THE $25,000,000,000 EIGENVECTOR
THE LINEAR ALGEBRA BEHIND GOOGLE
KURT BRYAN AND TANYA LEISE
Abstract. Googles success derives in large part from its PageRank algorithm, which ranks the importance
of webpages according to an eigenvector of a weighted link
CS 111 - Introduction to Computational Science
Homework 1
1. Consider the ODE describing a falling object under the gravity force and the drag force:
dv
cd 2
dt = g m v
,
v(0) = 0
(1)
where g = 9.81 m.s2 , m = 75 kg and cd = .25 kg.m1 . The exact solutio
CS 111 - Introduction to Computational Science
Homework 2
The goal of this homework is to simulate the motion of a ball bouncing off the walls of a closed
container, as described in class. The main force acting on the ball is the weight (fw = mg) so that
CS 111 - Introduction to Computational Science
Solving Diusion Problems
Diusion phenomena are ubiquitous in science and engineering. For example, diusion describes
the spread of particles through random motion from regions of higher concentration to regio
Electrical Networks
Gigantic linear system!
Need a computer to solve it.
Heat Distribution Around the Shuttle
Need to solve the heat equation
Heat Distribution Around the Shuttle
Need to solve the heat equation
Discretize the domain
Heat Distribution A
CS 111 - Introduction to Computational Science
Homework 6
Consider the diusion equation in the interval [0, 1]:
u
2u
= D 2,
t
x
(1)
where D is the diusion constant. Here, we will take D = .5. An exact solution for this equation is
u(x, t) = eDt cos(x) and
CS 111 - Introduction to Computational Science
Homework 5
Consider a domain = [1, 1] in one spatial dimension and the following linear advection equation:
u
u
+c
= 0,
t
x
(1)
where c is the velocity. We take the initial data for u to be u(x, 0) = cos(x).
CS 111 - Introduction to Computational Science
Homework 3
1. Recall that a system of ODEs can be written as:
dY
= F(t, Y, par),
(1)
dt
where t is the variable, par is a list of parameters and Y is a vector whose components are the
solutions of the system,
CS 111 - Introduction to Computational Science
Homework 2
The goal of this homework is to simulate the motion of a ball bouncing o the walls of a closed
container, as described in class. The main force acting on the ball is the weight (fw = mg) so that
th
CS 111 - Introduction to Computational Science
Homework 1
Consider the ODE describing a falling object under the gravity force and the drag force:
dv
dt
= g
v (0) = 0
cd 2
mv
,
(1)
where g = 9.81 m.s2 , m = 75 kg and cd = .25 kg.m1 . The exact solution is
UCSB
CS 111 - Introduction to Scientific Computing
Final Project 3 - Due on the nal exam day at 5pm
You are not allowed to work in group for the nal project.
The goal of this project is to implement the Navier-Stokes equations in two spatial dimensions.
T
UCSB
CS 111 - Introduction to Scientific Computing
Final Project 2 - Due on the nal exam day at 5pm
You are not allowed to work in group for the nal project.
Consider the Poisson equation in two spatial dimensions:
2
u = f,
with the exact solution given o
UCSB
CS 111 - Introduction to Scientific Computing
Final Project 1 - Due on the nal exam day at 5pm
You are not allowed to work in group for the nal project.
The goal of this project is to implement the nonlinear denoising algorithm of Perona and Malik.
C
CS 111: Introduction to Scientific
Computing
What is Scientific Computing?
One of the earliest, and still one of the largest, uses of computer was to solve problem in
science and engineering, and more specifically, to obtain solution of mathematical
model
UCSB
CS111: Introduction to Scientific Computing
Figure 1: Should I switch or stick?
Lets Make A Deal
Consider the following variation of the final round of the classic TV game show Lets Make A Deal.
There are three doors, and behind one of them there is