Math 375 Natalia Vladimirova (many ideas, examples, and excersises are borrowed from Profs. Monika Nitsche, Richard Allen, and Stephen Lau)

Transcription

1 Natalia Vladimirova (many ideas, examples, and excersises are borrowed from Profs. Monika Nitsche, Richard Allen, and Stephen Lau) January 24, 2010

2 Starting Under windows Click on the Start menu button Click on the menu entry Under Linux Use SSH to log onto linux.unm.edu Type matlab Starting Desktop Tools Getting Help Saving your work

3 Desktop Tools Command Window type commands Workspace view program variables double click on a variable to see it in the Array Editor Command History view past commands Current Directory view and select files in the current directory the current directory can be changed in the toolbar To switch to the default windows arrangement, go to menu Desktop > Desktop Layout > Default Starting Desktop Tools Getting Help Saving your work

4 Getting Help Using the Help Browser From command window >> help >> help plot Running demos >> demos >> help demos Starting Desktop Tools Getting Help Saving your work On the web

5 Saving your work in class, create your directory (folder) in T: drive files reside on the current directory or the search path >> addpath T:\myfolder >> addpath ~/matlab you can save a whole session using >> diary "diary_jan23.txt" >> diary off >> diary on Starting Desktop Tools Getting Help Saving your work bring your flash card (floppy, CD-RW) to save class work or upload your files to your UNM storage (?)

6 Variables creating a variable ( is case sensitive); semicolon suppresses output >> x = 5; >> x = 5 x = 5 checking variables >> whos Variables Build-in constants Arithmetic operators Logical operators Math functions Scripts deleting a variable >> clear x deleting all variables >> clear

7 Build-in constants Imaginary unit i = j = 1 and π >> clear >> pi pi = >> i i = 0 + 1i >> j j = 0 + 1i Variables Build-in constants Arithmetic operators Logical operators Math functions Scripts Build-in constants can be overridden >> i = 8 i = 8 >> clear >> i i = 0 + 1i

8 Arithmetic operators + addition subtraction * multiplication / division ˆ power complex conjugate and transpose.* element-by-element mult./ element-by-element div.ˆ element-by-element power. transpose Variables Build-in constants Arithmetic operators Logical operators Math functions Scripts

9 Relational and logical operators == equal = not equal < less than <= less than or equal > greater than >= greater than or equal & AND OR NOT Variables Build-in constants Arithmetic operators Logical operators Math functions Scripts

10 Mathematical functions Elementary functions (sin, cos, sqrt, abs, exp, log10, round) >> help elfun Advanced functions (bessel, beta, gamma, erf) Example >> help specfun >> help elmat >> x=2.3; a=sin(x), b=cos(x) Variables Build-in constants Arithmetic operators Logical operators Math functions Scripts a = b = >> c = ( sin(x) )^2 + ( cos(x) )^2 c =

11 scripts Example of script file (myscript.m) myscript.m % This is a comment. % convert cartesian coordinares (x,y) % to polar coordinates (r,theta) x = 4; y = 3; r = sqrt(x^2 + y^2) theta = atan(y/x) Variables Build-in constants Arithmetic operators Logical operators Math functions Scripts % continuation character is... Save it in your working directory Call it from the command window >> myscript

12 Creating vectors >> x = [ ] % separated by spaces x = >> x = [1, 2, 5, 3] % separated by commas x = >> y = [6; 7; 4; 8] >> y = x y = y = Creating vectors More ways... Creating matrices Addition Multiplication Accessing elements... more Matrix operations Matrix size Exercises Note: A = transpose ( A )

13 More ways of creating vectors Using increment x 1 : x : x n >> x = 0:0.2:1 x = Using linspace (x 1, x n, n) >> x = linspace(0,1,6) x = Creating vectors More ways... Creating matrices Addition Multiplication Accessing elements... more Matrix operations Matrix size Exercises Note: both points are included, x = xn x1 n 1

14 Creating matrices Explicitly >> A = [1 2 3; 4 5 6; 7 8 9] A = Using functions >> M = 2; N = 3; >> a = zeros(m, N) >> b = ones(m, N) >> c = rand(m, N) >> I = eye(n) Creating vectors More ways... Creating matrices Addition Multiplication Accessing elements... more Matrix operations Matrix size Exercises scalar is 1 1 matrix vector is 1 N or N 1 matrix you can create N 1 N 2... N n matrices

15 Matrix addition (or subtraction) element by element C = A + B : ( a11 a 12 a 21 a 22 ) ( b11 b + 12 b 21 b 22 C ij = A ij + B ij ) ( a11 + b = 11 a 12 + b 12 a 21 + b 21 a 22 + b 22 >> A = [1 2; 3 4] >> B = [5 6; 7 8] A = B = ) Creating vectors More ways... Creating matrices Addition Multiplication Accessing elements... more Matrix operations Matrix size Exercises >> C = A + B C = being added must be the same size

16 Matrix multiplication not element by element C = AB: ( a11 a 12 a 21 a 22 ) ( b11 b 12 b 21 b 22 C ij = n k=1 A ikb kj ) ( a11 b = 11 + a 12 b 21 a 11 b 12 + a 12 b 22 a 21 b 11 + a 22 b 21 a 21 b 12 + a 22 b 22 >> A = [1 2; 3 4] >> B = [4 3; 2 1] A = B = ) Creating vectors More ways... Creating matrices Addition Multiplication Accessing elements... more Matrix operations Matrix size Exercises >> C = A * B >> C = A.* B C = C = For element by element multiplication use.*

17 Accessing elements of a matrix >> A = [1 2 3; 4 5 6; 7 8 9] A = A(i, j) subscription - numbering starts from 1 (Fortran style) >> A(2,3) ans = 6 Creating vectors More ways... Creating matrices Addition Multiplication Accessing elements... more Matrix operations Matrix size Exercises whole row >> x = A(2,:) x = whole column >> y = A(:,3) y = 3 6 9

18 Accessing elements of a matrix (cont.) Matrix elements can be accessed by index array. Try the following: >> v = 10:10:80 >> ind = [3,4,7] >> v(ind) >> v(ind) = or by condition: >> v = 10:10:80 >> v>30 >> v(v>30) = -1 Creating vectors More ways... Creating matrices Addition Multiplication Accessing elements... more Matrix operations Matrix size Exercises

19 Matrix operations [ ] concatenation >> x = [ zeros(1,3) ones(1,2) ] x = ( ) subscription >> x = [ ]; >> y = x(2) y = 3 Creating vectors More ways... Creating matrices Addition Multiplication Accessing elements... more Matrix operations Matrix size Exercises >> y = x(2:4) y = 3 5 7

20 Finding matrix size Explain the output of the following command >> A = [1 2 3; 4 5 6] >> ndims(a) >> length(a) >> size(a) >> [n m] = size(a) >> sum(a) >> sum(a,1) >> sum(a,2) Creating vectors More ways... Creating matrices Addition Multiplication Accessing elements... more Matrix operations Matrix size Exercises

21 What happens when you run the following commands? Why? >> x = 0:1:5 >> y = x*x >> y = x.*x >> y = x^2 >> y = x *x >> y = x*x >> y = 2*x >> y = 1./x >> x = 0:0.1:1 >> y = x(4:8) >> y = x(5:) >> y = x([1,5,7,11]) Creating vectors More ways... Creating matrices Addition Multiplication Accessing elements... more Matrix operations Matrix size Exercises >> A = [1 2 3; 4 5 6; 7 8 9] >> B = A((2:3),(2:3)) >> C = sin([0, 0.5*pi, pi, 1.5*pi])

22 Simple plot plots arrays. Arrays must be created first! >> t = 1:1:7 >> plot(t, sin(t)) More points - smoother line >> x = 1:0.1:7; >> y = sin(x); >> plot(x, y) Simple plot Label you plots! Saving plots Markers Multiple Plots In class exercise Note: y(x) is plotted, not x(t) and y(t). Vectors x and y must be the same length. To create a graph in logaritmic coordimates use loglog, semilogx, or semilogy instead of plot.

23 Label you plots! Once we created the plot... >> x = linspace(0, 2*pi, 50); >> y = sin(x); >> plot(x, y, o-r ) % plot with red circles we can add features to it >> axis([0, 2*pi, -1.1, 1.1]) >> xlabel( x ); ylabel( y ) >> title( Plot of the sine function ) >> leg( y(x) = sin(x), Location, NorthEast ) >> grid on Simple plot Label you plots! Saving plots Markers Multiple Plots In class exercise You can edit figure using menu functions.

24 Saving plots to files Save your figure as *.fig - this way you can modify it later. Save PDF on letter-size paper. Here gcf is the handle to current figure ( get current figure ). The plot position on the paper is given as the coordinates of upper-left and lower-right corners >> set(gcf, papersize, [8.5, 11],... paperposition, [1, 1, 6, 4]) >> saveas(gcf, myplot.pdf, pdf ) Simple plot Label you plots! Saving plots Markers Multiple Plots In class exercise If figure to be inserted into a document, use paper size same as plot size >> set(gcf, papersize, [6, 4],... paperposition, [0, 0, 6, 4]) >> saveas(gcf, myplot.pdf, pdf )

25 Marker options (type help plot to remind) b blue. point - solid g green o circle : dotted r red x x-mark -. dashdot c cyan + plus -- dashed m magenta * star (none) no line y yellow s square k black d diamond w white v triangle (down) ^ triangle (up) < triangle (left) > triangle (right) p pentagram h hexagram Simple plot Label you plots! Saving plots Markers Multiple Plots In class exercise

26 Multiple graphs, panels, and figures Multiple graphs on the same plot (also see help hold) >> x = linspace(0, 2*pi, 100); >> y1 = cos(3*t).* exp(-0.5*t); >> y2 = sin(3*t).* exp(-0.5*t); >> plot(t,y1, x, t,y2, * ) To open new figure (new window) >> figure >> figure(1) %make figure 1 current figure Simple plot Label you plots! Saving plots Markers Multiple Plots In class exercise To divide figure to panels >> subplot(2,2,1) >> plot(t,y1) >> subplot(2,2,2) >> plot(t,y2) >> subplot(2,2,3) >> plot(y1,y2)

27 In class exercise By plotting y(t) below with different values of parameter a find a at which y(t) passes through given experimental points as close as possible y(t) = e t 2 sin(at) t = y = Simple plot Label you plots! Saving plots Markers Multiple Plots In class exercise

28 User defined functions Unlike scripts, functions can take arguments and return values Like scripts, functions reside in M-files (*.m) Name of the file = name of the function Example of function file mycomplex.m mycomplex.m function z = mycomplex(x,y) z = x + i*y; User defined more... more Function handles here x and y are arguments, z is returned value, and mycomplex is the name of the function Call it from command line or from another function >> c = mycomplex(a,b)

29 User defined functions (cont.) can take matrices as arguments can return multiple values You can put several functions in a single M-file. Example: function file myplot.m myplot.m function myplot [x,y] = mydata(n); plot(x,y); User defined more... more Function handles function [x,y] = mydata(n) x = linspace(0, 2*pi, n); y = sin(x); Can we call mydata(n) from command line?

30 User defined functions (cont.) Example: function file mycart2pol.m mycart2pol.m function[ radius, theta ] = mycart2pol(x, y) % convert cartesian coordinares (x, y) % to polar coordinates (r, theta) theta = atan2(y, x); radius = sqrt(x.^2 + y.^2); User defined more... more Function handles What happens when you call mycart2pol as following? >> mycart2pol(1,1) >> r = mycart2pol(1,1) >> [r th] = mycart2pol(1,1) >> [r th] = mycart2pol([ ], [0 0 0]) >> help mycart2pol

31 Function handles There are a number of useful functions which take function handle as an argument Example: to numerically evaluate integral cube.m 2 1 x 3 dx = 1 4 ( ) = 3.75 using function quad we first create file cube.m function y = cube(x) y = x.^3; User defined more... more Function handles and then pass the function handle (@cube) and integration limits (1,2) to quad >> quad(@cube, 1, 2) or create inline function, defined by the string >> f = inline( x.^3 ); >> quad(f, 1, 2)

32 For loop The for loop executes a statement or group of statements a predetermined number of times. for x = x 1 : x : x n statement The increment x can be omitted; default x = 1 for x = x 1 : x n statement For loop For loop: examples For loop: examples While loop If statement If-else statement If-else example Other statements Simple text I/O The index of a for loop can be an array. v=v 1 : v : v n for x = v statement For loops can be nested

33 For loop: examples >> format compact >> for n = 1:5 disp(n) >> for n = 0.5:0.1:0.8 disp(n) >> v = [2, 4, 8, 16]; >> for n = v disp(n) For loop For loop: examples For loop: examples While loop If statement If-else statement If-else example Other statements Simple text I/O >> for j = 1:3 for i = 1:3 A(i,j) = i+j;

34 For loop: examples Create vector [ ] in a classical way >> v = zeros(1,5) % allocate space >> for i=1:5 % cycle through the elements v(i)=i % assign value to each element Note: allocation is optional (but it makes the code run faster). Try the following: >> clear v >> for i=1:5 v(i)=i For loop For loop: examples For loop: examples While loop If statement If-else statement If-else example Other statements Simple text I/O >> v = zeros(1,5) >> a = v(8) >> v(8) = 8

35 While loop The while loop executes a statement or group of statements repeatedly as long as the controlling expression is true while condition statement >> x=1; >> while x > 0.01 x = x/2 For loop For loop: examples For loop: examples While loop If statement If-else statement If-else example Other statements Simple text I/O

36 If statement If evaluates a logical expression and executes a group of statements based on the value of the expression. if condition statement Example: find the maximum element in an array: >> v=rand(1,6) >> vmax = -1; >> for i=1:length(v) if v(i) > vmax vmax = v(i); >> disp(vmax) For loop For loop: examples For loop: examples While loop If statement If-else statement If-else example Other statements Simple text I/O But, of coarse, it is easier just to do >> vmax = max(max(v))

37 If-else statement if-else can take care for both branches of the fork... if condition1 statement1 else statement2... or multiple branches if condition1 statement1 elseif condition2 statement2 elseif condition3 statement3 else statement4 For loop For loop: examples For loop: examples While loop If statement If-else statement If-else example Other statements Simple text I/O

38 If-else example Script rps.m simulates a rock-paper-scissors player rpm.s % simulate a rock-paper-scissors player x = rand(1) % get random number between 0 to 1 x = x*3 % rescale x so that 0 < x < 3 if x<1 disp( rock! ) elseif x<2 disp( paper! ) else disp( scissors! ) For loop For loop: examples For loop: examples While loop If statement If-else statement If-else example Other statements Simple text I/O

39 Other program control statements continue passes control to the next iteration of the loop break terminates the execution of a loop return exits the currently running function switch executes statements based on the value of a variable or expression try, catch error control statements For loop For loop: examples For loop: examples While loop If statement If-else statement If-else example Other statements Simple text I/O Note: there is no goto statement in

40 Simple text input and output C-style output to the screen >> fprintf( square root of %d is %6.3f\n, 3, sqrt(3)) square root of 3 is The same output can be directed to a file >> fid=fopen( mydata.txt, wt ); >> fprintf(fid, %s\n, % k sqrt(k) ); >> for k=1:3 fprintf(fid, %d >> fclose(fid); %6.3f\n, k, sqrt(k)); For loop For loop: examples For loop: examples While loop If statement If-else statement If-else example Other statements Simple text I/O An easy way to load text data >> load mydata.txt >> disp(mydata)