Transcript Problems
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