Queen's College (Affiliated with the Memorial University)
NUMERICAL METHDS W/ APPL IN FINANCE
MATH 348

Fall 2014
Math 348: Numerical methods with applications in nance
Problem set # 1
January 15, 2008
1. (a) Determine the second order (n = 2) Taylor polynomial approximation for f (x) =
expanded about x0 = 0. Inc
Queen's College (Affiliated with the Memorial University)
NUMERICAL METHDS W/ APPL IN FINANCE
MATH 348

Fall 2014
Math 348: Numerical methods with applications in nance
Problem set # 2
January 17, 2008
1. Use Gauss elimination with backward substitution to solve the following linear systems (keep
all the correct
Queen's College (Affiliated with the Memorial University)
NUMERICAL METHDS W/ APPL IN FINANCE
MATH 348

Fall 2014
Math 348: Problem set #6, Monte Carlo Integration
1. Consider a congruential random number generator
xk+1 = (axk + b)
mod M
with b = 0, M = 8192 and a seed x0 = 1.
(a) What is the period of this gener
Queen's College (Affiliated with the Memorial University)
NUMERICAL METHDS W/ APPL IN FINANCE
MATH 348

Fall 2014
Math348: Numerical methods with applications in nance
Problem set#3
To solve some questions in this problem set, you need to use MATLAB. Below are a few hints and
recommendations on how to use matlab.
Queen's College (Affiliated with the Memorial University)
NUMERICAL METHDS W/ APPL IN FINANCE
MATH 348

Fall 2014
Math348: Numerical methods with applications in nance
Problem set#4
1. For each one of the data sets below use the polyt function of matlab to nd the least
square polynomial approximation of degree at
Queen's College (Affiliated with the Memorial University)
NUMERICAL METHDS W/ APPL IN FINANCE
MATH 348

Fall 2014
, , , (2on WWQSVQ/LOCAS
W M
,f
,  we MM , MrmC/VAQYKQ/Sa 34* , W ,
, , ago +9va w
x3
,
\crUA ,_
)q
_ £0 mtg x1 H01,
am,ol,_,. ,. So ,_/yz/L,ULOLL
I: CL, XL; 2 z (Loas\. Tole: wwsm
mm, q
Queen's College (Affiliated with the Memorial University)
NUMERICAL METHDS W/ APPL IN FINANCE
MATH 348

Fall 2014
Math 348: Problem set #7, Finite dierence methods for PDEs
1. Use the forward in time centered in space (FTCS) method (explicit scheme) to approximate
the solution to the following parabolic partiald
Queen's College (Affiliated with the Memorial University)
NUMERICAL METHDS W/ APPL IN FINANCE
MATH 348

Fall 2014
Math 348: Problem set #5, Numerical Intergration
1. Approximate the following integrals using the trapezoidal rule
1
a.
0.5
1/4
d.
1/4
1
1.5
2
dxb.
x4
x2 ex dx
(1)
e3x sin 2xdx.
x2 ln xdx
(2)
c.
0
1
/
Queen's College (Affiliated with the Memorial University)
NUMERICAL METHDS W/ APPL IN FINANCE
MATH 348

Fall 2014
Math 348: Problem set #8, Optimization
1. Determine whether each of the following functions is coercive on R2 .
(a) f (x, y) = x + y + 2,
(b)f (x, y) = x2 + y 2 + 2,
2 2xy + y 2 (d)f (x, y) = x4 2xy +
Queen's College (Affiliated with the Memorial University)
LINEAR ALGEBRA
MATH 341

Spring 2013
Math 341 HW 11:
April 26, 2013
1. For each of the following matrices
(a) Find all eigenvalues of A
(b) Determine the eigenspace corresponding to each eigenvalue
(c) If possible nd a basis of eigenvect
Queen's College (Affiliated with the Memorial University)
LINEAR ALGEBRA
MATH 341

Spring 2013
Math 341 HW 2:
February 4, 2013
1. Write a detailed proof that if V is a vector space and cfw_X V is some subset. Then
spancfw_X =
W
cfw_X W
2. Let W1 and W2 be two subspaces of V . Prove that a W1
Queen's College (Affiliated with the Memorial University)
LINEAR ALGEBRA
MATH 341

Spring 2013
Math 341 HW 4:
February 15, 2013
1. Let A and B be nonempty sets. Assume that f : A B is an injective function.
Show that there is a surjective function g : B A such that g(f (a) = a for all a A.
2. F
Queen's College (Affiliated with the Memorial University)
LINEAR ALGEBRA
MATH 341

Spring 2013
Homework 8: (Due April 5)
March 23, 2013
1. Prove that for any m n matrix A, rank(A)=0 A is the zero matrix.
2. For each of the following matrices compute the rank and the inverse if it exists.
1 1
2