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Universitat Duisburg-Essen Winter Term 2011/12Fakultat fur Mathematik 12. Oktober 2011Computational MechanicsDipl.-Math. Andreas FischleDipl.-Math. Alexander Heinlein

Introduction to Numerical Methods

Homework 1

This first set of homeworks is intended to introduce you to the Matlab program-ming language in a playful way.

It may be helpful to use the diary function to document the output of Matlab.

Exercise 1: (5 Points / Programming)

(i) Use Matlab to print the machine representations of the real numbers 43

,0.0000012345 and 1.2345e06 in the following formats

(1) short ; short e ; short g

(2) long ; long e ; long g

(3) rat

and explain the results. What is the meaning of e in this context?

(ii) What happens if you use the following commands in Matlab

(1) f1 = 3 3 ;

(2) f2 = 3 3 ; g1 = [3 , 1; 1 , 3]3

(3) f3 = 3. 2 g2 = [3 , 1; 1 , 3] . 3

(4) 3 + 3

(5) disp(Hello world!); disp(Introduction to Numerical Methods); What is the difference between g1 and g2, cf., (2) and (3)?

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Exercise 2: (5 Points / Programming)Let A = (aij)i,j=1...n be the matrix with entries aij = i/j and x the vector xj = jfor j = 1 . . . n, i.e.,

A =

1 12

1

3

1

n

2 12

3

.. .

2

n

3 32

. . . . . . 3n

.... . . . . . . . .

...n n

2 1

, x =

1

23...n

.

(i) Try three different ways to build the matrix A, i.e.,

(1) using 2 nested for-loops(for k=1... for l=1... A(k,l)=k/l; end end)

(2) using one vector and one for-loop

(k=1:n; for l=1... A(:,l) = k/l; end)

(3) using the product of two vectors(k=(1:n); l=ones(1,n)./k; A = k * l;)

(ii) Build the matrix A and the vector x in Matlab and compute the matrix-vector product y = Ax for n = 3, 5, 10, 100, 1000.Verify the solution by your theoretical knowledge on matrix-vector multi-plication.

Exercise 3: (6 Points / Programming)

(i) Write a MATLAB function function x = make inter(n) which takes anumber of nodes n and returns a vector x containing n equidistant nodeswith a constant spacing h = 1

n1in the interval [0, 2] R.

(ii) Plot the functions sin(x) and cos(x) separately on this interval with h =0.01. How many nodes do you need?

(iii) Plot the sin(x) function for every step size h = 1, 0.5, 0.1, 0.01 and supe-rimpose all of these plots to obtain a single combined plot.

Due date: Thursday, 20 October 2011

Please, follow the guidelines described in the Syllabus on the homepage whichdescribes how to turn in your homework. Thank you for your cooperation.