FEEG6002C1
UNIVERSITY OF SOUTHAMPTON FEEG6002C1
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SEMESTER 1 ASSESSMENT PAPER 2020/21
Advanced Computational Methods 1
DURATION – 24 hours
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This paper contains five questions. Answer all questions.
This is an online assessment. The solution document must be
submitted via Blackboard only. Please submit your answers to
all questions using a single word document.
Please use Quincy (free download from below link)
http://www.codecutter.net/too…
or use any of the following freely available on-line C compilers
to write, compile and execute C programmes.
https://www.onlinegdb.com/onl…
https://www.programiz.com/c-p…
A total of 100 marks are available for this paper.
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Question 1:
a). Why do you want to work on a remote system (or remote server)?
(3 marks)
b). What is a terminal?
(2 marks)
c). Why do you need a terminal in the Linux operating system?
(2 marks)
d). What is the motivation of using shell scripts in the Linux operating system?
(3 marks)
e). List three different Linux shell types?
(3 marks)
f). What is the motivation of using make files in the Linux operating system?
(3 marks)
g). Explain with steps how to create make files in the Linux operating system?
(4 marks)
h). What is a function in C programming?
(2 marks)
i). C programming offers two types of functions. What are they, and how are they
different?
(2 marks)
j). Show two methods for defining a symbolic constant named MAXIMUM that has a
value of 200.
(2 marks)
k). If a function does not return a value, what type should be used to declare it?
(2 marks)
l). In a one-dimensional array declared with n elements, what is the subscript of the
last element?
(2 marks)
(30 marks)
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Question 2:
Fill in the blanks in each of the following.
a). The first element of an array has index ______.
b). The __ operator returns the number of bytes used to store a variable type t.
c). To declare a pointer of type char, you need to use _______.
d). Memory allocation through _ takes place at run time.
e). C programs consist only of _______.
f). __ need to be given before the function can be called.
g). If you use constants in a C-program, it is good practice to define a _____.
h). __ is a special macro representing End Of File.
i). __ is a valuable function for allowing user input to a C program.
j). With the ______, C program can use alias names of data types that are machine
specific.
(10 Marks)
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Question 3:
In this question you will develop a set of functions to manipulate 3×3 matrices in
various ways.
a) Write a function print_matrix(…) which takes as an input a two-dimensional
array of size nxn of type double representing a matrix, and an integer n
corresponding to the size of this array. The function should print the elements of this
array to the standard output displaying it in the form of a matrix. For example, we
expect:
1 0
0 1
for a 2×2 array a with elements a0=1, a0=0, a1=0,
a1=1. The function print_matrix(…) does not return any value.
(2 marks)
b) Write a function transpose(…) which takes as an input a 3×3 array a of type
double representing a matrix, and returns a transpose of this matrix stored in the
same input variable a. Note the transpose of a square matrix 𝑎𝑖𝑗 is defined as:
(𝑎)𝑖𝑗
𝑇 = 𝑎𝑗𝑖.
(2 marks)
c) Write a function det2(…) which takes as an input a 2×2 array a of type
double representing a matrix, and returns a variable of type double equal to the
determinant of the input matrix:
|𝑎| = |
𝑎00 𝑎01
𝑎10 𝑎11
| = 𝑎00𝑎11 − 𝑎01𝑎10.
(2 marks)
d) Write a function cofactor(…) which takes as an input a 3×3 array a of type
double representing a matrix, and two integers i and j, and returns a variable of
type double equal to the cofactor 𝑐𝑖𝑗 of the matrix a. The cofactor 𝑐𝑖𝑗 is defined as:
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where 𝑚𝑖𝑗 is the minor of the element 𝑎𝑖𝑗 of the 3×3 matrix a, equal to the
determinant of the reduced 2×2 matrix obtained by removing all elements of the ith
row and jth column of the matrix a. Thus the notation 𝑎𝑘𝑙
(𝑖𝑗)
in the above equation
means the elements of the matrix remaining after removing the ith row and jth
column from the matrix a. Use function det2(…) developed above to simplify your
code.
(4 marks)
e) Write a function det3(…) which takes as an input a 3×3 array a of type
double representing a matrix, and returns a variable of type double equal to the
determinant of the input matrix a. To evaluate the determinant |𝑎| of a matrix 𝑎 use
the Laplace expansion method:
|𝑎| = ∑𝑎𝑘𝑖𝑐𝑘𝑖
2
𝑘=0
𝑖 = 0 𝑜𝑟 1 𝑜𝑟 2
where 𝑐𝑘𝑖 is the cofactor of the matrix 𝑎 as defined above. Note that the index i is
fixed and the result is independent of the choice of i. Use the function
cofactor(…) developed as part of the previous question to simplify your code.
(4 marks)
f) Write a function inverse(…) which takes as an input a 3×3 array a of type
double representing a matrix, and returns an inverse of this matrix stored in the
same input variable a. To calculate the inverse matrix 𝑎
−1 of a matrix 𝑎, note that the
elements of the inverse matrix can be expressed as:
where 𝑐𝑖𝑗 denotes the cofactor of the matrix a and |𝑎| is the determinant of the matrix
a. Use your functions det3(…) and cofactor(…) developed above to simplify
your code.
(4 marks)
g) Write a function multiply(…) which takes as an input two 3×3 arrays of type
double representing two matrices. The function should calculate the matrix product
of the two matrices and print it to the standard output using the function
print_matrix(…) developed above. The function multiply(…) does not
return any value.
(2 marks)
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Example of incomplete function main() could look like this:
int main(void)
{
…
…
printf(“\nOriginal matrix:\n”);
print_matrix(3, a);
printf(“Determinant of the matrix:\n”);
printf(“%f\n”, det3(a));
printf(“\nTranspose of the matrix:\n”);
transpose(a);
print_matrix(3, a);
printf(“Inverse of the matrix:\n”);
transpose(a); // transpose back to obtain original
matrix
for(i=0; i<N; i++)
for(j=0; j<N; j++)
ainvi = ai; // copy of a to invert
inverse(ainv);
print_matrix(3, ainv);
printf(“Check that the product gives unit
matrix.\n”);
multiply(a, ainv);
…
}
and assuming the given choice of the matrix the expected output if the missing parts
have been populated is:
Original matrix:
1.000000 0.000000 1.000000
0.000000 1.000000 1.000000
0.000000 0.000000 1.000000
Determinant of the matrix:
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1.000000
Transpose of the matrix:
1.000000 0.000000 0.000000
0.000000 1.000000 0.000000
1.000000 1.000000 1.000000
Inverse of the matrix:
1.000000 -0.000000 -1.000000
-0.000000 1.000000 -1.000000
0.000000 -0.000000 1.000000
Check that the product gives unit matrix.
1.000000 0.000000 0.000000
0.000000 1.000000 0.000000
0.000000 0.000000 1.000000
(20 marks)
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Question 4:
In this question you will evaluate the statistics of the United States presidential
election in 2012. You are given a file uselecton2012.txt which has 5 tabdelimited
columns:
Alabama 795696 0 1255925 9
Alaska 122640 0 164676 3
Arizona 1025232 0 1233654 11
Arkansas 394409 0 647744 6
…
…
The 1st column represents the state name, 2nd and 3rd columns the democratic popular
and electoral votes, and 4rd and 5th columns represent the republican popular and
electoral votes. Reference:
https://en.wikipedia.org/wiki…
a) Write a function count_lines(…) which takes as input a file name string and
returns an integer equal to the count of the number of lines in the input file with the
given name.
(2 marks)
b) Write a structure votes which has the following members:
char state[100]; // state name
long dempv; // democrats popular votes
long demev; // democrats electoral votes
long reppv; // republicans popular votes
long repev; // republicans electoral votes
(2 marks)
c) In function main(…) call the function count_lines(…) to determine the
number of lines nlines in the file uselection2012.txt. Allocate dynamically
the array arr to be nlines long with elements being of type structure votes.
(2 marks)
d) Write a function initialise_votes(…) which takes as input a file name
string, a pointer to an array of structures of type votes, and an integer
corresponding to the length of this array. The function will be called in main(…) for
example as:
initialise_votes(“uselection2012.txt”, arr, nlines);
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and populate the members of each structure given in the array arr by information
contained in each corresponding line in uselection2012.txt. Thus, after
calling the function initialise_votes(…) the members of the structure
arr[0] should be:
state=‘Alabama’; dempv=795696; demev=0; reppv=1255925;
repev=9;
the members of the structure arr[1] should be:
state=‘Alaska’; dempv=122640; demev=0; reppv=164676;
repev=3;
and so on. The function initialise_votes(…) does not return any value, and
the output should be available through the input pointer arr.
(4 marks)
e) Write a function print_list(…) which takes as an input a pointer to an array
of structures of type votes and an integer corresponding to the length of this array.
The function will be called in main(…) for example as:
print_list(arr, nlines);
The function does not return any value and will only print the content of the file to
the standard output. You can use this function to check if your initialisation of the
array of structures arr correctly reflects the format of the file
uselecton2012.txt, i.e.:
Alabama 795696 0 1255925 9
Alaska 122640 0 164676 3
Arizona 1025232 0 1233654 11
…
(2 marks)
f) Write a function print_vote_state(…) which takes as an input a pointer to
an array of structures of type votes and an integer corresponding to the length of
this array. The function will be called in main(…) for example as:
maxstate = print_vote_state(arr, nlines);
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where the output variable maxstate is a structure of type votes, and corresponds
to the state with the highest counts of the popular votes determined from among both
democrat and republican electoral votes.
(4 marks)
g) Write a function print_vote_total(…) which takes as an input a pointer to
an array of structures of type votes and an integer corresponding to the length of
this array. The function will be called in main(…) for example as:
print_vote_total(arr, nlines);
and will print to standard output four numbers corresponding to the total sum of
popular and electoral votes for democrats, and for republicans. The function
print_vote_total(…) should return a string“dem. win”or“rep. win”
depending on the higher value of the total values of electoral votes.
(4 marks)
An example of incomplete function main(…) could look like this:
int main(void){
…
nlines = count_lines(“uselection2012.txt”);
…
…
initialise_votes(“uselection2012.txt”, arr, nlines);
print_list(arr, nlines);
printf(“\n”);
maxstate = print_vote_state(arr, nlines);
printf(“%s wins with %ld (dem) and %ld (rep) popular
votes.\n”, maxstate.state, maxstate.dempv,
maxstate.reppv);
printf(“\n”);
printf(“%s\n”, print_vote_total(arr, nlines));
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…
…
}
and the expected output if the missing parts have been populated correctly is:
Alabama 795696 0 1255925 9
Alaska 122640 0 164676 3
Arizona 1025232 0 1233654 11
Arkansas 394409 0 647744 6
California 7854285 55 4839958 0
…
…
…
WestVirginia 238269 0 417655 5
Wisconsin 1620985 10 1407966 0
Wyoming 69286 0 170962 3
California wins with 7854285 (dem) and 4839958 (rep)
popular votes.
Dem. pop. vote total: 65915794
Dem. el. vote total: 332
Rep. pop. vote total: 60933504
Rep. el. vote total: 206
Dem. wins
(20 marks)
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Question 5:
This question will consider solving the predator-and-prey population dynamics
problem. The problem can be mathematically stated as follows. Let 𝑝1(𝑡) be the
number (population) of rabbits and 𝑝2(𝑡) the number of foxes. The time-dependence
of 𝑝1 and 𝑝2
is governed by the following conditions:
- Rabbits proliferate at a rate 𝑎, and per unit time a number 𝑎𝑝1
is born. - Number of rabbits is reduced by collisions with foxes. Per unit time 𝑐𝑝1𝑝2
rabbits are eaten. - Birth rate of foxes depends only on food intake in form of rabbits.
- Foxes die a natural death at a rate 𝑏.
This dynamics can be expressed by the following system of first order differential
equations (ODE):
𝑑𝑝1
𝑑𝑡 = 𝑎𝑝1 − 𝑐𝑝1𝑝2 = 𝑓1(𝑝1
, 𝑝2)
𝑑𝑝2
𝑑𝑡 = 𝑐𝑝1𝑝2 − 𝑏𝑝2 = 𝑓2(𝑝1
, 𝑝2)
The task is to solve this system of equations. To obtain the solution, assume: - Rabbit birth rate 𝑎 = 0.7
- Rabbit-fox collision rate 𝑐 = 0.007
- Fox death rate 𝑏 = 1
- Initial conditions 𝑝1
(0) = 70 and 𝑝2
(0) = 50 - Time interval from 0 to 30 (dimensionless)
a) Write a function fode2(…) which returns type void and takes as input two
arrays of type double. Each of the arrays has two elements. The first array stores the
variables 𝑝1
, 𝑝2
. The second array stores the values of the right hand sides 𝑓1 and 𝑓2
,
and serves as the output of the function fode2(…).
(5 marks)
b) Extend the function euler(…) given in lecture 8 as:
void euler(double t, double y, double y0,
double (*f)(double, double), int n)
{
int i;
float dt = t[1] – t[0]; // step -
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y[0] = y0; // initial condition
for (i = 0; i < n; i++)
{
y[i + 1] = y[i] + dt * f(t[i], y[i]);
}
}
to incorporate the arrays for populations of both rabbits and foxes, including the
function fode2(…) developed above.
(10 marks)
c) In main(…) run the function euler(…) with appropriate inputs, and export the
data for time 𝑡 and populations 𝑝1 and 𝑝2
to the file data.txt. Use your favourite
software to plot the data (see figure 1 below).
(5 marks)
An example of incomplete function main(…) could look like this:include<stdio.h>
include<stdlib.h>
include<math.h>
define Ti 0.0 // initial time
define Tf 30.0 // final time
define dT 0.001 // time step
define Y01 70.0 // initial condition 1
define Y02 50.0 // initial condition 2
define A 0.7
define C 0.007
define B 1
…
…
int main()
{
…
…
for(i=0; i<=N; i++) - FEEG6002C1
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{
t[i] = Ti + dT*i; // define time
}
euler(y1, y2, fode2, N); // y1, y2 to store
solutions
// no explicit time
dependence
// print to file
for(i=0; i<=N; i++)
{
fprintf(fw, “%f\t%f\t%f\n”, t[i], y1[i], y2[i]);
}
…
…
}
and when the missing parts have been populated the expected output is as shown in
Figure 1 below. You can use this figure to compare with your solution. - FEEG6002C1
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Figure 1 – Example of the plot calculated using the extended Euler algorithm
assuming parameters listed in the figure title. Populates of rabbits and foxes reach a
steady state with a phase lag.
(20 marks)
END OF PAPER