Chapter 1, Question 11: Describe two important data communications standards-making bodies. How
do they differ?
Answer:
The International Organization for Standardization (ISO) makes technical recommendations about data
communication interfaces. The Ameri
Summary Analysis of SecurityNow! podcasts Symmetric Block Ciphers and HMACs
SecurityNow: Symmetric Block Ciphers
Steve Gibson and Leo Laporte start the podcast with commentary on a previous episodes scenario
challenge: would you be able to double-encrypt
Numerical Linear Algebra
The two principal problems in linear algebra are:
Linear system
Given an n n matrix A and an n-vector b, determine x I n such that
R
Ax = b
Eigenvalue problem
Given an n n matrix A, nd a scalar (an
eigenvalue) and a nonzero vector
Approximation Theory
Function approximation is the task of constructing, for a given function, a simpler
function so that the dierence between the two functions is small and to then
provide a quantiable estimate for the size of the dierence.
Why would one
Clustering
What do we mean by clustering?
Clustering strives to identify groups/clusters of objects that behave similarly
or show similar characteristics. In common terms it is also called look-a-like
groups.
Similarity is often quantied through the use
The QR Decomposition
We have seen one major decomposition of a matrix which is A = LU (and its
variants) or more generally PA = LU for a permutation matrix P. This was
valid for a square matrix and aided us in solving the linear system Ax = b.
The QR de
Iterative Methods for Solving Ax = b
A good (free) online source for iterative methods for solving Ax = b is given in
the description of a set of iterative solvers called templates found at netlib:
http : /www.netlib.org/linalg/html templates/Templates.ht
Numerical Quadrature
When you took calculus, you quickly discovered that integration is much
more dicult than dierentiation. In fact, the majority of integrals can not
be integrated analytically. For example, integrals such as
b
b
sin x dx
a
2
ex dx
2
a
Interpolation & Polynomial Approximation
The next problem we want to investigate is the problem of nding a simple
function such as a polynomial, trigonometric function or rational function
which approximates either a discrete set of (n + 1) data points
(
Linear Least Squares
Suppose we are given a set of data points cfw_(xi, fi), i = 1, . . . , n. These
could be measurements from an experiment or obtained simply by evaluating
a function at some points. One approach to approximating this data is to
interp
Introduction to Boundary Value Problems
When we studied IVPs we saw that we were given the initial value
of a function and a dierential equation which governed its behavior
for subsequent times. Now we consider a dierent type of problem
which we call a bo
Introduction to Dierential Equations
Dierential equations arise in many disciplines such as engineering,
mathematics, sciences (e.g., physical, chemical and biological) as well
as economics. These dierential equations often result from modeling
some pheno
Lectures - Week 14
Vector Form of Taylors Series, Integration in Higher Dimensions,
and Greens Theorems
Vector form of Taylor Series
We have seen how to write Taylor series for a function of two independent variables, i.e.,
to expand f (x, y) in the neigh
Lectures - Week 7
Eigenvalues
The Algebraic Eigenvalue Problem
There are two major problems in linear algebra solving a linear system Ax = b and the
eigenvalue problem which is stated below.
Given an n n matrix A, nd a scalar and a nonzero vector x such t
Lectures - Week 4
Matrix norms, Conditioning, Vector Spaces, Linear Independence,
Spanning sets and Basis, Null space and Range of a Matrix
Matrix Norms
Now we turn to associating a number to each matrix. We could choose our norms analogous to the way we
Lectures - Week 11
General First Order ODEs & Numerical Methods for IVPs
In general, nonlinear problems are much more dicult to solve than linear ones. Unfortunately many phenomena exhibit nonlinear behavior. We want to look at the general form
of a rst o
Lectures - Week 5
Four Basic Spaces
1. The column space (or equivalently the range) of A, where A is m n matrix is all
linear combinations of the columns of A. We denote this by R(A).
By denition (because it contains all linear combinations and is thus c
Introductory Lecture
Many phenomena (physical, chemical, biological, etc.) are model by dierential equations.
Recall the denition of the derivative of f (x) and its physical and graphical interpretation.
Example Suppose we are told that the population p
Lectures - Week 10
Introduction to Ordinary Dierential Equations (ODES)
First Order Linear ODEs
When studying ODEs we are considering functions of one independent variable, e.g., f (x),
where x is the independent variable and f is the dependent variable.
Lectures - Week 12
Single step and Multistep Methods for First Order Initial Value Problems
Runge-Kutta Methods
To obtain a more accurate scheme than Forward Euler, we must do additional work such
as additional function evaluations. Single step methods ta
Lectures - Week 15
Line Integrals, Greens Theorems
and a Brief Look at Partial Dierential Equations
Line Integrals
Another type of integral that one encounters in higher dimensions is a line integral where
we want to integrate a quantity along a given cur
Lectures - Week 13
Two Point Boundary Value Problems and
Functions of Several Variables
We now want to briey look at a linear second order BVP which is sometimes called a
two-point BVP because we are specifying conditions at the two endpoints of our domai
Lecture 3 - Vectors and Matrices
Last time we saw that if we have n equations in n unknowns then there are n2
coecients (some may be zero) and n right hand side components. To eciently study
linear systems we need to write all linear systems in a generic
ISC 4933-?/5935-? - Mathematical Tools for Scientic Computing
Fall 2014
Instructor:
Website:
Oce Hours:
TA:
Professor Janet Peterson
email: [email protected]
oce: 444 DSL
phone: 850-644-1979
http:/www.sc.fsu.edu/jpeterson
T 11-12, R 3:30-4:00, other times
Lecture 5 - Triangular Factorizations & Operation Counts
LU Factorization
We have seen that the process of GE essentially factors a matrix A into LU . Now we
want to see how this factorization allows us to solve linear systems and why in many cases
it is
Lectures - Week 6
Linear Least Squares, Orthonormal vectors and the QR Decomposition
Linear Least Squares Method
As an example of an application where these interconnections between the spaces is useful,
we consider the linear least squares problem. Suppo
Lectures - Week 8
Eigenvalues & Numerical Methods for Finding a Single Eigenvalue
and the Singular Value Decomposition Theorem
Review
First we will summarize some of the facts we saw last week for the algebraic eigenvalue
problem of nding a nonzero vector
Lectures - Week 9
The Singular Value Decomposition Theorem
We know that eigenvalues are only dened for a square matrix. However, in this section
we want to dene an analogue of eigenvalues for a rectangular matrix. This will lead us
to our nal decompositio
ACS I - Problem Set # 1
Problems # 2 ,4,5,8 are to be turned in
Due: Thursday, 1/22
1. a. First determine if each of the sets is a vector space when the standard operations of addition and
scalar multiplication are used. If the space is nite dimensional d
Lab # 4 - ACS I
Goals. For homework you have written codes for evaluating a Lagrange interpolating polynomial which
you can use for this lab. In the rst exercise you are asked to provide code for a piecewise linear interpolant
and in the next two exercise