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Problems and solutions
Session 3
Problems
1. Write function Xn = mspolygon(X,x0,a) that scales the
INPUT polygon by a (a>0) and moves its center to point
x0, and draws both polygons in one image.
 The polygon is given by matrix X whose columns are the
nodes (corner points) of the polygon. The output Xn is the
nodes of new polygon.
 Define the centerpoint to be the average of the nodes.
 Test your function with P of Exercise 1/Session 2.
Introduction to MATLAB - Solutions 3
Problems
2. Write a function Xt = roundt(X,t) that rounds real
numbers to grid tZ = (…,-2t,-t,0,t,2t,…) and complex
numbers to grid tC = tZ+itZ.
 The input X can be a matrix and t>0.
 Test your function (real case) with X = -5:.01:5 and
t=sqrt(2)/2. Draw a picture.
 Test your function (complex case) with X =
randn(1,5)+2*i*randn(1,5) and t=0.5. Draw a picture.
Write both test cases in one m-file.
Introduction to MATLAB - Solutions 3
Problems
3. Continue the Triangle Exercise 7/Session 2.
a) Write a function xn = Qpoints(n) where the input
argument n is a vector
n(j) = number of random points in [0,1]x[0,1]
(e.g. n = 1000:1000:10000)
and xn is a cell array with
xn{j} = n(j) random points.
b) Call Qpoints many times to find an approximative
error when computing the area of T with different n’s.
Represent the results graphically.
Introduction to MATLAB - Solutions 3
Some solutions
1. function Xn = mspolygon(X,x0,a);
Xsc = a*X;
Xn = [Xsc(1,:)+x0(1);Xsc(2,:)+x0(2)];
CALL
Xn = mspolygon(P,[2;1],0.75);
plot(P(1,:),P(2,:),’b’,Xn(1,:),Xn(2,:),’r’)
2. function Xt = roundt(X,t);
Xt = t*round(X/t);
CALL
X = -5:.01:5;
Xt = roundt(X,sqrt(2)/2);
plot(X,Xt,’.’)
Z = round(1,5)+2*i*round(1,5);
Zt = roundt(Z,0.5);
plot(Z(1,:),Z(2,:),’o’,Zt(1,:),Zt(2,:),’rx’)
Introduction to MATLAB - Solutions 3
Some solutions
3. function xn = Qpoints(n);
nn=length(n);
xn = cell(n,1);
for j = 1:nn
xn{j} = rand(2,n(nn));
end
--A routine to compute the area of T:
---
function aT = areaT(xn)
% area of T computed with xn (cell)
nn = length(xn);
aT = zeros(nn,1);
for j = 1:nn
x = xn{j};
aT(j) = sum(x(2,:)<(1-x(1,:)))/size(x,2);
end
Introduction to MATLAB - Solutions 3
Main m-file:
% number of points –vector:
n = [1, 10, 100,1000,10000,100000];
nn = length(n);
% number of rounds:
N = 1000;
aTall = zeros(n,N);
for k = 1:N % compute the areas N times
xn = Qpoints(n);
aTall(:,k) = areaT(xn);
end
deviations = std(aTall’)
% histograms, nn even
for k = 1:nn
subplot(2,nn/2,k)
hist(aTall(k,:))
end