1. Introduction to MATLAB Session 5 Simopekka Vänskä, THL 2010

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

3. Graphics 2 2-3 dimensional graphs

4. Introduction to MATLAB - Session 5 Graphics 2 2-3 dimensional plots The graphical library of MATLAB is large.  See >> help graph2d >> help graph3d >> help specgraph When representing your computations, use some time to think effective visualization techniques Besides the image itself, control the axis, labels, ticks, grids, etc.  Nothing too much, nothing missing.

5. Introduction to MATLAB - Session 5 Line plot extensions plot3 draws the line in 3D grid puts the grid mode on/off area(x,Y) fills the columns of Y and draws them one upon theorher  Practical when showing how some partition developes polar plots the line in the polar coordinates Try the following >> t = (0:.001:1)’; >> plot3(cos(10*t),sin(10*t),t) >> grid >> s = [sqrt(t), t, t.^2]; >> subplot(1,2,1) >> area(t,s); >> subplot(1,2,2) >> plot(t,cumsum(s’)) >> polar(2*pi*t,cos(4*pi*t))

6. Introduction to MATLAB - Session 5 Scalar functions on plane - meshgrid A scalar function on plane: z = f(x,y). Meshgrid is needed for drawing: If x and y are vectors, we need to create all pairs (x(i),y(j)). >> [X,Y] = meshgrid(x,y); Try the following: >> x = 0:.1:1; >> y = 0:.2:2; >> [X,Y] = meshgrid(x,y); >> X >> Y >> plot(X,Y,’o’) Remark X and Y are constant in different coordinates.

7. Introduction to MATLAB - Session 5 Scalar functions on plane Some commands contour draws the contour plot mesh and surf draw the surface of the function (unfilled and filled), surfc both surface and contours  shading interp clears the surface lines  colorbar displays the scale  view sets the view angle  colormap sets the color scale Image, imagesc for pixel plots Try the following >> x = -1:.01:1; >> y = x; >> [X,Y] = meshgrid(x,y); >> Z = X.^2 + Y.^3; >> contour(x,y,Z) >> contour(x,y,Z,20) >> mesh(x,y,Z) >> surf(x,y,Z) >> shading interp; >> colorbar >> view(2) >> view([0 -1 1]) >> colormap hot >> colormap default >> surfc(x,y,Z) >> imagesc(x,y,Z)

8. Introduction to MATLAB - Session 5 Vectorfields on plane or in 3D A vector field on plane: F = (f 1 (x,y),f 2 (x,y)) quiver(X,Y,U,V) draws an arrow (U(j),V(k)) to each grid point (X(j),Y(k)) Similarly quiver3 in 3D Try the following: >> x = -1:.1:1; >> y = x; >> [X,Y] = meshgrid(x,y); >> U = X; >> V = Y.^2; >> quiver(X,Y,U,V);

9. Introduction to MATLAB - Session 5 Some other plotting functions hist(X) makes a histogram of each column of X  Returns also the number of elements falling in the intervals. bar(X) draws the columns of X as vertical bars pie(x) draws the pie plot of vector x (normalized) fill(X,Y,C) fills the polygons defined by the columns of X and Y with color C isosurface(X,Y,Z,F,V) draws the isosurface F(X,Y,Z) = V Try the following >> hist(randn(10000,2),20); >> N = hist(randn(1000,3),20); >> bar(rand(2,4)) >> pie(1:4) >> fill([0 1 2 1 0],[0 1 -1 -1 0],’r’) >> x = -1:.1:1; >> [X,Y,Z] = meshgrid(x,x,x); >> FXYZ = X.^2+2*Y.^2 - 3*Z.^2; >> isosurface(X,Y,Z,FXYZ,1)

10. Introduction to MATLAB - Session 5 Changing general figure properties Figure and axes properties, and many others, can be changed: General principle: Set a handle to current figure/axes with gcf/gca (get current figure/axes) To see the properties, use get To change the property, use set For more, see Handle Graphics Properties of help clf for clearing the figure Try the following >> bar(rand(2,4)) >> h = gca; >> get(h) >> set(h,’XtickLabel’,{’a’,’bcd’}) >> set(h,’YTick’,[])

11. More linear algebra

12. Introduction to MATLAB - Session 5 Matrix decompositions We finish the course with matrix decompositions, or, with the most important one SVD (for other decompositions, see HELP): Every matrix A has the singular value decomposition A = U*S*V’ >> [U,S,V]=svd(A); where  U and V are unitary matrices (the columns define an orthonormal basis)  S is a diagonal matrix with positive and decreasing elements.  Statisticians, compare to the principal component analysis. Every matrix has a pseudoinverse pinv(A) pinv(A) = V*R*U’ where R is a diagonal matrix with R(j,j) = 1/S(j,j) for nonzero S(j,j) and zero otherwise.  x = pinv(A)*y is the best nonregularized solution for all linear equations Ax=y.

13. Problems Session 5

14. Introduction to MATLAB - Session 5 Problems Draw the contour plot of function F(x,y) = x 1.5 y 2 exp(-(x-2)-(y-0.5)) on [0,5]x[0,5] and its gradient field with quiver command in the same picture.  Use meshgrid to create the X,Y –grid.  Compute the derivative by hand or by using your earlier differf.m function. Make a histogram of points randn(10000,1).

15. Homeworks

16. Introduction to MATLAB - Session 5 How to do your homework You can co-operate with other students  …but do not just copy the code. Add graphics (into the code) which demonstrates the calculations. How to return:  Write a m-file called runthis.m which calls your functions.  Put the m-files and data files in the same directory, and make sure that runthis.m file works.  Send all homework files to the lecturer, who will copy your program files to one directory and test it by executing runthis – command. If the program works well enough, you pass the homework. Lecturer contacts you by e-mail.

17. Introduction to MATLAB - Session 5 Homework Topics Personal finance – how your savings accumulate SIR – model for measles Homeworks are described in more details in pdf-files that are located at the course website.  If you find them very easy, do both. But return only one.

18. >> I hope you …have learned some MATLAB. And wish you a successful career with computational math!