MATH3090: Computational Mathematics
Assignment 2
Due on Wednesday, November 5 (in-class)
1. Prove that the Markov transition matrix P has at least one eigenvalue equal to 1.
2. The following graph of another six-node subset of the Web. In this example, th
Chapter 2
Linear Equations
One of the problems encountered most frequently in scientic computation is the
solution of systems of simultaneous linear equations. This chapter covers the solution of linear systems by Gaussian elimination and the sensitivity
hmzhu
Page 1
12/1/2014
MATH3090: Practice Questions
1. Analyze the discretization error of improved Eulers method in the context of our
well-stirred coffee model, i.e., Newtons cooling model. Denote the time step size as
h = T/K.
a)
The continuous model i
4.6 Sampling Theory: the use of
Fourier Transform in reality
Math3090, HM Zhu
1.Effect of Finite Sampling
H(u)
h(t)
t
-L
X
w(t)
L
W(u)
u
1
T
t
u
Effect of Finite Sampling
Time
Frequency
The convolution of the Fourier spectrum with a sinc function
results
Chapter 2
Systems of Linear Equations
2.5 Application: Google PageRank
(Ref: C. ROUSSEAU, How Google Works)
Math3090, HM Zhu
Google PageRank: the problem
2
Math3090, HM ZHU
What is Google PageRank? (from wiki)
1. A link analysis algorithm used by Google S
Topics for Assignment 2
1. Mathematics and Cooking:
Harvard Lecture Series on Science & Cooking:
Gelation & Heat Transfer
Related Article:
http:/books.google.ca/books?id=oGq4v4e7rG8C&pg=P
A43&lpg=PA43&dq=heat+transfer+equation+cooking&so
urce=bl&ots=yOUIa
4. Mathematics, Music and More
4.4. The Fourier Transforms
Jean Baptist Joseph Fourier
(1768-1830)
Mathematician, Physicist,
Historian, and Politician
Best known for the Fourier transform
Auxerre, France
In 1807, his introductory
manuscript Theory of the
Chapter 2
Systems of Linear Equations
(Moler, Chapter 2)
Math3090, HM Zhu
An example
Solve the linear systemAx = b, where
1
1
1 1
1
4
3
2
2
1 2 13 1 4 15
A
, b A
3
13 1 4 15 1 6
1
1
1
1
4
4
7
6
5
1. Numericall y Solve Ax b using M ATLAB
2. Nu
5. Mathematics and Music
Math
3090, HM Zhu
Modern Technology and Math?
MRI
2
4. Mathematics and Music
4.1 Pure Tones, Pitch, Frequency
Math
3090, HM Zhu
Simple Harmonic Motion
Simple Harmonic Motion Simulation
Math
3090, HM ZHU
4
Period and Frequency
The
Chapter 3
Matrices and Digital Images
3.1. Digital image 101
Math 3090, HM Zhu
Taking good photos of interesting things
CT and Solving Linear Systems
Math 3090, HM ZHU
Learning Objectives
Understand digital images
Understand the basic math concepts behi
Chapter 3
Matrices and Digital Imaging
3.2 A Simple Model for CT Image Reconstruction
MATH 3090, HM Zhu
Beginning of X-Ray (Radiography)
2
MATH 3090 HM ZHU
X-Ray
3
MATH 3090 HM ZHU
Failure of X-Ray
4
MATH 3090 HM ZHU
Computed (Axial) Tomography (CT)
How C
Learning Objectives in MATH 3090
Learn the fundamental thinking process of solving a
real-world problem
Master some basic numerical programming and
analysis skills
Train you the teamwork, presentation, and writing skills
Make you realize by examples that
MATH3090: Computational Mathematics
Assignment 1
Due on Friday, October 3 (in-class)
1. Modify the coffee cooling model to account for a room temperature that starts at 20 Celsius and
increase at a constant rate to maximum 26 Celsius in 2 hours and then s