M ATLAB Tutorial Covered by: Toby Heyn University of Wisconsin, Madison

Before getting started … Acknowledgement: Almost entirely, this tutorial compiled from bits of information gathered from various internet sources  It is available for download from SBEL website in PPT format  Please let me know of any mistakes you find me at The right frame of mind: You will not be able to say at the end of workshop that you know MATLAB but rather that you have been exposed to MATLAB (I don ’ t know MATLAB myself, I ’ m just using it … ) Use MATLAB ’ s “ help ”, this is your first stop Second stop: search the web for examples that come close to what you need  You learn how to use MATLAB by using it, that ’ s why the start might be slow and at times frustrating

Contents  What is Matlab?  MATLAB Components  MATLAB Desktop  Matrices  Importing and Exporting Data  Elementary math with MATLAB  M-file Programming  Functions vs. Scripts  Variable Type/Scope  Debugging MATLAB functions

What is MATLAB? Integrated Development Environment (IDE) Programming Language Collection of Toolboxes A package that excels in Linear Algebra support

MATLAB as an IDE Integrated development environment (IDE)  Write your own code for computation  Good visualization (plotting) tools  Easy-to-use environment Command Window Command History Help Browser Workspace Browser Editor/Debugger

MATLAB as Programming Language High-level language Data types Functions Control flow statements Input/output Graphics Object-oriented programming capabilities

Toolboxes Collections of functions to solve problems from several application fields. DSP (Digital Signal Processing) Toolbox Image Toolbox Wavelet Toolbox Neural Network Toolbox Fuzzy Logic Toolbox Control Toolbox Multibody Simulation Toolbox And many many other … ( Visit for instance amazing number of toolboxes available: if you need something, it ’ s out there somewhere available for download)

MATLAB for [Linear] Algebra

Calculations at the Command Line »a = 2; »b = 5; »a^b ans = 32 »x = 5/2*pi; »y = sin(x) y = 1 »z = asin(y) z = »a = 2; »b = 5; »a^b ans = 32 »x = 5/2*pi; »y = sin(x) y = 1 »z = asin(y) z = Results assigned to “ans” if name not specified () parentheses for function inputs Semicolon suppresses screen output MATLAB as a calculator Assigning Variables A Note about Workspace: Numbers stored in double-precision floating point format » -5/( )^2 ans = » (3+4i)*(3-4i) ans = 25 » cos(pi/2) ans = e-017 » exp(acos(0.3)) ans = » -5/( )^2 ans = » (3+4i)*(3-4i) ans = 25 » cos(pi/2) ans = e-017 » exp(acos(0.3)) ans =

General Functions whos: List current variables and their size clear: Clear variables and functions from memory cd: Change current working directory dir: List files in directory pwd: Tells you the current directory you work in echo: Echo commands in M-files format: Set output format (long, short, etc.) diary(foo): Saves all the commands you type in in a file in the current directory called “ foo ”

Getting help help command ( >>help ) lookfor command ( >>lookfor ) Help Browser ( >>doc ) helpwin command ( >>helpwin ) Search Engine Printable Documents “ Matlabroot\help\pdf_doc\ ” Link to The MathWorks

Handling Matrices in Matlab

Matrices Entering and Generating Matrices Subscripts Scalar Expansion Concatenation Deleting Rows and Columns Array Extraction Matrix and Array Multiplication NOTE: we don ’ t have time to carefully look at all these topics. I want you to be aware that these facilities exist in MATLAB, and that you can access them when needed by first doing a “ help ” on that command

»a=[1 2;3 4] a = »b=[-2.8, sqrt(-7), (3+5+6)*3/4] b = i »b(2,5) = 23 b = i »a=[1 2;3 4] a = »b=[-2.8, sqrt(-7), (3+5+6)*3/4] b = i »b(2,5) = 23 b = i Any MATLAB expression can be entered as a matrix element (internally, it is regarded as such) In MATLAB, the arrays are always rectangular Entering Numeric Arrays NOTE: 1) Row separator semicolon (;) 2) Column separator space OR comma (,) Use square brackets [ ]

The Matrix in MATLAB Rows (m) Columns (n) A = A (2,4) A (17) Rectangular Matrix: Scalar:1-by-1 array Vector:m-by-1 array 1-by-n array Matrix:m-by-n array

»w=[1 2;3 4] + 5 w = »x = 1:5 x = »y = 2:-0.5:0 y = »z = rand(2,4) z = »w=[1 2;3 4] + 5 w = »x = 1:5 x = »y = 2:-0.5:0 y = »z = rand(2,4) z = Scalar expansion Creating sequences: colon operator (:) Utility functions for creating matrices. Entering Numeric Arrays

Numerical Array Concatenation (Tiling) »a=[1 2;3 4] a = »cat_a=[a, 2*a; 3*a, 4*a; 5*a, 6*a] cat_a = »a=[1 2;3 4] a = »cat_a=[a, 2*a; 3*a, 4*a; 5*a, 6*a] cat_a = Use [ ] to combine existing arrays as matrix “elements” Row separator: semicolon (;) Column separator: space / comma (,) Use square brackets [ ] Note: The resulting matrix must be rectangular 4*a

Array Subscripting / Indexing A = A(3,1) A(3) A(1:5,5) A(:,5) A(21:25) A(4:5,2:3) A([9 14;10 15]) A(1:end,end) A(:,end) A(21:end)’

Deleting Rows and Columns »A=[1 5 9; ; i+1] A = i »A(:,2)=[] A = i »A(2,2)=[] ??? Indexed empty matrix assignment is not allowed. »A=[1 5 9; ; i+1] A = i »A(:,2)=[] A = i »A(2,2)=[] ??? Indexed empty matrix assignment is not allowed. “:” is a VERY important construct in MATLAB

Matrix Multiplication »a = [ ; ]; »b = ones(4,3); »c = a*b c = »a = [ ; ]; »b = ones(4,3); »c = a*b c = [2x4] [4x3] [2x4]*[4x3] [2x3] a(2nd row).b(3rd column) »a = [ ; ]; »b = [1:4; 1:4]; »c = a.*b c = »a = [ ; ]; »b = [1:4; 1:4]; »c = a.*b c = c(2,4) = a(2,4)*b(2,4) Array Multiplication (componentwise operation)

Matrix Manipulation Functions zeros : Create an array of all zeros ones : Create an array of all ones eye: Identity Matrix rand: Uniformly distributed random numbers diag: Diagonal matrices and diagonal of a matrix size: Return array dimensions fliplr: Flip matrices left-right flipud: Flip matrices up and down repmat: Replicate and tile a matrix

Matrix Manipulation Functions transpose ( ’ ): Transpose matrix rot90: rotate matrix 90 tril: Lower triangular part of a matrix triu: Upper triangular part of a matrix cross: Vector cross product dot: Vector dot product det: Matrix determinant inv: Matrix inverse eig: Evaluate eigenvalues and eigenvectors rank: Rank of matrix

Elementary Math

Logical Operators Math Functions Polynomial and Interpolation

Logical Operations » Mass = [-2 10 NaN Inf 31]; » each_pos = Mass>=0 each_pos = » all_pos = all(Mass>=0) all_pos = 0 » all_pos = any(Mass>=0) all_pos = 1 » pos_fin = (Mass>=0)&(isfinite(Mass)) pos_fin = » Mass = [-2 10 NaN Inf 31]; » each_pos = Mass>=0 each_pos = » all_pos = all(Mass>=0) all_pos = 0 » all_pos = any(Mass>=0) all_pos = 1 » pos_fin = (Mass>=0)&(isfinite(Mass)) pos_fin = = = equal to > greater than < less than >= Greater or equal <= less or equal ~ not & and | or isfinite(), etc.... all(), any() find Note: 1 = TRUE 0 = FALSE

Elementary Math Function abs, sign: Absolute value and Signum Function sin, cos, asin, acos…: Triangular functions exp, log, log10: Exponential, Natural and Common (base 10) logarithm ceil, floor: Round to integer, toward +/-infinity fix: Round to integer, toward zero

Elementary Math Function round: Round to the nearest integer gcd: Greatest common divisor lcm: Least common multiple sqrt: Square root function real, imag: Real and Image part of complex rem: Remainder after division

Elementary Math Function Operating on Arrays max, min: Maximum and Minimum of arrays mean, median: Average and Median of arrays std, var: Standard deviation and variance sort: Sort elements in ascending order sum, prod: Summation & Product of Elements trapz: Trapezoidal numerical integration cumsum, cumprod: Cumulative sum, product diff, gradient: Differences and Numerical Gradient

END MATLAB for [Linear] Algebra

Programming and Application Development

Topics discussed … The concept of m-file in MATLAB Script versus function files The concept of workspace Variables in MATLAB Type of a variable Scope of a variable Flow control in MATLAB The Editor/Debugger

Function M-file function r = ourrank(X,tol) % rank of a matrix s = svd(X); if (nargin == 1) tol = max(size(X)) * s(1)* eps; end r = sum(s > tol); function r = ourrank(X,tol) % rank of a matrix s = svd(X); if (nargin == 1) tol = max(size(X)) * s(1)* eps; end r = sum(s > tol); function [mean,stdev] = ourstat(x) [m,n] = size(x); if m == 1 m = n; end mean = sum(x)/m; stdev = sqrt(sum(x.^2)/m – mean.^2); function [mean,stdev] = ourstat(x) [m,n] = size(x); if m == 1 m = n; end mean = sum(x)/m; stdev = sqrt(sum(x.^2)/m – mean.^2); Multiple Input Arguments use ( ) Multiple Output Arguments, use [ ] »r=ourrank(rand(5),.1); »[m std]=ourstat(1:9);

Basic Parts of a Function M-File function y = mean (x) % MEAN Average or mean value. % For vectors, MEAN(x) returns the mean value. % For matrices, MEAN(x) is a row vector % containing the mean value of each column. [m,n] = size(x); if m == 1 m = n; end y = sum(x)/m; Output Arguments Input Arguments Function Name Online Help Function Code

Script and Function Files Script Files Work as though you typed commands into MATLAB prompt Variable are stored in MATLAB workspace Function Files Let you make your own MATLAB Functions All variables within a function are local All information must be passed to functions as parameters Subfunctions are supported

The concept of Workspace At any time in a MATLAB session, the code has a workspace associated with it The workspace is like a sandbox in which you find yourself at a certain point of executing MATLAB The “Base Workspace”: the workspace in which you live when you execute commands from prompt Remarks: Each MATLAB function has its own workspace (its own sandbox) A function invoked from a calling function has its own and separate workspace (sandbox) A script does not lead to a new workspace (unlike a function), but lives in the workspace from which it was invoked

Variable Types in MATLAB Local Variables In general, a variable in MATLAB has local scope, that is, it’s only available in its workspace The variable disappears when the workspace ceases to exist Recall that a script does not define a new workspace – be careful, otherwise you can step on variables defined at the level where the script is invoked Since a function defines its own workspace, a variable defined in a function is local to that function Variables defined outside the function should be passed to function as arguments. Furthermore, the arguments are passed by value Every variable defined in the subroutine, if to be used outside the body of the function, should be returned back to the calling workspace

Variable Types in MATLAB Global Variables These are variables that are available in multiple workspaces They have to be explicitly declared as being global Not going to expand on this, since using global variables is a bad programming practice A note on returning values from a function Since all variables are local and input arguments are passed by value, when returning from a function a variable that is modified inside that function will not appear as modified in the calling workspace unless the variable is either global, or declared a return variable for that function

if ((attendance >= 0.90) & (grade_average >= 60)) pass = 1; end; if ((attendance >= 0.90) & (grade_average >= 60)) pass = 1; end; eps = 1; while (1+eps) > 1 eps = eps/2; end eps eps = 1; while (1+eps) > 1 eps = eps/2; end eps if Statement while Loops Flow Control Statements

a = zeros(k,k) % Preallocate matrix for m = 1:k for n = 1:k a(m,n) = 1/(m+n -1); end a = zeros(k,k) % Preallocate matrix for m = 1:k for n = 1:k a(m,n) = 1/(m+n -1); end method = 'Bilinear';... (some code here)... switch lower(method) case {'linear','bilinear'} disp('Method is linear') case 'cubic' disp('Method is cubic') otherwise disp('Unknown method.') end Method is linear method = 'Bilinear';... (some code here)... switch lower(method) case {'linear','bilinear'} disp('Method is linear') case 'cubic' disp('Method is cubic') otherwise disp('Unknown method.') end Method is linear for Loop: switch Statement: Flow Control Statements

Editing and Debugging M-Files The Editor/Debugger Debugging M-Files Types of Errors (Syntax Error and Runtime Error) Using keyboard and “ ; ” statement Setting Breakpoints Stepping Through  Continue, Go Until Cursor, Step, Step In, Step Out Examining Values  Selecting the Workspace  Viewing Datatips in the Editor/Debugger  Evaluating a Selection

Debugging Select Workspace Set Auto- Breakpoints tips

Importing and Exporting Data Using the Import Wizard Using Save and Load command load fname load fname x y z load fname -ascii load fname -mat load fname load fname x y z load fname -ascii load fname -mat save fname save fname x y z save fname -ascii save fname -mat save fname save fname x y z save fname -ascii save fname -mat

Read formatted data, reusing the format string N times. Import and Exporting Numeric Data with General ASCII delimited files »[A1…An]=textread(filename,format,N) Input/Output for Text File » M = dlmread(filename,delimiter,range) Suppose the text file stars.dat contains data in the following form: Jack Nicholson 71 No Yes 1.77 Helen Hunt 45 No No 1.73 Read each column into a variable [firstname, lastname, age, married, kids, height] = textread('stars.dat','%s%s%d%s%s%f');

Input from Text File fopen: Open a file for input/output fgetl: Read line from file fclose: Close one or more open files » fid= fopen( ' myFile.txt ', ‘ r ' ); » str1= fgetl(fid); » fclose (fid); » fid= fopen( ' myFile.txt ', ‘ r ' ); » str1= fgetl(fid); » fclose (fid);

Other Tidbits

The “inline” Utility inline function » f = inline('3*sin(2*x.^2)','x') f = Inline function: f(x) = 3*sin(2*x.^2) » f(2) ans = » f = inline('3*sin(2*x.^2)','x') f = Inline function: f(x) = 3*sin(2*x.^2) » f(2) ans = Use char function to convert inline object to string Numerical Integration using quad » Q = quad('1./(x.^3-2*x-5)',0,2); » F = inline('1./(x.^3-2*x-5)'); » Q = quad(F,0,2); » Q = quad('myfun',0,2) » Q = quad('1./(x.^3-2*x-5)',0,2); » F = inline('1./(x.^3-2*x-5)'); » Q = quad(F,0,2); » Q = quad('myfun',0,2) function y = myfun(x) y = 1./(x.^3-2*x-5); Note : quad function use adaptive Simpson quadrature

Symbolic Math symbolic variable symbolic function » syms t; » func1=sym('3*cos(10*t)'); » syms t; » func1=sym('3*cos(10*t)'); » syms t; » func1=sym('3*cos(10*t)'); » value=subs(func1,t,2.3) value = » diff(func1) ans = (-30)*sin(10*t) » syms t; » func1=sym('3*cos(10*t)'); » value=subs(func1,t,2.3) value = » diff(func1) ans = (-30)*sin(10*t) substitution symbolic differentiation

Graphics Fundamentals

Graphics and Plotting in MATLAB Basic Plotting plot, title, xlabel, grid, legend, hold, axis Editing Plots Property Editor Mesh and Surface Plots meshgrid, mesh, surf, colorbar, patch, hidden Handle Graphics

2-D Plotting Syntax: Example: plot(x1, y1, 'clm1', x2, y2, 'clm2',...) x=[0:0.1:2*pi]; y=sin(x); z=cos(x); plot(x,y,x,z,'linewidth',2) title('Sample Plot','fontsize',14); xlabel('X values','fontsize',14); ylabel('Y values','fontsize',14); legend('Y data','Z data') grid on x=[0:0.1:2*pi]; y=sin(x); z=cos(x); plot(x,y,x,z,'linewidth',2) title('Sample Plot','fontsize',14); xlabel('X values','fontsize',14); ylabel('Y values','fontsize',14); legend('Y data','Z data') grid on color line marker

Sample Plot TitleYlabelXlabelGridLegend

Displaying Multiple Plots Nomenclature: Figure window – the window in which MATLAB displays plots Plot – a region of a window in which a curve (or surface) is displayed Three typical ways to display multiple curves in MATLAB (other combinations are possible … ) One figure contains one plot that contains multiple curves  Requires the use of the command “ hold ” (see MATLAB help) One figure contains multiple plots, each plot containing one curve  Requires the use of the command “ subplot ” Multiple figures, each containing one or more plots, each containing one or more curves  Requires the use of the command “ figure ” and possibly “ subplot ”

Subplots Syntax: » subplot(2,2,1); » … » subplot(2,2,2) »... » subplot(2,2,3) »... » subplot(2,2,4) »... » subplot(2,2,1); » … » subplot(2,2,2) »... » subplot(2,2,3) »... » subplot(2,2,4) »... subplot(rows,cols,index)

The “ figure ” Command Use if you want to have several figures open for plotting The command by itself creates a new figure window and returns its handle >> figure If you have 20 figures open and want to make figure 9 the default one (this is where the next plot command will display a curve) do >> figure(9) >> plot( … ) Use the command close(9) if you want to close figure 9 in case you don ’ t need it anymore

Data Types

Numeric Arrays Multidimensional Arrays Structures and Cell Arrays

Multidimensional Arrays » A = pascal(4); » A(:,:,2) = magic(4) A(:,:,1) = A(:,:,2) = » A(:,:,9) = diag(ones(1,4)); » A = pascal(4); » A(:,:,2) = magic(4) A(:,:,1) = A(:,:,2) = » A(:,:,9) = diag(ones(1,4)); Page N Page The first references array dimension 1, the row. The second references dimension 2, the column. The third references dimension 3, The page.

Structures Arrays with named data containers called fields. » patient.name='John Doe'; » patient.billing = ; » patient.test= [ ; ; ]; » patient.name='John Doe'; » patient.billing = ; » patient.test= [ ; ; ]; Also, Build structure arrays using the struct function. Array of structures » patient(2).name='Katty Thomson'; » Patient(2).billing = ; » Patient(2).test= [ ; ; ]; » patient(2).name='Katty Thomson'; » Patient(2).billing = ; » Patient(2).test= [ ; ; ];

Cell Arrays Array for which the elements are cells and can hold other MATLAB arrays of different types. » A(1,1) = {[1 4 3; 0 5 8; 7 2 9]}; » A(1,2) = {'Anne Smith'}; » A(2,1) = {3+7i}; » A(2,2) = {-pi:pi/10:pi}; » A(1,1) = {[1 4 3; 0 5 8; 7 2 9]}; » A(1,2) = {'Anne Smith'}; » A(2,1) = {3+7i}; » A(2,2) = {-pi:pi/10:pi}; Using braces {} to point to elements of cell array Using celldisp function to display cell array

Conclusion  Matlab is a language of technical computing.  Matlab, a high performance software, a high- level language  Matlab supports GUI, API, and …  Matlab Toolboxes best fits different applications  Matlab …

Getting more help Contact You can find more help and FAQ about mathworks products on this page. Contact comp.soft-sys.matlab Newsgroup Use google to find more information (like the content of this presentation, in the first place)

? ?

Exercise 1 (10 minutes) Define a matrix A of dimension 2 by 4 whose (i,j) entry is A(i,j)=i+j Extract two 2 by 2 matrices A1 and A2 out of the matrix A. A1 contains the first two columns of A, A2 contains the last two columns of A Compute the matrix B to be the sum of A1 and A2 Compute the eigenvalues and eigenvectors of B Solve the linear system Bx=b, where b has all the entries equal to 1 Compute the determinant of B Compute the inverse of B Compute the condition number of B NOTE: Use only MATLAB native functions for all operations

Exercise 3: A debug session (10 minutes) Use the function demoBisect provided on the next slide to run a debug session Save the MATLAB function to a file called demoBisect.m in the current directory Call once the demoBisect.m from the MATLAB prompt to see how it works >>help demoBisect >>demoBisect(0, 5, 30) Place some breakpoints and run a debug session  Step through the code, and check the values of variables  Use the MATLAB prompt to echo variables  Use dbstep, dbcont, dbquit commands

function xm = demoBisect(xleft,xright,n) % demoBisect Use bisection to find the root of x - x^(1/3) - 2 % % Synopsis: x = demoBisect(xleft,xright) % x = demoBisect(xleft,xright,n) % % Input: xleft,xright = left and right brackets of the root % n = (optional) number of iterations; default: n = 15 % % Output: x = estimate of the root if nargin<3, n=15; end % Default number of iterations a = xleft; b = xright; % Copy original bracket to local variables fa = a - a^(1/3) - 2; % Initial values of f(a) and f(b) fb = b - b^(1/3) - 2; fprintf(' k a xmid b f(xmid)\n'); for k=1:n xm = a + 0.5*(b-a); % Minimize roundoff in computing the midpoint fm = xm - xm^(1/3) - 2; % f(x) at midpoint fprintf('%3d %12.8f %12.8f %12.8f %12.3e\n',k,a,xm,b,fm); if sign(fm)==sign(fa) % Root lies in interval [xm,b], replace a a = xm; fa = fm; else % Root lies in interval [a,xm], replace b b = xm; fb = fm; end