Introduction to MATLAB Session 2

1. Introductionto MATLABSession 2 Simopekka Vänskä, THL Department of Mathematics and Statistics University of Helsinki 2010

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

3. Basic commands

4. MATLAB commands/functions • Genaral syntax: [OUTPUT parameters] = functionname(INPUT parameters); • Functions do not change the INPUT parameters • Number of parameters can variate • Help function: >> help functionname INPUT parameters OUTPUT parameters Function Introduction to MATLAB - Session 2

5. Operate on each element of the input matrix A: size(OUTPUT) = size(A) Examples >> A = [1 0; pi pi/2]; >> sin(A) ans = 0.8415 0 0.0000 1.0000 >> round(A) ans = 1 0 3 2 Some common functions: sin, cos, tan, asin, acos, atan exp, log, sqrt abs, conj, imag, real, angle fix, floor, ceil, round, sign For more, see >> help elfun Try the following: >> asin(1) >> asin(5) >> exp(1) >> exp(709) >> exp(710) >> log(exp(50)) >> v = [-1.2, 0.5, 1.2]; >> round(v) >> fix(v) >> floor(v) >> ceil(v) >> sign(v) >> sign(0) Some elementwise functions Introduction to MATLAB - Session 2

6. Operate on each column of the input matrix A: size(OUTPUT) = size(A,2) Exception: raw vectors Example >> A = [1 0; pi pi/2]; >> sum(A) ans = 4.1416 1.5708 Operating dimension choosable >> sum(A,2) ans = 1.0000 4.7124 Examples sum, prod, cumsum, cumprod mean, median, std, min, max sort Try the following >> A = [1 3 3; 2 2 4] >> mean(A) >> mean(A’) >> min(A) >> [X,I] = min(A) >> cumsum(A) >> cumprod(A) >> sort(A) Some columnwise functions Introduction to MATLAB - Session 2

7. Polynomials and interpolation • Finding the roots of the polynomial: roots(C) returns the roots of the polynomial whose coefficients are given by vector C • P(x) = C(1)*X^N + ... + C(N)*X + C(N+1). • Recall: Polynomial of degree n has exactly n complex roots. • Polynomial fitting: polyfit(x,y,n) fits a polynomial of degree n to the data points (X,Y) and returns the coefficients. • Simple interpolation interp1 interp1(X,Y,X0) interpolates the data points (X,Y) to given interpolation points X0. • interp2, interp3 for 2- and 3-dimensional data. Introduction to MATLAB - Session 2

8. Logical expressions

9. Logical datatype False: 0 True: 1 (nonzero) Relational operations: == equal ~= not equal >=, >, <=,< Logical operations: & (and), | (or), ~ (not), xor, any, all isempty, isnan Try the following >> v = 1:5; >> v==3 >> (v>=2)&(v<5) >> (v>=2).*(v<5) >> f = v==3 >> f = (v==3) >> whos f >> all(f) >> any(f) >> ~f >> g = (v>=2) >> xor(g,f) >> g|f Logical expressions and functions Introduction to MATLAB - Session 2

10. find(X) returns the indeces of non-zero (true) entries of X >> I = find(X) returns the vector indeces >> [I,J] = find(X) returns the matrix indeces >> I = find(X,k) return the k first indeces Restricting vectors with find. Optional notations >> X(find(X<4)) and >> X(X<4) are the same. Try the following >> X = rand(4) >> I = find(X<0.4) >> [I,J] = find(X<0.4) >> Y = X(X<0.4) >> I = find(X<0.4) >> X(I) FIND Introduction to MATLAB - Session 2

11. Graphics 1 - PLOT

12. No. 1 plotting command - plot Basic idea: • Command plot(x,y) connects the points (x(1),y(1)), (x(2),y(2)) , …, (x(n),y(n)) with lines. • Into current figure and axis (if does not exist, creates). • For matrices X and Y: plot(X,Y) acts columnwise (exceptions exist). Introduction to MATLAB - Session 2

13. Basic line properties Plot symbols: ., o, x, +, *, … Color: b, g, r, c,… Line style: -, :, -., -- For more, see >> help plot How to use: >> plot(x,y,’bo’) Multiple data in one plot: >> plot(x1,y1,’bo’,x2,y2,’r’,…) OR use hold command: >> plot(x1,y1,’bo’) >> hold on >> plot(x2,y2,’r’) >> hold off Additional properties: Graphics 2 session. Remark: Text within ’ ’ is a string A string is a char array, a char valued matrix, e.g. >> J = ’the Lord’ Try the following >> x = (-3:.1:3)’; >> plot(x,exp(x)) >> c = 1:5; >> plot(x,x.^2*c) >> plot(x,x,’rx’,x,x+2,’b-’) Plot line properties Introduction to MATLAB - Session 2

14. Basic plot editing commands Example >> plot(x,y,’r’,x0,y0,’bo’) • Title for the image >> title(’Linear fit’) • Labels for the axes >> xlabel(’Month’) >> ylabel(’Cases’) • Setting the axis limits >> axis([0 6.5 0 15]) • Labeling of the plot lines >> legend(’fit’,’data’) Here, an additional property ’Location’ was used: >> legend(’fit’,’data’,’Location’,’NorthWest’) OR move with the mouse in the figure window. Introduction to MATLAB - Session 2

15. >> figure(n) sets figure n as current figure or, creates figure n if it does not exist Example: Numbering subplots. >> subplot(m,n,k) Breaks the figure window to subfigures with m raws and n columns and sets current axis to axis number k For more, see >> help subplot Try the following: >> subplot(1,2,1) >> plot(rand(3)) >> subplot(2,2,2) >> plot(rand(10)) >> subplot(2,2,4) >> plot(0:.1:5,sqrt(0:.1:5)) Figure and subplot 1 2 3 4 5 6 Introduction to MATLAB - Session 2

16. ProblemsSession 2

17. Problems Write your solutions to m-files • Check how matrix A = a) [2 0; b) [2 0; c) [-1 0; d) [ 1 1; e) [1 -1; f) [2 1; 0 1]; 0 -1]; 0 1]; -1 1]; 1 1]; 0 2]; maps the points P = [0, 4, 4, 3, 3, 2.5, 2.5, 2, 0, 0; 0, 0, 3, 4, 5, 5, 4.5, 5, 3, 0]; by plotting P and points A*P. To plot P you can use plot(P(1,:),P(2,:)). • Plot functions y=sin(x) and y = cos(x) on interval [0,4p] in the same figure but with different colors. Introduction to MATLAB - Session 2

18. Problems • Draw the unit circle in R2. • Draw the unit circle so that the line is green for x>0 and black for x<0. • Map the unit circle to the ellipse with major axes u = [2;1], minor axes v = [-1/2;1], and center (1,1). Draw the ellipse in the same picture with the unit circle. • Hint: Map linearly and transport. • Draw the image of the mapping f: 1 + [-i,i]  C, • f(z) = log(z), • f(z) = z^2, in the complex plane. Hint: Real plane. Introduction to MATLAB - Session 2

19. Mortality fitting • In this exercise we consider mortality in Finland at 2007 (data loaded from Tilastokeskus website). • Copy kuolleisuus.xls (at the wikipage of the course) to your working directory. Load it to MATLAB (start your m-file with M = xlsread(’kuolleisuus.xls’);). The file contains matrix M with • M(:,1) = age • M(j,2) = mortality for mails at age(j) [1/1000] • M(j,3) = mortality for femails at age(j) • Fit polynomials of degree 2 and 3 to the mortality data. • Fit an exponential function to the mortality data, i.e., fit a polynomial of degree 1 to the log(mortality) –data. • Present your fit graphically. Use subplots, colors, titles, legends, and axis labels. Introduction to MATLAB - Session 2

20. Computing area with random points • Compute the area of the unit triangle T = span((0,0),(1,0),(0,1)) with uniformly distributed random numbers as follows: • Generate N uniformly distributed random points x =(x1,x2) in the unit square • Find the fraction of the points falling in T. Illustrate this graphically, plot the random points and T. Plot the points in T and the points out T with different colors. • Approximate area of T. Test the accuracy with different number of points N. Introduction to MATLAB - Session 2

21. >>quit …to exit MATLAB.