Computing piby using Monte Carlo Methods

1 Introduction

Please carry out all sections and document the code with appropriate comments. Some tables are

required, these can be included as comments in the source code or submitted as a separate file.

2 Computing piby using Monte Carlo Methods

The Monte Carlo method uses a set of random points (x,y). If d is the distance from the origin then ndis the number of points where d <= 1. So pican be estimated by nd/nt , where nt is the total number ofpoints. Use the code ran2.c  to generatethe random numbers. Carry out the following parallelisation steps:

1. Construct a code that generates two random numbers x,y between 0????1.

2. Calculate the distance of the point (x,y) from the origin.

3. Use the result to estimate pi, using the formula above.

4. Parallelize it by using OpenMP constructs but when updating nd use a syncronous directive.

5. Make the function ran2 thread safe by adding OMP directives only. You should not need to alter

the code. So ignore the Stackoverflow response.

6. You may find that by changing the number of threads changes the result, why?

3 Steady State

You have a 2D array w of 100×100. The first row is 0.0 the last row, first and last columns are 100 and

these values remain fixed. Initially all other elements are set to 75.

w(1,2 : 99) = 0.0 (1)

w(100,1 : 100) = 100.0

w(1 : 100,1) = 100.0

w(1 : 100,100) = 100.0

w(2 : 99,2 : 99) = 75.0

Repeatedly update the interior values of w(2 : 99,2 : 99) with the equation below;

wnew(i, j) = (w(i+1, j)+w(i−1, j)+w(i, j+1)+w(i, j−1))/4 (2)

Stop when the maximum difference between w and wnew is less than 10 to the power -4

1. Create the serial code first.

2. Parallelize the initialisation and update of w, using OpenMP constructs.

       3. Note that each iteration wnew is independent of the previous w.

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