1. Problems and solutions Session 2

2. Introduction to MATLAB - Solutions 2 Problems Write your solutions to m-files 1. 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,:)). 2. Plot functions y=sin(x) and y = cos(x) on interval [0,4  in the same figure but with different colors.

3. Introduction to MATLAB - Solutions 2 Problems 3. Draw the unit circle in R 2.  Draw the unit circle so that the line is green for x>0 and black for x<0. 4. 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. 5. Draw the image of the mapping f: 1 + [-i,i]  C, a)f(z) = log(z), b)f(z) = z^2, in the complex plane. Hint: Real plane.

4. Introduction to MATLAB - Solutions 2 Mortality fitting 6. 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 males at age(j) [1/1000]  M(j,3) = mortality for females 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.

5. Introduction to MATLAB - Solutions 2 Computing area with random points 7. 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.

6. Introduction to MATLAB - Solutions 2 Some solutions 1. A = [2 0; 0 1]; Q = A*P; subplot(2,3,1) plot(P(1,:),P(2,:),’b’,Q(1,:),Q(2,:),’r’) A = [2 0; 0 -1]; Q = A*P; subplot(2,3,2) plot(P(1,:),P(2,:),’b’,Q(1,:),Q(2,:),’r’) etc. 2. x = (0:.01:(4*pi))’; plot(x,cos(x),’b’,x,sin(x),’g’) OR plot(x,[cos(x),sin(x)]) 3. t = 0:.01:(2*pi); x = cos(t); y = sin(t); plot(x(x>0),y(x>0),’b’,... x(x<=0),y(x<=0),’r’) 4. A = [2 -.5; 1 1]; t = 0:.01:(2*pi); x = [cos(t);sin(t)]; y = A*x; plot(x(1,:),x(2,:),’b’,… y(1,:)+1,y(2,:)+1,’r’) 5. z = 1 + i*((-1):.01:1); fz = log(z); gz = z.^2; plot(real(fz),imag(fz),’b’,… real(gz),imag(gz),’r’)

7. Introduction to MATLAB - Solutions 2 Some solutions subplot(1,2,2) plot(a,morm,’b.’,a,morf,’r.’,… a,exp(Lfits(:,1)),’b’,… a,exp(Lfits(:,2)),’r’) 7. N = 5000; x = rand(2,N); T = (x(2,:) < (1-x(1,:))); inT = find(T); outT = find(~T); plot(x(1,inT),x(2,inT),’k.’,… x(1,outT),x(2,outT),’g.’) areaTapprx = length(inT)/N 6. a = M(:,1); morm = M(:,2); % male mortality morf = M(:,3); p2m = polyfit(a,morm,2); p2f = polyfit(a,morf,2); fits = [p2m(1)*a.^2+p2m(2)*a+p2m(3),… p2f(1)*a.^2+p2f(2)*a+p2f(3)]; figure(1) plot(a,morm,’b.’,a,morf,’r.’,… a,fits(:,1),’b’,a,fits(:,2),’r’) % exponential fit: Lmorm = log(max(morm,.05)); Lmorf = log(max(morf,.05)); % max for not taking log(0) Lpm = polyfit(a,Lmorm,1); Lpf = polyfit(a,Lmorf,1); Lfits = [Lpm(1)*a+Lpm(2),Lpf(1)*a+Lpf(2)]; figure(2) subplot(1,2,1) plot(a,Lmorm,’b.’,a,Lmorf,’r.’,… a,Lfits(:,1),’b’,a,Lfits(:,2),’r’)