•Matlab: Matrix Laboratory
•Matlab is both a computer programming language and a software environment for using language effectively.
•Matlab has a number of add-on software modules, called toolboxes, that perform more specialized computations, dealing with applications such as
–Image and signal processing
–Financial analysis
–Control systems design
–Communication systems
–Fuzzy logic
–Neural networks
–Filter design
–Optimization
–Partial differential equations
–Statistics
–System identification
–Virtual reality
–Wavelet
–…
•Matlab Windows and Menus
•Basic Operations and Arrays
•Files, Functions
•Plotting with Matlab
•Programming with Matlab
•Numerical Methods for Differential Equations
Matlab
• Enter commands and expressions at the prompt position (>>) in command window.
• Ex: r = 8/10
• Matlab supports many mathematical functions, most of which are used in the same way you write them mathematically.
• abs(x) Absolute value |x|
• sign(x) Sign, returns −1if x<0, 0 if x =0 ,1if x>0
• exp(x) Exponential
• log(x) Natural logarithm ln(x)
• log10(x) Common (base 10) logarithm
• sqrt(x) Square root
• rem(x,y) Remainder of x/y. For example, rem(100,21) is 16. Also called the modulus function.
| • Row vector • >> r = [2,4,10]; • >> n = [2,4,10] • n = • 2 4 10 • >> n = [2 4 10] • n = • 2 4 10 • Column vector • >> r = [2; 4;10] • r = 2 • 4 • 10 • >> y = [ 2 4 10]’ • y = 2 • 4 • 10 | • Ex: r =[ 2 4 10] • >> v = 5*r • v = [10 20 50] • >>w = r + v • w = [12 24 60] • >> u = [r, w] • u = [2 4 10 12 24 60] • Ex: Creat a row vector from 5 to 30, incremented by 0.1. • >> t = [5:0.1:30] • t = 5, 5.1,5.2,…, 29.8,29.9,30. |
| • Matrix • >> a = [2,4,10;16,3,7] • a = • 2 4 10 • 16 3 7 • >> A = [6, -2;10,3;4,7], B = [9,8;-5,12]; • >> A*B • ans • 64 24 • 75 116 • 1 116 | · Array operations · Ex: a =[1:5]; b = [3:7]; · >> a = [1:5] · a = · 1 2 3 4 5 · >> b = [3:7] · b = · 3 4 5 6 7 · >> c = a .* b · c = 3 8 15 24 35 · >> c = a ./ b · c = 0.333 0.5 0.6 0.667 0.7143 · >> c = a .^ b · c = 1 16 243 4096 78125 |
• Two special functions in Matlab can be used to generate new vectors containing all zeros or all
• ones.
• zeros(m,1) Returns an m-element column vector of zeros
• zeros(1,n) Returns an n-element row vector of zeros
• ones(m,1) Returns an m-element column vector of ones
• ones(1,n) Returns an n-element row vector of ones
• Examples:
• >> x = zeros(3,1)
• x=
• 0
• 0
• 0
• >> y = ones(1,3)
• y=
• 1 1 1
• Basic Operations and Arrays
• 
Ex:
Ex:•
• >> x = [0:0.01:1];
• >> y = exp(-x) .* sin(x) ./sqrt(x .^2 +1);
• >>plot(x,y)
| Matrixes and arrays • > >f=[ 123 ;456 ] • f= • 123 • 456 • > >g=f ’ %transpose % • g= • 14 • 25 • 36 • >> x = 0:4; • >> y = 5:5:25; • >> A = [x’ y’] • A= • 05 • 11 0 • 21 5 • 32 0 • 42 5 | • > >A=[ 123 ;456 ] • A= • 123 • 456 • >> s = size(A) • s= • 23 • >> [r,c] = size(A) • r= • 2 • c= • 90 • 3 • >> r = size(A,1) • r= • 2 • >> c = size(A,2) • c= • 3 |
Files and Functions
• Matlab uses two types of M-files: script files and function files.
• A script file contains a sequence of Matlab commands and is useful when you need to use many commands or arrays with many elements.
•
• A function file is useful when you need to repeat a set of commands several times.
• Example of a simple script file computes the matrix C=AB and displays it on the screen.
• % Program ‘prod_abc.m’
• %This program computes the %matrix product C = A * B,and %displays the result.
• A=[1,4;2,5];
• B=[3,6;1,2];
• C=A*B
• In the Matlab command window type the script file’s name prod_abc to execute the program.
• >>prod_abc
• C=
• 7 14
• 11 22
• Function file: All the variables in a function file are local, which means their values are available only within the function. Function files are useful when you need to repeat a set of commands several times. Its syntax is as follows:
• function [output variables] = function_name(input variables);
• Note:
• (i) output variables are enclosed in square brackets, the square brackets are optional when there is only one output.
• (ii) input variables are enclosed with parentheses
• (iii) function_name must be the same as the filename in which it is saved (with the .m extension)
• (iv) function is called by its name.
• Ex: , find the value of x that gives a minimum of y for .
• function y=f2(x)
• y=1-x .* exp (-x);
• Save it as f2.m
• In command window, type x=fmin(‘f2’, 0, 5). The answer is x = 1. To find the minimum value of y, type y = f2(x).
• XY plotting functions
• Ex: >>x = [0:0.1:52];
• >>y = 0.4*sqrt(1.8*x);
• >>plot(x,y)
• >>xlabel(‘Distance (miles)’)
• >>ylabel(‘Height (miles)’)
• >>title(‘Tocket height as a function of downrange distance’)
• The function plot command fplot
• Syntax
• fplot(‘string’,[xmin xmax])
• or
• fplot(‘string’,[xmin xmax ymin ymax])
• Ex: >> f = ‘cos(tan(x))-tan(sin(x))’;
• >>fplot(f, [1 2])
| Plotting with Matlab • Ex: plot the hyperbolic sine and tangent • x = [0:0.01:2]; • y = sinh(x); • z = tanh(x); • plot(x,y,x,z,’- -‘), xlabel(‘x’), … • ylabel(‘Hyperbolic sine and tangent’),… • legend(‘sinh(x)’, ‘tanh(x)’) | • Data markers, line types and colors • Data markers: . * x o + • Line types: - -- -. : • Colors: black(k) Blue(b) Green(g) Red(r) Yellow(y) |
• Surface mesh plots (z=f(x,y))
• [X,Y]=meshgrid(x,y), if x=[xmin:xspacing:xmax], y=[ymin:yspacing:ymax]
• The meshgrid function generates the grid point in the xy plane, and then evaluate the function f(x,y) at these points.
 |
• Ex: with a spacing of 0.1.
• >>[X,Y]=meshgrid(-2:0.1:2);
• >>Z=X.*exp(-((X-Y.^2).^2+Y.^2));
• >>mesh(X,Y,Z), xlabel(‘x’), ylabel(‘y’),zlabel(‘z’)
• Contour plots
• Ex:
• >>[X,Y]=meshgrid(-2:0.1:2);
• >>Z=X.*exp(-((X-Y.^2).^2+Y.^2));
• >>contour(X,Y,Z), xlabel(‘x’), ylabel(‘y’)
| Programming with Matlab • Relational operators • < less than • <= less than or equal to • > greater than • >= greater than or equal to • = = equal to • ~= not equal to • Logical operations • ~ not • & and • | or • xor(a,b) exclusive or | Conditional statements (1) if logical expression statements end Ex: if x >= 0 y = sqrt(x) end or it can be written on a single line if x >= 0, y = sqrt(x), end |
| (2) Nest if statements if logical expression 1 statement group 1 if logical expression 2 statement group 2 end end | Nested if Statements • if statements may be nested, as shown in the following example: If d<5 0 count = count + 1; disp(d); If b>d b=0 ; end end |
| (3) The else statement if logical expression statement group 1 else statement group 2 end Ex: if x >= 0 y= sqrt(x) else y = exp(x) – 1 end | (4) The elseif statement if logical expression 1 statement group 1 elseif logical expression 2 statement group end Ex: y = lnx if x >= 5, y = sqrt(x) if 0<= x < 5 if x >= 5 y = log(x) elseif x >= 0 y = sqrt(x) end |
| Loops for loop variable = m:s:n Statements end m:s:n=>initial value m, incremented by the value s and terminated by the value n. Note: s may be negative, k=10:-2:4 produce k = 10,8,6,4. if s is omitted, the step value defaults to one. While loop is used when the looping process terminates because a specified condition is satisfied, and thus the number of passes is not known in advance. while logical expression statements End Ex: x = 5; while x <25 disp (x) x = 2*x-1; end The results displayed by the disp statement are 5, 9, and 17. For the while loop to function properly, the following two conditions must occur; • The loop variable must have a value before the while statement is executed. • The loop variable must be changed somehow by the statements. | Ex: implied loops x= [0:5:100]; y = cos(x); To achieve the same result using a for loop, we must type for k = 1: 21 x = (k-1) *5; y(k) = cos(x); end Switch structure: Anything programmed using switch can also be programmed using if structures. However, for some applications the switch structure is more readable than code using the if structures. switch input expression (scalar or string) case value1 statement group 1 case value2 statement group 2 … otherwise statement group n end |
| Ex: variable angle switch angle case 45 disp(‘Northeast’) case 135 disp(‘Southeast’) case 225 disp(‘Southwest’) case 315 disp(‘Northwest’) otherwise disp(‘Direction Unknown’) end | • Consider writing a user-defined function that searches a matrix input argument for the element with the largest value and returns the indices of that element. • function [r,c] = indmax(x) • % INDMAX returns the row and column indices of • % the maximum-valued element in x • [m n] = size(x); • xmax = x(1,1); • r=1; c=1; • for k=1:m for l=1:n if x(k,l)> xmax xmax = x(k,l); r=k; c=l; end end • end |
| • Testing this function: • >> A=3; B=[2 4 3]; C=[3 2; 6 1]; • >> [r c]=indmax(A) • r= • 1 • c= • 1 • >> [r c]=indmax(B) • r= • 1 • c= • 2 • >> [r c]=indmax(C) • r= • 2 • c= • 1 | |
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