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Summary - final exam review
Intro to Computational Math (University Of Waterloo)
Distributing prohibited | Downloaded by Peter Liao ([email protected])
lOMoARcPSD
Winter 2012
CM271.CS371.AMATH341 Final Exam Review
LATEXer:
1
W. Kong
Summary
Convergence
. Let = - .
Suppose
converges
to
If > 0, > 0 such that
lim
=
converge to
=
with order q and
then,
is said to
asymptotic error constant .
(Note: terminology in deSterck is equivalent, but different)
Note that:
Higher order q wil
Quick Review: Least Squares Problem
For matrix A (mxn), and vector b (mx1), find
vector x (nx1) that solves
min 2
We determined x s.t. ATAx = ATb is a solution to
the least squares problem when ATA is positive
definite.
ATAx = ATb are called the Nor
More on QR
There are other ways to calculate QR factors
Householder transformations
Givens rotations
We discussed what Matlab calls the 'economy'
QR factorization, where Q is mxn orthogonal, R is
nxn upper triangular.
More generally, it is written as
Developing an error bound for
Simpson's rule
Expand f at m, using third Taylor's polynomial
There exists some [a,b] such that
=
+ +
+
()
( )( )
+
Integrate both sides over [a,b]
CS371/AM242 Winter 2014
21
Where
= + () + ()
() =
()()
+
()
Google PageRank Algorithm:
An application of
Numerical Linear Algebra
Module 3.5
An interesting simplification
of the algorithm (seriously simplified)
06/02/2014
Winter 2014 CS371/AM242
1
A search engine must
Access and index all web pages
Determine im
With these methods, more points
means higher order polynomial
Is this always better? higher degree
more fluctuations in p(x)
Less "cost effective" evaluations
Alternatives?
"closest" fit with lower order polynomial over full
interval
Divide full inte
Module 08:
Computing in Science and
Engineering: Top 10 Algorithms of
the Twentieth Century
Monday, March 31, 2014
(borrowing greatly from Wikipedia today)
30/03/2014
1
Overview
The editors goal was "to assemble the 10
algorithms with the greatest influe
Lagrange Example: Population example
Years since
1995
1
6
11
16
Pop
(millions)
28.85
30.01
31.61
33.48
=
( 6)( 11)( 16)
=
(5)(10)(15)
=
=
12/02/2014
()()()
CS371/AM242 Winter 2014
12
Lagrange Example: Population example
Years since
1995
1
6
1
Convergence of Fixed Point Methods
Let g be continuous over interval , ,
g(x) , for all , ,
x* is a fixed point of g , ,
. . is continuous on [ , + ]
Define = ( ! )
Then
If |g'(x*)| < 1, s.t. cfw_xk converges to x* for
|x0-x*| <
If |g'(x*)|>1
Newton-Cotes Methods for
Approximate f(x) by a polynomial of degree n
Integrate the interpolating polynomial
Evaluate f(x) at n+1 evenly-spaced points
Can we reduce the error further by consider
other choices of points and approaches?
CS371/AM242 Winter 2
Discrete Fourier Transform (DFT)
f(t) not usually known exactly
Periodic data, period T, f(tT) = f(t)
If not periodic, we can "make" periodic by
repeating the information
N evenly spaced observations at
tj = j (T/n), for j=0:N-1
Let fj = f(tj), j =
Module 05:
Numerical Integration
Starting Friday, February 28, 2014
28/02/2014
CS371/AM242 Winter 2014
1
The Definite Integral
Given a continuous function f(x) and an interval
[a,b], determine:
=
An exact solution exists:
Find F such that
=
=
28/02
From last day
= / ,
Assume N = 2m:
= / ,
/
= 1
1
=
+ +
+
1
+ +
+
+
(develop rest of algorithm and running time on board)
21/03/2014
36
Example of FFT in action for N=8 (1)
, , , , , , , = 1,2,3,4,5,6,7,8
To calculate [
CS371/AM242
Winter 2016
Assignment 2
Due: Thursday, February 11, 2016 @ 10pm
Submit to MarkUs
Updated: January 30, 2016
Please note the following:
All assignments must be completed individually.
Unless stated otherwise, write any numeric solutions with
CS371/AM242
Introduction to
Computational Mathematics
Winter 2016
Module 01
Starting Tuesday, January 5, 2016
05/01/2016
CS371/AM242 Winter 2016
1
Course Information
Basic Information on course web page
https:/www.student.cs.uwaterloo.ca/~cs371/
Most in
Module 02:
Finding roots
Starting Monday, January 13
13/01/2014
CS371/AM242 Winter 2014
1
Given a function f(x), computationally
find x* such that f(x*) = 0.
Possible computational problems:
fl(x*) may not be exact
fl(f(x*) may not be exactly 0
Additi
AMATH 242 / CS 371
Spring 2015: Assignment 1
Due Date: Wednesday May 20, beginning of class
Total Marks: 20
Q-1: Let x > 0 satisfy the binary format discussed in the class
x = .2e
x
Consider a computer using a positive binary oating-point representation w
CROWDMARK
AM 242 / CS 371 - Midterm Examination
Instructor: L. Case
Please print in pen:
Date
Waterloo Student ID Number:
Friday, February 26, 2016
Examination
Duration of Exam
1.5 hours
Midterm
Number of Test Pages
8 pages (4 pages double-sided)
Winter 2
Exercises: Floating Point, Interpolation, Numerical Integraiton
1. Consider a fictitious floating number system composed of the following numbers:
b1 .b2 b3 2y : b2 , b3 , y cfw_0, 1
S =cfw_
and b1 = 1 unless b1 = b2 = b3 = y = 0 .
i.e. each number is no
INTRODUCTION TO
COMPUTATIONAL
MATHEMATICS
Course Notes for
CM 271 / AMATH 341 / CS 371
Fall 2008
2007
Fall
Instructor: Prof.
Prof.Keith
Justin
Wan
Geddes
School of Computer Science
University of Waterloo
Course notes by Prof. Hans De Sterck
Design and typ
INTRODUCTION TO
COMPUTATIONAL
MATHEMATICS
Course Notes for
AMATH 242 / CS 371
University of Waterloo
November 28, 2013
These notes have been funded by
c
2006-2013
Hans De Sterck and Paul Ullrich
2
Contents
Preface
5
1 Errors and Error Propagation
1.1 Sour
Module 04:
Numerical Linear Algebra
Starting
Thursday, February 11, 2016
05/04/2016
Winter 2016 CS371/AM242
1
Google PageRank Algorithm:
An application of
Numerical Linear Algebra
We will look at a very simplified version of this
algorithm
Involves seve
AMATH242/CS371
Outline
Matlab Overview
Useful Commands
Matrix Construction and Flow Control
Script/Function Files
Basic Graphics
Guidelines for Assignment Questions
2 / 54
AMATH242/CS371
Getting to Matlab
Everyone who is registered in the course sho
CS371/AM242
Winter 2016
(complete) Assignment 3
Due: Thursday, March 10, 2016 @ 10pm
Submit to MarkUs
Updated: Friday, February 19, 2016
Please note the following:
Additional questions related to Numerical Linear Algebra are now included as Questions 4-6
Module 03: Fourier Analysis
Working with periodic data and
some cool algorithms
Starting Tuesday, January 26, 2016
11/02/2016
1
Fourier Analysis
The process of approximating periodic
functional data (from observations or a known
function) with a (possibl
Module 03: Linear Systems
Starting: Wednesday, January 22
02/02/2014
Module 03: Linear Systems
1
Ax = b, where
11 12
21 22
=
1 2
1
2
1
2
b=
1
2
And x =
02/02/2014
Module 03: Linear Systems
2
Solving a lower triangular system
%
A
b
n
specific example
=