Date assigned: May 20, 2016
Date due: May 31, 2016
Numerical Methods
Homework 5
1) The differential equation dp/dt = kmax (1-p/pmax) p can be used to model a population. pmax
is a constant that equals the maximum population and kmax is a constant that equ

6.3 The Secant Method
Approximates the derivative in the NR Method by a
backward finite divided difference
NR:
Secant:
False-position
Secant
Modified Secant Method
Example 6.8
6.4 Brents Method
Hybrid- clever algorithm
Speedy open method wherever possible

MAT202E Lecture VI
Part III- Linear
Algebraic Equations
Dr. Elin TAN
Matrix Notation
Matrix Operating Rules
Representing Linear
Algebraic Equations in
Matrix Form
Chapter 9
Gauss Elimination
Solving Small Numbers
of Equations
Graphical Method
Cramers Ru

C H A P T E R 11
Gauss-Seidel
The Gauss-Seidel method is the most commonly used iterative method. Assume that we are
given a set of n equations:
\A]\X] = \B)
Suppose that for conciseness we limit ourselves to a 3 x 3 set of equations. I f the diagonal
ele

CHAPTER 3
Approximations and Round-Off Errors
Although the numerical technique yielded estimates that were close to the exact
analytical solution, there was a discrepancy, or error, because the numerical method
involved an approximation.
Actually, we were

CHAPTER 5
Bracketing Methods
This chapter on roots o f equations deals with methods that exploit the fact that a function
typically changes sign in the vicinity of a root.
These techniques are called bracketing methods because two initial guesses for the

CHAPTER 1
Mathematical Modeling and Engineering Problem Solving
1.1. A SIMPLE MATHEMATICAL MODEL
A mathematical model can be broadly defined as
* a formulation or
* equation
that expresses the essential features of a physical system or process in mathemat

C H A P T E R
10
L U Decomposition and Matrix Inversion
This chapter deals with a class of elimination methods called LU decomposition techniques.
The primary appeal of LU decomposition is that the time-consuming elimination step can be
formulated so that

C H A P T E R
17
Least-Squares Regression
17.1 L I N E A R R E G R E S S I O N
The simplest example of a least-squares approximation is fitting a straight line to a set of paired
observations: ( x yi), (x , y ), . . , (x , y ).
b
2
2
n
n
The mathematical

CHAPTER 7
Roots o f Polynomials
In this chapter, we will discuss methods to find the roots of polynomial equations of the general
form
m Jf "fc
=
.
*
ac_ - | - a | j r + &2
X
I
g
4- - * +
^
a^r
where n = the order of the polynomial and the a's = constant

CHAPTER 6
Open Methods
For the bracketing methods in Chap. 5, the root is located within an interval prescribed by a
lower and an upper bound.
Repeated application of these methods always results in closer estimates of the true value of the
root.
Such met

CHAPTER 4
Truncation Errors and the Taylor Series
Truncation errors are those that result from using an approximation in place of an exact
mathematical procedure.
For example, in Chap. 1 we approximated the derivative of velocity of a falling parachutist

C H A P T E R
9
Gauss E l i m i n a t i o n
This chapter deals with simultaneous linear algebraic equations that can be represented generally
as
mm + max* H + atumm
mtixi+a&m +
=
where the a's are constant coefficients and the 6's are constants.
The techn

Chapter 18
~Curve Fitting ~
Interpolation
These notes are only to be used in class presentations
1
Spline Interpolation
There are cases where polynomials can lead to erroneous results
because of round off error and overshoot.
The function (black line) has

clear
clc
% n = 10;
% A = rand(n);
% b = rand(n,1);
A = [3 -0.1 -0.2;0.1 7 -0.3;0.3 -0.2 10];
% A = [2 4 1;5 2 1;1 2 1];
% b = [-5;12;3];
n = size(A,1);
B = eye(n);
X = zeros(n);
U = A;
L = eye(n);
for i = 1:n;
L(i+1:n,i) = (U(i+1:n,i)/U(i,i);
U(i+1:n,i

MAT202E Numerical Methods
Lecture III
Dr. Elin TAN
Chapter 3
Approximations and
Round-Off Errors
Errors
How much error is present in our
calculations?
Is it tolerable?
IDENTIFICATION
QUANTIFICATION
MINIMIZATION
1. Round-Off Errors
Computers can represent

MAT202E Numerical Methods
Lecture II
Dr. Elin TAN
Chapter 1Mathematical Modeling
and
Engineering Problem
Solving
A simple mathematical
model
Engineering
Problem
Solving
Process
<- Terminal Velocity
Numerical
Solution
Analytical
Solution
Conservation Laws

MAT202E Numerical Methods Lecture V
Dr. Elin TAN
Chapter 6
Open Methods
Bracketing
Methods
The root is located with in a interval
prescribed by a lower and an upper
bound
Re peate d applicatio n of the se
methods always result in closer
estimates of the t

Numerical ODE SOLUTIONS
Initial-Value Problems
Programming Euler Method
f=inline('1-2*v^2-t','t','v')
h=0.01
t=0
v=1
T(1)=t;
V(1)=v;
for i=1:100
v=v+h*f(t,v)
t=t+h;
T(i+1)=t;
V(i+1)=v;
end
dv
1 2v 2 t.
dt
v(0) 1
for ti 0.01i,
i 1,2,.,100
Definition of th

Date assigned: March 22, 2016
Date due: April 4, 2016
Numerical Methods
Homework 3
1) Given the four points (1,2), (3,4), (5,3), (9,8), write the cubic in Lagrangian form that
passes through them. Multiply out each term to express in the standard form, as

Date assigned: April 14, 2016
Date due: April 21, 2016
MAT202E NUMERICAL METHODS
HOMEWORK 4
1) The following table has values for f(x). a) Integrate between x=1.0 and x=1.7 by using the
trapezoidal rule (h=0.1). b) Repeat the integration this time using a

Math 523: Numerical Analysis I
Solution of Homework 3. Numerical Quadrature
Problem 1. Consider the integral
with n panels.
R1
0
ex dx. We divide the interval [0, 1] into a uniform partition
(a) Apply the composite Trapezoid rule for n = 1, 2, 4, 8, 16, 3

Jim Lambers
MAT 460/560
Fall Semeseter 2009-10
Lecture 29 Notes
These notes correspond to Section 4.5 in the text.
Romberg Integration
Richardson extrapolation is not only used to compute more accurate approximations of derivatives,
but is also used as th

S EC . 6.1
A PPROXIMATING THE D ERIVATIVE
323
Central-Difference Formulas
If the function f (x) can be evaluated at values that lie to the left and right of x, then
the best two-point formula will involve abscissas that are chosen symmetrically on both
si