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5
False position method
f(x)
f(xu
)
xr
f(xu
)
xl
f(x l)
xl
xt
xu x
f(x l)
xr
xu
Figure 8: False position method to nd the roots of an equation
Idea: if f (xl ) is closer to zero than f (xn), then the root is more likely to be closer to xl than to
xu. (NOT

Example:
Find the roots of function f (x) = x3 2x2 + 0.25x + 0.75.
Solution:
To nd the exact roots of f (x), we rst factorize f (x) as
f (x) = x3 2x2 + 0.25x + 0.75
= (x 1)(x2 x 0.75)
= (x 1) (x 1.5) (x + 0.5)
Thus, x = 1, x = 1.5 and x = 0.5 are the exac

CoE 3SK3 Exercises: Computer Arithmetic
1. Convert the following base-2 numbers to base-10: (a) 101101, (b) 101.101, and (c) 0.01101.
2. The innite series
n
f (n) =
i=1
1
i4
4 /90
converges on a value of f (n) =
as n approaches innity. Write a program in

2
Truncation errors in oating point representation
When a real value x is stored using its oating point representation f l(x), truncation error occurs. This is because of the need to represent an innite number
of real values using a nite number of bits.
E

3
3.1
Approximating a function using a polynomial
McLaurin series
Assume that f (x) is a continuous function of x, then
i
f (x) =
ai x =
i=0
i=0
f (i)(0) i
x
i!
is known as the McLaurin Series, where ais are the coefcients in the polynomial
expansion give

CoE 3SK3 Assignment 3
1. For the following set of equations
2x2 + 5x3 = 9
2x1 + x2 + x3 = 9
3x1 + x2 = 10
(a) Compute the determinant of the coefcient matrix.
(b) Use Cramers rule to solve for the xs.
(c) Manually solve the equation set using Gauss elimin

Chapter 3: Linear Algebraic Equations
General form of a system of linear algebraic equations
a11x1 + a12x2 + + a1nxn = b1
a21x1 + a22x2 + + a2nxn = b2
an1x1 + an2x2 + + annxn = bn
which can be rewritten as
a11 a12 a1n
x1
b1
a21 a22 a2n x2 b2
=
an1 a

5 Cholesky Decomposition
Cholesky decomposition is another (efcient) way to implement LU decomposition for symmetric matrices.
Consider AX = b, A = [aij ]nn, and aij = aji (A = A).
Chokesky decomposition: A = LL , where
l11 0 . . . 0
l21 l22
L=
. 0
.
ln

7
Error Analysis for Solving a Set of Linear Equations
Consider AX = b,
When |A| = 0, A is non-singular, there is a unique solution.
When |A| = 0, A is singular, there is no solution or an innite number of
solutions.
When |A| 0, the solution is sensiti

Chapter 2: Roots of Equations
Problem: given f (x) = 0, nd x.
In general, f (x) can be any function. For some forms of f (x), analytical solutions are available.
However, for other functions, we have to design some methods, or algorithms to nd either exac

Chapter 1: Computer Arithmetic
Outline:
Positional notation binary representation of numbers
Computer representation of integers
Floating point representation
IEEE standard for oating point representation
Truncation errors in oating point representati

Answerforassignment2(3SK3)
(a)
x +
1 0
2
E (k , c) =
[cos x (kx + c)] dx
2 x0
x0 +
E
1
=
(cos x kx c) xdx = 0
k
x
0
x0 +
E
1
(cos x kx c)dx = 0
=
c
x
0
x0 +
(cos x) xdx k
x0
x0 +
x0
x0 +
x dx c
x0
(cos x)dx k
x0 +
x0 +
2
xdx c
x0
xdx = 0
x0
x0 +

Name_
Student Number_
CoE 3SK3 Computer Aided Engineering
Midterm Test
February 26, 2014
McMaster University
Department of Electrical and Computer Engineering
Instructor:
Xiaolin Wu
([email protected])
THIS EXAMINATION PAPER INCLUDES 2 PAGES AND 4

CoE 3SK3: Course Project (Part 1)
Template Matching Algorithms
Weight: 12%
Individual work required
Due: February 19, 2017
Template matching, a fundamental problem in computer vision and image
processing, is the task of locating the best matched patch in

CoE 3SK3 Computer Aided Engineering
Midterm Test
February 7, 2010
McMaster University
Department of Electrical and Computer Engineering
Instructor:
Xiaolin Wu
([email protected])
1. This test is nominally 120 minutes long.
2. A cheat-sheet (letter-

Roots of Equations (Chapters 5 and 6)
Problem: given f (x) = 0, find x.
In general, f (x) can be any function. For some forms of f (x), analytical solutions are available.
However, for other functions, we have to design some methods, or algorithms to find

Bicubic Interpolation
Electrical and Computer Engineering
McMaster University, Canada
February 1, 2014
(ECE @ McMaster)
Bicubic Interpolation
February 1, 2014
1 / 26
Interpolation
Definition
Interpolation is a method of constructing new data points within

COE 3SK3 Final Project
Fast Screening Algorithm For
Template Match
TA: Bolin Liu
Contact: [email protected]
Office: ITB A103
Office hour: Monday 2:30pm-5:30pm
About image:
A m*n grayscale image can be considered as a matrix with m rows
and n columns.
E

Tutorial for project part 2
TA: Bolin Liu
Contact: [email protected]
Office: ITB A103
Office hour: Monday 2:30pm-5:30pm
Step 1: establish 4 summed area tables
Where I(i, j) is the pixel value in original image.
Step 2: compute 4 patch features
You MUST