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Introduction to Matlab 7 Part I Daniel Baur ETH Zurich, Institut für Chemie- und Bioingenieurwissenschaften ETH Hönggerberg / HCI F128 – Zürich E-Mail: [email protected] http://www.morbidelli-group.ethz.ch/education/snm/Matlab Daniel Baur / Introduction to Matlab Part I 1 File System Your home directory is mapped to Y:\ The «my documents» folder points to Y:\private File reading and writing can take longer than usual since this is a network drive Always save your data in your home directory!! If you save it locally on the computer, it might be lost. Daniel Baur / Introduction to Matlab Part I 2 Accessing your Data from Home To access your home directory from outside the ETH, connect to the ETH VPN and map the folder \\d.ethz.ch\dfs\users\all\<Login-Name> Windows: Map network drive (right-click on computer) Mac: Go to / Connect to server Unix: smbmount Log in as d\<Login-Name> Daniel Baur / Introduction to Matlab Part I 3 Introduction What is Matlab? Matlab is an interactive system for numerical computation What are the advantages of Matlab? Quick and easy coding (high level language) Procedural coding and Object oriented programming are supported Minimal effort required for variable declaration / initialization Simple handling of vectors and matrices (MATrix LABoratory) High quality built-in plotting functions Full source-code portability Strong built-in editing and debugging tools Extremely diverse and high quality tool boxes available Large community that contributes files and programs (mathworks file exchange website) Extensive documentation / help files Daniel Baur / Introduction to Matlab Part I 4 Introduction (Continued) What are the weaknesses of Matlab? Not optimal for symbolic calculations (especially on the output side), use Maple or Mathematica instead Not as fast as C++ or Fortran, especially for computationally demanding problems Very expensive (except for students) Where to get Matlab? ETH students have free access to Matlab Go to http://www.ides.ethz.ch/ and search for Matlab in the catalogue You might have to set a password on the ides-website in order to log in Remember to choose the correct operating system Map the web-drive \\ides.ethz.ch\<Login-Name> to download / install Matlab Daniel Baur / Introduction to Matlab Part I 5 Matlab environment (Try it out!) Variable Inspector / Editor File Structure Workspace (Variable List) Command Prompt Command History File Details Daniel Baur / Introduction to Matlab Part I 6 Where to get help If you know which command to use, but not how: Type help command in the command window for quick help Type doc command in the command window to open the help page of the command Right click on a word and select «help on selection», or click the word and press F1 If you do not know which command to use: There are extensive forums and other sources available on the internet, google helps a lot! Type doc or use the menu bar to open the user help and search for what you need Send me an email Daniel Baur / Introduction to Matlab Part I 7 What if something goes wrong? The topmost error message is usually the one containing the most useful information The underlined parts of the message are actually links that you can click to get to the place where the error happened! If a program gets stuck, use ctrl+c to terminate it Daniel Baur / Introduction to Matlab Part III 8 Variables in Matlab Try: Valid examples: Invalid examples: a = 1 speed = 1500 Cost_Function = a + 2 String = 'Hello World' 2ndvariable = 'yes' First Element = 1 Rules Variable names are case sensitive («NameString» ≠ «Namestring») Maximum 63 characters First character must be a letter Letters, numbers and underscores «_» are valid characters Spaces are not allowed Daniel Baur / Introduction to Matlab Part I 9 Variables in Matlab (Continued) Try out these commands: a = 2 b = 3; c = a+b; d = c/2; d who whos clear who TestString = 'Hello World' Note that every variable has a size (all variables are arrays!) No need to declare variables or specify variable types! Daniel Baur / Introduction to Matlab Part I 10 Variables in Matlab (Continued) Variable assignments a b c a a = = = + = 2; 3; a + b; b b = 2; The result is stored in «c» The result is stored in «ans» This produces an error By pressing the up and down arrows, you can scroll through the previous commands A semicolon «;» at the end of a line supresses command line output By pressing the TAB key, you can auto-complete variable and function names Daniel Baur / Introduction to Matlab Part I 11 Vectors in Matlab Vector handling is very intuitive in Matlab (try these!): Row vector: Column vector: Vector with defined spacing: Vector with even spacing: Transpose: a a b c d e f = = = = = = = [1 2 3] [1, 2, 3] [1; 2; 3] 0:5:100 (unit: 0:100) linspace(0, 100, 21) logspace(0, 3, 25) e' You should see Daniel Baur / Introduction to Matlab Part I 12 Vector arithmetics Try these out: a = [1, 2, 3] b = [1; 2; 3] Operations with constants c = 2*a d = 2+a Vector addition f = a + c Element-by-Element operations a.^2 d = d./a Functions using element-byelement operations (examples) b = sqrt(b) c = exp(c) d = factorial(d) Vector product A = b*a A is a (3,3) matrix! a*a Error! (1,3)*(1,3) a^2 Daniel Baur / Introduction to Matlab Part I Operations with scalar constants (except power) are always element-by-element. 13 Vector arithmetics (Continued) Notes on vector multiplication a = [1, 2, 3] a 1 2 3 b = [1; 2; 3] 1 b 2 3 c = a*b d = b*a (1,3)*(3,1) = (1,1) Scalar (dot product) (3,1)*(1,3) = (3,3) Matrix e = a.*a f = a.*b (1,3).*(1,3) = (1,3) Vector (element-by-element) Error! Vectors must be the same size for element-by-element operations Remember the rules for vector / matrix addition, subraction and multiplication! Daniel Baur / Introduction to Matlab Part I 14 Matrices in Matlab Creating matrices (try these out!) Direct: Matrix of zeros: Matrix of ones: Random matrix: Normally distributed: A = [1 2 3; 4 5 6; 7 8 9] B = zeros(3); B = zeros(3,2); C = ones(3); C = ones(3,2); R = rand(3); R = rand(3,2); RD = randn(3) Matrix characteristics Size Largest dimension Number of elements [nRows, nColumns] = size(A) nColumns = size(A,2) maxDim = length(A) nElements = numel(A) Creating vectors Single argument calls create a square matrix, therefore use commands like v = ones(3,1); to create vectors Daniel Baur / Introduction to Matlab Part I 15 Accessing elements of vectors / matrices Try: Vectors Single element: Multiple elements: Range of elements: Last element: All elements: Matrices a = (1:5).^2 a(:) always returns a A = a'*a; column vector. Single element: Submatrix: Entire row / column: Multiple rows / columns: Last element of row / column: All elements as column vector: Daniel Baur / Introduction to Matlab Part I 16 Arithmetics with matrices Try these out: A = rand(3) Operations with constants B = 2*A C = 2+A Matrix addition; Transpose D = A+C D = D' Deleting rows / columns C(3,:) = [] D(:,2) = [] Matrix multiplication C*D D*C Not commutative! A^2 Element-by-element operations A.^2 E = 2.^A Ei,j = 2^Ai,j sqrt(A) Functions using matrices sqrtm(A) sqrtm(A)^2 inv(A) Daniel Baur / Introduction to Matlab Part I 17 Matrix divison Consider the following A = rand(3); B = rand(3); A*C = B C = A-1*B = inv(A)*B Matrix inversion is one of the most computationally expensive operations overall, so what should we do instead? Matlab has more sophisticated built-in algorithms to do matrix divisions which are called left- and right divide; They are symbolized by the operators \ and /, respectively. inv(A)*B = A-1*B A\B; A*inv(B) = A*B-1 A/B; Daniel Baur / Introduction to Matlab Part I 18 More matrix manipulations Try: Matrices in block form B = [ones(3); zeros(3); eye(3)] From matrices to vectors b = B(:) From vectors to matrices b = 1:12; B = zeros(3,4); B(:) = b B = reshape(b, 3, 4) C = repmat(b, 5, 1) Diagonal matrices b = 1:12; D = diag(b) Meshes [X, Y] = meshgrid(0:2:10, 0:5:40) Daniel Baur / Introduction to Matlab Part I 19 More Matrix Manipulations (Continued) Daniel Baur / Introduction to Matlab Part I 20 Operators for matrices Consider the operators: [nRows, nColumns] = size(A); [maxValue, Position] = max(A,[],dim); sum(A,dim); sum(A(:)); Also: min(A) Also: mean(A), var(A), std(A), ... det(A); inv(A); eig(A); cond(A); norm(A,p); p norm( A, p) xi i 1 n Daniel Baur / Introduction to Matlab Part I 1 p 21 Exercise 1. Compute the approximate value of exp(1) Hints: Define a vector of length 20 for the first 20 elements of the summation, then sum it up; The ! operator is factorial() xk e k 0 k ! x 2. Compute the approximate value of exp(2) 3. Compute the cross product of u = [1, 3, 2] and v = [-1, 1, 2] u2 v3 u3v2 u v u3v1 u1v3 u v u v 1 2 2 1 Daniel Baur / Introduction to Matlab Part I 22 Solution of Linear Algebraic Systems (Exercise) 1. Write the following system of equations in Matrix form: 2 x1 4 x2 8 x3 2 3 x1 2 x2 2 x3 5 A x b x 3 x x 4 2 3 1 2. Is this system singular? 3. How would you solve this system? Computing the inverse of a matrix is very expensive. Use left division instead! Daniel Baur / Introduction to Matlab Part I 23 Exercise (Continued) 1. Solve the system AX B 2 4 8 2 14 26 3 2 2 X 5 5 9 1 3 1 4 8 2 2. Now solve this system: A 0 1 0 0 4 0 0 B 38 25 0 0 22 x 5 234 9 68 Daniel Baur / Introduction to Matlab Part I 24