n2,..., a nn ] Matrix manipulation : The same with vectors Finding the ijth element : A(i, j) Finding the ith column vector : A(:, i) Finding the ith row vector : A(i, :) Diagonal of a A : diag(a) The identity matrix I n : eye(n) Inverse (A 1 ) : inv(a) Exponential of a matrix (e A ) : expm(a) Solving a linear system of equations (Ax = b) : x = A/b Eigenanalysis of A : [U, S] = eig(a) 4

5 4 Plotting Suppose that we would like to plot f(x) = x 2 for x [ 2, 2]. First we define a vector x = linspace( 2, 2, 100) (or using the other way described earlier). Then we define f = x. 2 and we now have two vectors x and f. Now we can plot f(x) using the command : plot(x, f). You can add or change pretty much everything in your figure and there are two ways to do that : Either by using the various menus in the Figure window, or by using various commands. Both will be shown in section, but only the latter will be described in these notes. Two of the parameters you can add to the plot command are the color of the curve and the style. The default color is blue, but you can also choose other colors by adding the color parameter to plot : plot(x, f, color ), where color is : r for red, k for black, g for green, y for yellow, m for magenta. Inside the color parameter and before the color string, you can also add the style parameter to have a dashed ( ), dotted ( : ), or dash-dotted ( -. ) line. So for example the command plot(x, f, : r ) plots a dotted red curve. You can also add labels for the x and y axis, add a text somewhere in the Figure and finally 5

6 a title by using the commands : xlabel( your label for x axis ) ylabel( your label for y axis ) gtext( add text ) (as soon as you type this command and move the cursor towards the Figure window, the cursor becomes two lines that mark the place you would like to add the text) title( your title ) Finally you can change the axis in the figure, by using the command : axis([min x value max x value min y value max y value]) 5 For loops When you would like to repeat a block of code, say n times, you can put it inside a for loop as shown here : fori = 1 : n; block of code Example : Suppose that you want to define a function f that depends on two variables x and y. Say for example that f(x, y) = sin(x) sin(y). Define first the two vectors x = 6

7 linspace( 2, 2, 100), y = linspace( 2, 2, 100). Then you can define a matrix that its rows correspond to y variations and columns correspond to x variations. That is the f ij element of your matrix f corresponds to the value of f(x, y) at x j and y i. So the ijth element of f should be : f(i, j) = sin(x(j)) sin(y(i)). But since you want this for every i,j you have to put this command into two consecutive for loops : for i = 1 : length(y); for j = 1 : length(x); f(i, j) = sin(x(j)) sin(y(i)); Note that this is not the most effective way to define the elements of f timewise but we just wanted a simple example for the use if for loops. 6 If loops If you would like to execute a block of code when a specific condition is satisfied, you can put it inside an if loop as shown here : if condition ; block of code 7

8 else; (that is if the above condition is not satisfied) Example : We build on our previous example and suppose that we would like now to separate the positive values of f from the negative values. In order to do that we define two matrices f + and f and inside the for loop we set the conditions that : f p (i, j) = f(i, j) if f(i, j) > 0 and f n (i, j) = f(i, j) if f(i, j) < 0. We can do that by typing the following block of code : for i = 1 : length(y); for j = 1 : length(x); if f(i, j) > 0; f p (i, j) = f(i, j); else; f n (i, j) = f(i, j); 8

9 7 Integrals Matlab has a built-in function called quad that computes the integral of a function f (int b af(x)dx) : quad( function, a, b). Let us choose a specific example, say 2 0 x2 dx. You have to first define f = x 2 as a function (.m file) as follows : function y = myfun(x) y = x.ˆ 3; and save this as an.m file (Matlab will choose the default name myfun for you). Then you can calculate the integral using quad : int = quad(@myfun, 0, 2) There is a second way to calculate the integral yourself by using the simple Simpson rule : ( ) b f(x)dx = f(x i ) f(a)/2 f(b)/2 x a i To translate this into a Matlab code, we first define the vector x = linspace(0, 2, 100) and f : f = x.ˆ 2. Now x = x(2) x(1) and i f(x i ) = sum(f). So the integral will be given by : int = (sum(f) f(1)/2 f(length(x))/2) (x(2) x(1)). 9

10 8 Differential equations Matlab has a built-in function called ode that computes the solution of the differential equation dy dt = f(y, t). In fact there are several versions of the same function (ode45, ode23, ode113, etc) that use a different method for integrating forward : [t, y] = ode113(@ function, [t 0 t final ], [ initial conditio Let us choose a specific example, say to calculate the solution to dx dt = x, x(0) = 1, 10 t 0 Again you will have to define f as a function (.m file) as follows : function y = myfun(x) y = x; and then calculate the solution using ode : [t, y] = ode113(@myfun, [0 10], [1]) Now, t is a vector of a certain length (Matlab has already chosen the length according to the integration scheme) and y is a matrix with 2 columns and length(t) rows. The first column is the solution y(t) and the second the time derivative of y. You can plot the solution by typing : plot(t, y(:, 1)) There is a second way to integrate the system yourself by using the simple Euler integration 10

11 scheme : dy dt y t+δt y t δt = f(y, t) (2) y t+δt = y t + f(y, t)δt (3) To translate this into a Matlab code, we choose a specific example, say to calculate the solution to dx dt = x, x(0) = 1 We first define the vector of time T = linspace(0, 10, 100) and the vector x = zeros(length(t ), 1). We plug in the initial condition : x(1) = 1 and we can now perform the integration : for it = 1 : length(t ) 1; x(it + 1) = x(it) + x(it) (T (2) T (1)); and finally plot the solution along with the analytically calculated one (x(t) = e t ): plot(t, x, T, exp(t )) 11