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. Include the remainder term.
x+1
(b) Use the polynomial app
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 digits shown on your calculator)
a.
4x1 x2 + x3 = 8,
2x
Queen's College (Affiliated with the Memorial University)
NUMERICAL METHDS W/ APPL IN FINANCE
MATH 348

Fall 2014
Math 348, 8017 Spring 2008 Midterm # 2 Page 1
MATHEMATICS 348 [S01]
Midterm Examination #2
March 27, 2008
Instructor: Boualem Khouider
Duration: 50 minutes.
Last name: _ _
First name: _ _
Student #: _ _
o This emamination has 9 questions worth a tot
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 generator if a = 2? Hint: recall that the period is the smal
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.
Normally both the Windows and the UNIX versions of MAT
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 most one, two, three, and four. Plot both
the data (us
Queen's College (Affiliated with the Memorial University)
NUMERICAL METHDS W/ APPL IN FINANCE
MATH 348

Fall 2014
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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 partialdierential equations. Find the exact solution
and compar
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
/4
/4
2x
dxe.
24
x
x sin xdx
0
0
2. Determine the exact
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 + y 4 .
(c)f (x, y) = x
2. Determine whether each of the
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 eigenvectors of A
(d) If possible nd an invertible matrix P and
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 W2 is a subspace of V if
and only if W1 W2 or W2 W1 .
3
Queen's College (Affiliated with the Memorial University)
LINEAR ALGEBRA
MATH 341

Spring 2013
Math 341 HW 3:
February 11, 2013
1. Show that
X=
1 0 0
0 0 0
,
0 1 0
0 0 0
,
0 0 1
0 0 0
,
0 0 0
1 0 0
,
0 0 0
0 1 0
,
0 0 0
0 0 1
is a basis for M23 (F)
2. Recall that Pn (R) is the vector space of polynomials with coecients in R with degree
less than or
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. Find a bijection from N to Z2 .
3. Let W1 and W2 be two
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 3
(a)
1 1 1
(b) 2 3 1
0 2 0
3. For each of the followi