1. Introduction to MATLAB Session 1 Simopekka Vänskä, THL 2010

2. About this course

3. Introduction to MATLAB - Session 1 Focus of this course...in learning to use MATLAB software  Not in numerical methods (but some examples) …to get some idea of the possibilities of MATLAB  We will not go through all properties of MATLAB.  To get into a position for learning application specific features.

4. Introduction to MATLAB - Session 1 Schedule and passing Schedule Five 3-hour sessions Homework Session working style Introduction to the topic + exercises, learning by doing Homework Estimated work load 5-10 hours More information later Passing the course (2 credits, passed/failed) Attendance min 3/5 sessions + passing the homework

5. Introduction to MATLAB - Session 1 Contents of this course Session 2 Some matrix commands Logical expressions Graphics 1 Session 3 My functions + strings, cells Controlling program flow Session 4 Function functions Session 5 Graphics 2 More linear algebra Starting homework Session 1 Genaral Matrices M-files

6. MATLAB

7. Introduction to MATLAB - Session 1 What is MATLAB? Matrix Labratory  Array/Matrix is the basic data element Environment for numerical computing >> quad(@(x) exp(sqrt(x.^2+1)), 0, 1) ans = 3.1769  not for symbolic calculus Library of mathematical functions Programming language Application-specific toolboxes

8. Introduction to MATLAB - Session 1 MATLAB desktop view

9. Introduction to MATLAB - Session 1 Getting started Basic idea Workspace  Matrices and other data elements are storaged in the workspace  Commands who and whos for listing the variables Execute commands (functions) to manipulate the matrices in the workspace  Matlab interpretes the commands, no compiler 1. Start MATLAB 2. Create a working directory  Under your personal home directory 3. Set ”Current directory” to your working directory 4. Create a variable v by typing >> v = 5 in a command window and try who and whos. 5. Write >> exp(v) and check the workspace.

10. MATRICES

11. Introduction to MATLAB - Session 1 Matrices (and vectors and scalars) Matrix: the basic data element (n,m) array Vector (n,1) matrix = column vector (1,m) matrix = raw vector Scalar (1,1) matrix Two ways to create matrices: 1. List the elements  = is the substitution  Use [ ] brackets 2. Built-in functions (Load from a file) Try the following: >> A = [1 2 3; 5,6,7] >> [1 2 3; 5,6,7] >> B = [1 2 3; 5,6,7]; >> B >> C=[A; B] >> D = ones(2,4) >> E = zeros(3) >> E2 = zeros(1000) >> F = zeros(1,4) >> G = rand(10,1) >> H = randn(10,1) >> I = eye(5) >> J = [ ]

12. Introduction to MATLAB - Session 1 More matrices Creating vectors with :, the colon command  a:b from a to b with step one  a:d:b from a to b with step d Multi-dimensional matrices Some scalars  i and j : complex number  eps : small number  pi :    Inf, NaN Hint: Use ; for not showing the result on the command window Hint: Command size(A) returns the size of matrix A Try the following >> 1:4 >> 0:5:20 >> v = 20:-5:0 >> rand(2,3,4) >> i >> 5+3*i >> i = 5 >> 5+3*i >> eps >> pi >> 5e3 >> 5.4e-3 >> beta >> help beta

13. Introduction to MATLAB - Session 1 Matrix arithmetics A+B, A-B A+c, A-c Matrix multiplication A*B Transpose A.’ and adjoint A’ Matrix power A^c  For noninteger c with spectral calculus \ left matrix divide  Ax=y  x=A\y = inv(A)*y / right matrix divide Array arithmetics A+B, A-B A+c, A-c Array multiplication A.*B Array power A.^c, A.^B Array divide A./B A, B matrices, c scalar – matrix sizes must match !

14. Introduction to MATLAB - Session 1 Reffering to vector elements Let v = [v1 v2... vn]. Refer to element vj with v(j) v(J)  J can be a matrix of indeces  v(J) is a matrix of size(J) and of elements defined by J Hint: Command length(v) returns the length of vector v Try the following: >> v = 0:5:30 >> v([1 1 3]) >> v(2) = 15 >> v(2:4) = 3:5 >> J = [1 1; 2 2]; >> v(J) >> v(10) >> v(10)=100;

15. Introduction to MATLAB - Session 1 Reffering to matrix elements Some properties Refer to entire column/raw with the colon :  A(:,5)  A(2,:) A(:) is the matrix A as a vector Referring with end –command  A(3,5:end)  A(2:end, : ) Let A = [a11 a12... a1m a21 a22... a2m  an1 an2... anm]. Refer to element ajk with  A(j,k) or  A(n*(k-1)+j) Consider matrix as a column vector (columns consecutively) A(J,K) with index matrices J,K

16. Introduction to MATLAB - Session 1 …Reffering to matrix elements Try the following >> A = [1:4;5:8;9:12] >> A(2,4) >> A(11) >> A(2,1:3) >> A(2,:) >> A([2 3],[3 1]) >> A(2:end,3) >> A(5:end) >> reshape(A,4,3) >> help reshape

17. M-FILES

18. Introduction to MATLAB - Session 1 MATLAB m-files Not practical to write all commands in the command window  use m-files  Text files of type *.m  For example, test1.m  Each line of an m-file is a MATLAB command line MATLAB executes the lines of an m-file by writing the name of the file in the command window, (or F5 from the editor)  >> test1  Visibility: m-file has to be in the MATLAB’s current directory (or in the MATLAB root) Can be written with any text editor  …but the MATLAB editor is preferable  File  New  m-file (blank)

19. Problems Session 1

20. Introduction to MATLAB - Session 1 Problems 1. Go throw the previous ”Try the following” exercises 2. Are the following vectors the same?  a = [1 -2 3]  b = [1 - 2 3]  c = [1-2 3] 3.How much memory does the matrix I=ones(1000) need? How about I=ones(10000)? Clear memory with ”clear” command. 4.Set a=1 and b=i. Check the memory usage (whos). 5.Compute a) log(-1), b) sqrt(-1),c) 1/0. 6.See help format, and try: >> format long>> format short>> pi

21. Introduction to MATLAB - Session 1 Problems Matrix manipulation – write your answers to m-files 7.Create (5,5) –matrix of zeros.  Substitute random numbers (with rand –command) to raws 3-5.  Delete the first and the last columns (=substitute empty matrix).  Create a vector [0 1 0 1 … 0 1] of length 20.  Create a vector [1,3,5,...,999,2,4,6,...,1000].  Create a vector [2,1,4,3,6,5,...,998,997,1000,999]. 8.Create a (100,100) -matrix 1 2 3 4 5 … 100 … 1 2 3 4 5 … 100 Hint: Multiplication with matrix ones(100,1).

22. Introduction to MATLAB - Session 1 Problems 9.Let A = [1 2; 0 3];  Compute A*ej for e1 = [1;0] and e2=[0;1];  Compute A^2 and A.^2.  Solve equation Ax=y for a) y = [1;0];b) y= [2;3];c) y = [ i ; 0]; 10.Create a ( :,1000) matrix a)X2 whose columns are R 2 rand vectors in the unit square, b)X3 whose columns are R 3 rand vectors in the unit cube.  Compute the lengths of the random vectors.  Calculate the average length of the random vectors. 11.Let p be a vector of annual interest rates% and let s be a vector of initial investments. Create a table A for the values of the investments after 10 years, A(j,k) = s(k)*(1+p(j)/100) 10. You can use e.g., p=.5:.5:5 and s = 2000:2000:10000.

23. Introduction to MATLAB - Session 1 Problems 12.Study sparse matrices from the MATLAB help >> help sparse 13. Create a sparse matrix S of size 2000x2000 with all diagonal elements 1, S(j,j)=1, and random lower diagonal S(j+1,j) = random number, j=1,…,1999.  Check that S is correct with full(S(1:10,1:10)).  Set y = ones(2000,1); and S2=full(S);  Compare with tic toc –commands (see help tic) the speed of solving Sx=y by  tic; x=S\y; toc  tic; x=S2\y; toc  tic; x=inv(S)*y; toc (see help inv)  Can you explain the result?

24. >> quit …to exit MATLAB.