计算机建模| FIT3139 Computational Modelling and Simulation – PAPER 1

本次澳洲写主要为计算机建模相关的限时测试

Question 1 [10 marks = 5 + 5 ]
This question is about errors and computer arithmetic.
A) Determine an expression that approximates the condition number for the following
function:
f(x) = e3x2+1
B) Explain what are the implications of the condition number being large.

Question 2 [15 marks = 10 + 5 ]
This question is about linear systems.
A) Compute the matrices L and U that correspond to the LU factorisation of A (i.e.,
A = LU). Explain step by step your computation, stating the elimination matrices
you use at each step.
A =
0
B @
2 1 2
10 6 12
4 6 15
1
C A

B) Sometimes an LU decomposition of A is expressed as PA = LU. This is known as
partial pivoting. What is the purpose of partial pivoting and how is it performed?

Question 3 [10 marks]
This question is about solving non-linear equations. Using pseudocode write down an
algorithm to nd the root of a non-linear function f(x) using the Secant method. Your
algorithm should use initial guesses x0 and x1, and stop when the absolute error is less
than .

Question 4 [10 marks = 5 + 5 ]
This question is about solving second-order dierence equations.
A) Consider the second-order dierence equation with:
Xt = 7Xt 1 6Xt 2; X0 = 0; X1 = 1
Write this system as a rst-order matrix dierence equation. That is, nd the
matrix, M, such that ”
Xt
Xt 1
#
= M
”
Xt 1
Xt 2
#
:

B) Using an eigendecomposition of M, nd an explicit solution to the system above.