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CSE 3802 / ECE 3431
Numerical Methods in Scientific Computation
Jinbo BiDepartment of Computer Science & Engineering
http://www.engr.uconn.edu/~jinbo
1
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
2
Numerical Methods,Lecture 1 3
• 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.
Class Meetings
Prof. Jinbo Bi CSE, UConn
4
• 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:• Homework 30%• Exam 1 and 2 40%• Final Exam 30%
Class Assignments
Numerical Methods Lecture 1
Prof. Jinbo Bi CSE, UConn
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• MATH 2110Q Multivariate Calculus• Taylor series
• MATH 2410Q Introduction to Differential Equations • Integration
• MATH 2210 Linear Algebra• Equation systems
Mathematical Background
Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
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• Languages to be used:• Matlab, C, C++
• CSE 1100/1010 programming experience
• Any OS is acceptable
Computer Background
Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
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• Go over the course syllabus• Course website
http://www.engr.uconn.edu/~jinbo/Fall2013_Numerical_Methods.htm
Syllabus
Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
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Today’s Class:
• Introduction to numerical methods• Basic content of course and class
expectations• Mathematical modeling
Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
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Introduction
• What are numerical methods?• “… techniques by which mathematical
problems are formulated so that they can be solved with arithmetic operations.” (Chopra and Canale)
• What type of mathematical problems?• Roots, Integration, Optimization, Curve
Fitting, Differential Equations, and Linear Systems
Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
Introduction
• How do you solve these difficult mathematical problems?
• Example: What are the roots of x2-7x+12?
• Three general non-computer methods• Analytical• Graphical• Manual
10Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
• This is what you learned in math class• Gives exact solutions• Example:
• 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
Analytical Solutions
)4)(3(1272 xxxx
11Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
Graphical Solution
12Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
• Using pen and paper, slide rulers, etc. to solve an engineering problem
• Very time consuming• Error-prone
Manual Solution
13Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
Numerical Methods
• What are numerical methods?• “… 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)
14Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
• What is this course about?• Using numerical methods to solve
mathematical problems that arise in engineering
• Most of the focus will be on engineering problems
Course Overview
15Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
• Introduction• Programming• Mathematical Modeling• Error Analysis
• Mathematical Problems• Roots, Linear Systems, Integration,
Optimization, Curve Fitting, Differential Equations
Basic Materials
16Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
A mathematical model is the formulation of a physical or engineering system in mathematical terms.
• Empirical• Theoretical
Mathematical Modeling
17Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
Mathematical Modeling
18Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
• A mathematical model is represented as a functional relationship of the form
Dependent independent forcingVariable = f 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
A simple example:• In an electrical circuit, I = V/R; The
current, I, is dependent on resistance parameter, R, and forcing voltage function, V.
Mathematical Modeling
19Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
Example 1
20Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
• 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 = m a
F=net force acting on the body (N)
m=mass of the object (kg)
a=its acceleration (m/s2)
• What is the velocity of a falling object?• First step is to model the system• Newton’s second law
• Total force is gravity and air resistance
Example 1
m
F
dt
dv
m
FamaF
cvmgFFF AirGravity
21Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
• First order differential equation• Analytical solution
Example 1
vm
cg
m
cvmg
m
F
dt
dv
)1()( tmc
ec
gmtv
22Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
• m=68.1kg, c=12.5 kg/s
Example 1
23Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
• What if we can’t find an analytical solution?
• How do you get a computer to solve the differential equation?
• Use numerical methods
Example 1
24Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
• Use the finite divided difference approximation of the derivative
• The approximation becomes exact as Δt → 0
Euler’s Method
ii
ii
tt
tvtv
dt
dv
1
1 )()(
25Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
• Using Euler’s method, we can approximate the velocity curve
Euler’s Method
)()()(
1
1i
ii
ii tvm
cg
tt
tvtv
dt
dv
)()()()( 11 iiiii tv
m
cgtttvtv
26Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
• Assume Δt=2
Euler’s Method
0)0( v
6.19)0(2)0()2(
v
m
cgvv
0.32)2(2)2()4(
v
m
cgvv
……
27Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
Euler’s method
28Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
• 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?
Euler’s Method
29Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
Overview of the problems
30Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
Overview of the problems
31Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn
Next class
• Programming and Software• Read Chapters 1 & 2
32Numerical Methods, Lecture 1
Prof. Jinbo Bi CSE, UConn