Docsity
Docsity
Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity

Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips
Guidelines and tips
Sell on Docsity
Sell on Docsity
Docsity AI
Docsity AI
Log inSign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search for your university

Find the specific documents for your university's exams

Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents
download point icon20 Points

For each uploaded document

Answer questions
download point icon5 Points

For each given answer (max 1 per day)

All the ways to get free points
Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

Simultaneous Equations - Computational Science - Solved Exam, Exams of Computational Physics

Alliance UniversityComputational Physics

This is the Solved Exam of Computational Science which includes Matrix Manipulation Functions, Acceleration Due to Gravity, Normal Distribution, Normalised Expression, Exponential with Several Bins, Acceptable Code, Normal Distribution etc. Key important points are: Simultaneous Equations, Combination of Eigenvectors, Coefficients of Eigenvectors, Long-Term Population, Circle of Radius, Matrix Equation, Electron Number Density, Input Variable

Typology: Exams

2012/2013

Uploaded on 02/20/2013

salagame
salagame 🇮🇳

4

(3)

36 documents

1 / 3

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
COSC1001/1901 (Semester Two, 2009) Exam Solutions and Marking Scheme
Computational Science in MATLAB
COSC1001 (Normal) & COSC1901 (Advanced)
Exam Solutions and Marking Scheme
1. (a) (2 marks) count counts the number of experiments where x==1, which is his home. So
the probability is 1-0.2476=0.7524.
(b) (2 marks) 0.2453, 0.2510 etc. It would not be reasonable to expect 0.16.
(c) (3 marks) The answer follows from an argument like:
P(round(a+rand) == 1) = P(a+rand >0.5) (1)
=P(rand >0.5−a) (2)
= 0.5 + a(3)
(d) (3 marks) a= 0.3
x = x + 2*round(0.3+rand) - 1
2. (a) (2 marks) αand βare the survival rates, and a maximum of 100% of the population
can survive at each generation.
(b) (2 marks) A = [0 1 0.5; 0.8 0 0; 0 0.5 0];
(c) (4 marks) The ratio children:adults:retirees should be 0.7454:0.5963:0.2981, and the
population will remain stable over time. This is because the eigenvalue with largest
magnitude is 1.0. Any initial starting population can be decomposed into a linear
combination of the three eigenvectors, and the coefficients of eigenvectors 2 and 3 will
gradually decay over time, leaving only the first eigenvector.
(d) (4 marks) If the long-term population is stable, then we must have:
(A−I)e= 0
for some eigenvector e=e1, e2, e3. This reduces to three simultaneous equations:
−e1+r2e2+r3e3= 0 (4)
αe1=e2(5)
βe2=e3(6)
This is trivial to solve:
e1(−1 + αr2+αβr3) = 0,
and the identity follows.
3. (a) (3 marks) Begin with 104sets of 2 random numbers between -1 and 1. The fraction of
these numbers that land within the circle of radius 1 is the ratio of the area of the unit
circle (i.e. π) to the area of the square defined by −1< x < 1 and −1< y < 1 (i.e. 4).
(b) (2 marks) x=2*rand(10000,1)-1
(c) (4 marks)
1
pf3

Partial preview of the text

Download Simultaneous Equations - Computational Science - Solved Exam and more Exams Computational Physics in PDF only on Docsity!

Computational Science in MATLAB

COSC1001 (Normal) & COSC1901 (Advanced)

Exam Solutions and Marking Scheme

  1. (a) (2 marks) count counts the number of experiments where x==1, which is his home. So the probability is 1-0.2476=0.7524. (b) (2 marks) 0.2453, 0.2510 etc. It would not be reasonable to expect 0.16. (c) (3 marks) The answer follows from an argument like:

P (round(a + rand) == 1 ) = P (a + rand > 0. 5 ) (1) = P (rand > 0. 5 − a) (2) = 0 .5 + a (3)

(d) (3 marks) a = 0. 3 x = x + 2*round(0.3+rand) - 1

  1. (a) (2 marks) α and β are the survival rates, and a maximum of 100% of the population can survive at each generation. (b) (2 marks) A = [0 1 0.5; 0.8 0 0; 0 0.5 0]; (c) (4 marks) The ratio children:adults:retirees should be 0.7454:0.5963:0.2981, and the population will remain stable over time. This is because the eigenvalue with largest magnitude is 1.0. Any initial starting population can be decomposed into a linear combination of the three eigenvectors, and the coefficients of eigenvectors 2 and 3 will gradually decay over time, leaving only the first eigenvector. (d) (4 marks) If the long-term population is stable, then we must have:

(A − I)e = 0

for some eigenvector e = e 1 , e 2 , e 3. This reduces to three simultaneous equations:

−e 1 + r 2 e 2 + r 3 e 3 = 0 (4) αe 1 = e 2 (5) βe 2 = e 3 (6)

This is trivial to solve:

e 1 (−1 + αr 2 + αβr 3 ) = 0,

and the identity follows.

  1. (a) (3 marks) Begin with 10^4 sets of 2 random numbers between -1 and 1. The fraction of these numbers that land within the circle of radius 1 is the ratio of the area of the unit circle (i.e. π) to the area of the square defined by − 1 < x < 1 and − 1 < y < 1 (i.e. 4). (b) (2 marks) x=2*rand(10000,1)- (c) (4 marks)

function pi_est=estimatepi(n) %This function computes a monte-carlo estimate for pi x=rand(n,1); y=rand(n,1); pi_est=mean(x.^2+y.^2 < 1)*4; end (d) i. (2 marks) 5 (as a = [-1 1 3 5 7]) ii. (2 marks) 7 7 iii. (2 marks) 8 9

  1. (a) (4 marks) The 4 equations are as follows, with the variables we are solving for being nH, nHe, nH+ and nHe+.

nH − fH (T )nenH+ = 0 (7) nHe − fHe(T )nenHe+ = 0 (8) nH + nH+ − 10 nHe − 10 nHe+ = 0 (9) nH+ + nHe+ = ne (10)

These obviously have to be written as a matrix trivially for full marks. The students also need a little inspiration to realize that the electron number density can not be part of the linear system, and needs to be considered an input variable in order to write a matrix equation (actually, that part of the full problem is non-linear). (b) (4 marks) Something like the folloing, which was actually designed for 2 ionization states of He...

function ns=saha(n_e, T) %This function returns a vector nH, nHplus, nHe and nHeplus m = zeros(5); m(1,1:2)=[1 -fH(T)n_e]; m(2,3:4)=[1 -fHe(T)n_e]; m(3,:)=[1 1 -0.1 -0.1]; m(4,:)=[0 1 0 1]; b=[0 0 0 0 n_e]’; ns=m/b; end

(c) (2 marks) function f=myfunc(xyz) x=xyz(1); y=xyz(2); z=xyz(3); f = cos(x+y) + (sin(x+z)).^2; end (d) (2 marks) function xyz=randomsphere xyz=2.*rand(3,1)-1;