CSE 3802 / ECE 3431
Numerical Methods in Scientific
Computation
Jinbo Bi
Department of Computer Science & Engineering
http://www.engr.uconn.edu/~jinbo
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The Instructor
• Ph.D in Mathematics
• Previous professional experience:
– Siemens Medical Solutions Inc.
– Department of Defense, Bioanalysis
• Research interests: biomedical informatics, machine
learning, data mining, optimization, mathematical
programming,
• Apply machine learning techniques in biological
data, medical image analysis, patient health records
analysis
• Homepage is at http://www.engr.uconn.edu/~jinbo
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Class Meetings
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Numerical Methods,
Lecture 1
Lectures are Tuesday and Thursday,
12:30 –1:45 pm
No specific lab time, but significant
computer time expected.
Computers are available in ITEB C25
and C27.
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Prof. Jinbo Bi
CSE, UConn
Class Assignments
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Homework will be assigned once every
week or two and due usually the
following week.
You may collaborate on the homework,
but your submissions should be your
own work.
Grading:
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Numerical Methods
Lecture 1
Homework
Exam 1 and 2
Final Exam
30%
40%
30%
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Prof. Jinbo Bi
CSE, UConn
Mathematical Background
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MATH 2110Q Multivariate Calculus
• Taylor series
• MATH 2410Q Introduction to Differential
Equations
• Integration
• MATH 2210 Linear Algebra
• Equation systems
Numerical Methods,
Lecture 1
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Prof. Jinbo Bi
CSE, UConn
Computer Background
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Languages to be used:
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Numerical Methods,
Lecture 1
Matlab, C, C++
CSE 1100/1010 programming
experience
Any OS is acceptable
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Prof. Jinbo Bi
CSE, UConn
Syllabus
• Go over the course syllabus
• Course website
http://www.engr.uconn.edu/~jinbo/Fall2013_
Numerical_Methods.htm
Numerical Methods,
Lecture 1
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Prof. Jinbo Bi
CSE, UConn
Today’s Class:
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Numerical Methods,
Lecture 1
Introduction to numerical methods
Basic content of course and class
expectations
Mathematical modeling
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Prof. Jinbo Bi
CSE, UConn
Introduction
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What are numerical methods?
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What type of mathematical problems?
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Numerical Methods,
Lecture 1
“… techniques by which mathematical
problems are formulated so that they can be
solved with arithmetic operations.” (Chopra
and Canale)
Roots, Integration, Optimization, Curve
Fitting, Differential Equations, and Linear
Systems
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Prof. Jinbo Bi
CSE, UConn
Introduction
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How do you solve these difficult
mathematical problems?
Example: What are the roots of x27x+12?
Three general non-computer methods
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Numerical Methods,
Lecture 1
Analytical
Graphical
Manual
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Prof. Jinbo Bi
CSE, UConn
Analytical Solutions
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This is what you learned in math class
Gives exact solutions
Example: x 2  7 x  12  ( x  3)( x  4 )
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Numerical Methods,
Lecture 1
Roots at 3 and 4
Not always possible for all problems and
usually restricted to simple problems
with few variables or axes
The real world is more complex than the
simple problems in math class
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Prof. Jinbo Bi
CSE, UConn
Graphical Solution
Numerical Methods,
Lecture 1
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Prof. Jinbo Bi
CSE, UConn
Manual Solution
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Numerical Methods,
Lecture 1
Using pen and paper, slide rulers, etc. to
solve an engineering problem
Very time consuming
Error-prone
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Prof. Jinbo Bi
CSE, UConn
Numerical Methods
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What are numerical methods?
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Numerical Methods,
Lecture 1
“… techniques by which mathematical
problems are formulated so that they can be
solved with arithmetic operations.” (Chopra
and Canale)
Arithmetic operations map into computer
arithmetic instructions
Numerical methods allow us to formulate
mathematical problems so they can be
solved numerically (e.g., by computer)
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Prof. Jinbo Bi
CSE, UConn
Course Overview
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What is this course about?
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Numerical Methods,
Lecture 1
Using numerical methods to solve
mathematical problems that arise in
engineering
Most of the focus will be on engineering
problems
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Prof. Jinbo Bi
CSE, UConn
Basic Materials
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Introduction
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Mathematical Problems
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Numerical Methods,
Lecture 1
Programming
Mathematical Modeling
Error Analysis
Roots, Linear Systems, Integration,
Optimization, Curve Fitting, Differential
Equations
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Prof. Jinbo Bi
CSE, UConn
Mathematical Modeling
A mathematical
model is the
formulation of a
physical or
engineering system
in mathematical
terms.
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Numerical Methods,
Lecture 1
Empirical
Theoretical
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Prof. Jinbo Bi
CSE, UConn
Mathematical Modeling
• A mathematical model is represented as a functional
relationship of the form
Dependent
Variable
=f
independent
forcing
variables, parameters, functions
• Dependent variable: Characteristic that usually reflects the
state of the system
• Independent variables: Dimensions such as time and space
along which the systems behavior is being determined
• Parameters: reflect the system’s properties or composition
• Forcing functions: external influences acting upon the system
Numerical Methods,
Lecture 1
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Prof. Jinbo Bi
CSE, UConn
Mathematical Modeling
A simple example:
• In an electrical circuit, I = V/R; The
current, I, is dependent on resistance
parameter, R, and forcing voltage
function, V.
Numerical Methods,
Lecture 1
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Prof. Jinbo Bi
CSE, UConn
Example 1
• Newton’s 2nd low of Motion states that “the
time rate change of momentum of a body is
equal to the resulting force acting on it.”
• The model is formulated as
F=ma
F=net force acting on the body (N)
m=mass of the object (kg)
a=its acceleration (m/s2)
Numerical Methods,
Lecture 1
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Prof. Jinbo Bi
CSE, UConn
Example 1
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What is the velocity of a falling object?
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First step is to model the system
Newton’s second law
F  ma  a 
F
m
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
dv
dt

F
m
Total force is gravity and air resistance
F  FGravity  F Air  mg  cv
Numerical Methods,
Lecture 1
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Prof. Jinbo Bi
CSE, UConn
Example 1
dv
dt
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
F
m

mg  cv
m
 g
c
v
m
First order differential equation
Analytical solution
v (t ) 
gm
c
t
m
(1  e )
c
Numerical Methods,
Lecture 1
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Prof. Jinbo Bi
CSE, UConn
Example 1
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Numerical Methods,
Lecture 1
m=68.1kg, c=12.5 kg/s
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Prof. Jinbo Bi
CSE, UConn
Example 1
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Numerical Methods,
Lecture 1
What if we can’t find an analytical
solution?
How do you get a computer to solve the
differential equation?
Use numerical methods
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Prof. Jinbo Bi
CSE, UConn
Euler’s Method
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Use the finite divided difference
approximation of the derivative
dv
dt
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Numerical Methods,
Lecture 1

v ( t i 1 )  v ( t i )
t i 1  t i
The approximation becomes exact as Δt
→0
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Prof. Jinbo Bi
CSE, UConn
Euler’s Method
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Using Euler’s method, we can
approximate the velocity curve
dv
dt

v ( t i 1 )  v ( t i )
t i 1  t i
m
v ( t i 1 )  v ( t i )  ( t i 1
Numerical Methods,
Lecture 1
 g
c
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v (ti )
c


 ti ) g 
v (ti ) 
m


Prof. Jinbo Bi
CSE, UConn
Euler’s Method
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Assume Δt=2
v (0)  0
c


v ( 2 )  v ( 0 )  2  g  v ( 0 )   19 . 6
m


c


v ( 4)  v (2 )  2 g 
v ( 2 )   32 . 0
m


……
Numerical Methods,
Lecture 1
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Prof. Jinbo Bi
CSE, UConn
Euler’s method
Numerical Methods,
Lecture 1
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Prof. Jinbo Bi
CSE, UConn
Euler’s Method
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Numerical Methods,
Lecture 1
Avoids solving differential equation
Not an exact formula of the function
Gets more exact as Δt→0
How do we choose Δt? Dependent on
the tolerance of error.
How do we estimate the error?
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Prof. Jinbo Bi
CSE, UConn
Overview of the problems
Numerical Methods,
Lecture 1
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Prof. Jinbo Bi
CSE, UConn
Overview of the problems
Numerical Methods,
Lecture 1
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Prof. Jinbo Bi
CSE, UConn
Next class
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Numerical Methods,
Lecture 1
Programming and Software
Read Chapters 1 & 2
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Prof. Jinbo Bi
CSE, UConn
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