introduction to matlab session 5 simopekka vänskä, thl 2010

Introduction to MATLABSession 5

Simopekka Vänskä, THL2010

Introduction to MATLAB - Session 5

Contents of this courseSession 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

Graphics 22-3 dimensional graphs


Graphics 22-3 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.


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))


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

>> plot(X,Y,’o’)

Remark X and Y are constant

in different coordinates.


Scalar functions on planeSome 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)


Vectorfields on plane or in 3D

A vector field on plane:F = (f1(x,y),f2(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;


>> U = X;

>> V = Y.^2;

>> quiver(X,Y,U,V);


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)


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’,[])

More linear algebra


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.

ProblemsSession 5


ProblemsDraw the contour plot of function

F(x,y) = x1.5y2exp(-(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).

Homeworks


How to do your homeworkYou 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.


Homework Topics

Personal finance – how your savings accumulateSIR – 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.

>> I hope you

…have learned some MATLAB.

And wish you a successful career with computational math!

introduction to matlab session 5 simopekka vänskä, thl 2010

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