CODEWITHSUNDEEP
matlabiterationnumerical-analysis
user11199583
user11199583

Reputation: 21

(Matlab) Exam questions about iteration result

I am preparing for a written test in numerical analysis.

There are multiple-answer questions of the type:

The code below gives a print-out closest to...

N=100;
dt=0.5/N;
x=1;
for n=1: N-1
x=x+dt*x*x;
end
display(x);

Correct option: 2

The code below gives a print-out closest to...

N=100;
x=1;
y=1;
for n=1: N-1
x = x - (x-exp(-x))/(1+exp(-x));
y = y - (y-exp(-y));
end
display(x-y)

Correct option: 0

The code below has a print-out that most often will be...

s=0.0;
N=10000;
for n=1:N
x=rand(1);
s=s+3*x*x+x;
end
display(s/N)

Correct option: 1.5

For the first question I assumed it was some use of Eulers method but I could not arrive at the number 2. I'm not sure how to tackle the second and third one.

Is there some kind of general strategy I can use to figure out what iterations like this should converge towards (without the use of a computer) ?

Upvotes: 1

Views: 144

Answers (1)

Ramashalanka
Ramashalanka

Reputation: 8854

The general strategy is to look at what the iteration that is being implemented is doing.

First question

This is dx/x^2=dt, i.e. 1/x0-1/x=t, i.e. x=x0/(1-x0*t)=1/(1-1*0.5)=2.

Second question

The equation for x is Newton's method for f(x)=x-exp(-x), i.e. f'(x)=1+exp(-x), so the solution is the root of x=exp(-x).

The equation for y is fixed point iteration for y=exp(-y). These both have the same solution x=y=0.5671 (4sf).

Third question

rand(1) is the uniform distribution on [0,1] with mean 1/2 and variance 1/12 so <x^2>=1/12+1/4=1/3 and 3<x^2>+<x>=3/3+1/2=1.5.

Upvotes: 3

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