Integration Rules:
b
I. To approximate the integral
f ( x)dx
using a
a
A. Single Application of the Trapezoid Rule
IT =
(b a)
[ f ( a ) + f (b ) ]
2
B. Single Application of Simpsons (1/3) Rule
IS =
(b a )
a +b
f ( a) + f 2 + f (b)
6
b
II. To approxima

Curve Fitting
Linearization of Non-Linear Models
Linearization of Non-Linear Models
It is possible to linearize some non-linear
equations so that you can use linear regression
to fit a model to the data.
For example, (page 453) we can linearize
exponentia

Curve Fitting 3
Linear Interpolation and Newtons
Divided Difference Interpolation
Polynomials
Linear Interpolation
Sometimes we need precise information
between data points. The most common way to
do this is to fit a polynomial:
f ( x) ao a1 x a2 x 2 . an

Curve Fitting
Regression
Methods of Curve Fitting
Data comes at discreet values. Often, wed like to know information about the
sample between data points either to build a model or simply to get required
values between two points. Two concepts well explor

Curve Fitting
Regression
Methods of Curve Fitting
Data comes at discreet values. Often, wed like to know information about the
sample between data points either to build a model or simply to get required
values between two points. Two concepts well explor

Curve Fitting 4
Lagrange Interpolating Polynomials
Lagrange Interpolating Polynomials
n
f n x Li x f xi
i 0
If n=1 First order polynomial
f1 x Lo x f xo L1 x f x1
n
Li x
j 0
j i
order
x xj
xi x j
product of
First Order Lagrange
For a first order:
i
x x

Taylor Series and Error
Approximation and round-off errors.
Understanding Error
Understanding error is fundamental to numerical methods.
In Numerical Methods, we dont always know the solution and cannot know the true
error. Therefore, we have methods to

Root Finding
Numerical methods for finding roots of equations
Root Finding
A root is where a function or equation crosses zero.
Roots
(can be either real or complex)
R1
R2
Root Finding
We can find roots:
Directly
Graphically
Using Numerical Methods
Brack

Taylor Series and Errors
Taylors Theorem and its Applications
Announcements
Homework Due Wednesday
No lab on Monday finish homework/study
for test
Test Wednesday on Linear Algebra
Deep thoughts:
Its a privilege to be here. Gratitude goes a long way.

Root Finding
Fixed-Point Iteration, Newton-Raphson
Method, and the Secant Method (and
Modified Secant Method)
Root Finding Methods
Closed Methods (Ch. 5)
Upper and lower interval limits
are selected such that the root
lies within the interval.
Bisection

Taylor Series
Numerical Differentiation
Taylor Series Numerical Differentiation
For:
f '( xi ) 2
f ( xi 1 ) f ( xi ) f '( xi )h
h .
2!
f ( xi 1 ) f ( xi ) f '( xi ) 2
h .
h
2!h
f ( xi 1 ) f ( xi ) f '( xi )
f '( xi )
h HOT
h
2!
f '( xi )
We can rewrite

Linear Algebra 1
Grammar of linear algebra
A Matrix
A Matrix is a 2D array with m rows and n
columns
For example, matrix M is a 3 row by 3 column
matrix or a 3x3 matrix
A11
M A21
A31
A12
A22
A32
A13
A23
A33
A Vector
A vector is a 1D array. In enginee

Linear Algebra 4
Matrix Rank and examples of
systems of linear equations
To date what I expect you to know
Matrix
Vector
Square Matrix
Upper/Lower Triangular
Matrix
How to represent a
linear system as Ax = b
What x is in geometrical
terms
Matrix propertie

Linear Algebra 2
Vectors, cross product, dot product
Vectors
A vector is an nx1 or 1xn array. Vectors commonly occur in
engineering and show up as physical quantities such as force
and position.
Vectors have a magnitude and a direction.
For example, we

EMCH 201: Introduction to Numerical Methods
Linear Algebra
A. Motivation:
1. Why are we doing this?
i. Numerous times in engineering, one is required to solve a set of simultaneous
linear equations.
ii. We need an abbreviated way to represent and solve th

Cardiovascular System with Pumps and Pipes
Set up the matrix equation for the system of equations presented in class modeling the
Cardiovascular System, using the parameters given in the table below. Solve for the
unknowns using MathCad to invert the matr

EMCH 201/PHYS 311: Introduction to Applications of Numerical Methods
Instructor: Riaz Ahmed, Instructor, Department of Mechanical Engineering, USC
Mid-Term Exam - 03
Time: 45 Minutes
Full Marks: 100
Exam date: 16th November 2016, at 10:55 AM
(Note: You ca

EMCH 201/PHYS 311: Introduction to Applications of Numerical Methods
Instructor: Riaz Ahmed, Instructor, Department of Mechanical Engineering, USC
Mid-Term Exam - 01
Time: 40 Minutes
Full Marks: 100
Exam date: 26th September 2016, at 10:50 AM
(Note: You c