Introduction to MATLABSession 2

Simopekka Vänskä, THLDepartment of Mathematics and StatisticsUniversity of Helsinki 2010

Introduction to MATLAB - Session 2

Contents of this course

Session 1 General

Matrices

M-files

Session 3 My functions + stings, cells

Controlling program flow

Session 2Some matrix commands

Logical expressions

Graphics 1

Session 4 Function functions

Session 5 Graphics 2

More linear algebra

Starting homework

Basic commands

Introduction to MATLAB - Session 2

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

FunctionINPUT

parametersOUTPUT

parameters

Introduction to MATLAB - Session 2

Some elementwise functions 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 03 2

Some common functions:sin, cos, tan, asin, acos, atanexp, log, sqrtabs, conj, imag, real, anglefix, 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)

Introduction to MATLAB - Session 2

Some columnwise functions 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, cumprodmean, median, std, min, maxsort

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)

Introduction to MATLAB - Session 2

Polynomials and interpolationFinding 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.

Logical expressions

Introduction to MATLAB - Session 2

Logical expressions and functions 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

Introduction to MATLAB - Session 2

FIND 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)

Graphics 1 - PLOT

Introduction to MATLAB - Session 2

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

Plot line properties 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-’)

Introduction to MATLAB - Session 2

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

Figure and subplot

>> 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))

1 2 3

654

ProblemsSession 2

Introduction to MATLAB - Session 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.

Introduction to MATLAB - Session 2

Problems

3. Draw the unit circle in R2. 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.

Introduction to MATLAB - Session 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 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

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.

>> quit

…to exit MATLAB.