Problems and solutions Session 2

7 pages
12 views

Please download to get full document.

View again

of 7
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Share
Description
Problems and solutions Session 2. 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
Transcript
Problems and solutionsSession 2ProblemsWrite 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 - Solutions 2Problems
  • 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 - Solutions 2Mortality 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 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.
  • Introduction to MATLAB - Solutions 2Computing 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 - Solutions 2A = [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.x = (0:.01:(4*pi))’; plot(x,cos(x),’b’,x,sin(x),’g’) OR plot(x,[cos(x),sin(x)])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’)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’)z = 1 + i*((-1):.01:1); fz = log(z); gz = z.^2; plot(real(fz),imag(fz),’b’,… real(gz),imag(gz),’r’)Some solutionsIntroduction to MATLAB - Solutions 2 subplot(1,2,2) plot(a,morm,’b.’,a,morf,’r.’,… a,exp(Lfits(:,1)),’b’,… a,exp(Lfits(:,2)),’r’)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)/Na = 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’)Some solutionsIntroduction to MATLAB - Solutions 2
    Related Search
    Similar documents
    View more...
    We Need Your Support
    Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

    Thanks to everyone for your continued support.

    No, Thanks