1. CSE 3802 / ECE 3431 Numerical Methods in Scientific Computation
Jinbo Bi
Department of Computer Science & Engineering

2. 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

3. Class Meetings 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.
Numerical Methods,
Lecture 1
Prof. Jinbo Bi CSE, UConn

4. Class Assignments
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%
Numerical Methods
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Prof. Jinbo Bi CSE, UConn

5. Mathematical Background
MATH 2110Q Multivariate Calculus
Taylor series
MATH 2410Q Introduction to Differential Equations
Integration
MATH 2210 Linear Algebra
Equation systems
Numerical Methods,
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Prof. Jinbo Bi CSE, UConn

6. Computer Background Languages to be used:
Matlab, C, C++
CSE 1100/1010 programming experience
Any OS is acceptable
Numerical Methods,
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Prof. Jinbo Bi CSE, UConn

7. Syllabus Go over the course syllabus Course website
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Prof. Jinbo Bi CSE, UConn

8. Today’s Class: Introduction to numerical methods
Basic content of course and class expectations
Mathematical modeling
Numerical Methods,
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Prof. Jinbo Bi CSE, UConn

9. 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,
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Prof. Jinbo Bi CSE, UConn

10. 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
Let us use some very simple examples to demonstrate the idea the concept. So what we mean numerical methods, numerical solutions.
X2-7x+12, we are looking for the roots of this function.
You could do three things, these three methods do not require computers.
1 so called analytical solutions
2, graphical way to find solutions
3, you manually calculate the solutions.
Numerical Methods,
Lecture 1
Prof. Jinbo Bi CSE, UConn

11. Analytical Solutions 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
Analycial solutions are the stuff you learn from your math class, algebra or calculus, differential equations.
For the given function, you could factorize the function into the product of these two terms. Based on the examination of the two terms, you will know if you set x=3 or 4, this function will reach value of 0. So you know the roots are 3 and 4.
Numerical Methods,
Lecture 1
Prof. Jinbo Bi CSE, UConn

12. Graphical Solution Numerical Methods, Prof. Jinbo Bi CSE, UConn
The graphical solution give you a lot intuitive idea about the problem you are trying to solve. You specify some x values, then you calculate the function values of these x, then you draw them as points in this figure, and connect these points
Numerical Methods,
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Prof. Jinbo Bi CSE, UConn

13. Manual Solution
Using pen and paper, slide rulers, etc. to solve an engineering problem
Very time consuming
Error-prone
Numerical Methods,
Lecture 1
Prof. Jinbo Bi CSE, UConn

14. 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)
Numerical Methods,
Lecture 1
Prof. Jinbo Bi CSE, UConn

15. Course Overview 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
Numerical Methods,
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Prof. Jinbo Bi CSE, UConn

16. Basic Materials Introduction Mathematical Problems Programming
Mathematical Modeling
Error Analysis
Mathematical Problems
Roots, Linear Systems, Integration, Optimization, Curve Fitting, Differential Equations
Numerical Methods,
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Prof. Jinbo Bi CSE, UConn

17. Mathematical Modeling
A mathematical model is the formulation of a physical or engineering system in mathematical terms.
Empirical
Theoretical
Mahtemtaicla modeling involves a deep understanding of the engineering problem that you need to solve. You want to fix a car, no matter what fantastic tools you may have, if you don’t understand how the car works, you can not do it.
Numerical Methods,
Lecture 1
Prof. Jinbo Bi CSE, UConn

18. Mathematical Modeling
A mathematical model is represented as a functional relationship of the form
Dependent independent forcing
Variable = 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
In general, a mathematcial model is represented as a funcational relationship of the following form
The dependent variable usually reflects the state or a chracteristics of the system;
Independenct variables are usually dimensions such as time, space, available resources, the kind of variables on which the system’s state can be determined.
Parameters are reflective of the system’s properties, sometimes come from a theory.
Numerical Methods,
Lecture 1
Prof. Jinbo Bi CSE, UConn

19. 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
Prof. Jinbo Bi CSE, UConn

20. 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 = m a
F=net force acting on the body (N)
m=mass of the object (kg)
a=its acceleration (m/s2)
Numerical Methods,
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Prof. Jinbo Bi CSE, UConn

21. Example 1 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
The gravity is the downward force which is paramterized by gravitational constant, which is approximately 9.8 m/s2
Numerical Methods,
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Prof. Jinbo Bi CSE, UConn

22. Example 1 First order differential equation Analytical solution
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Prof. Jinbo Bi CSE, UConn

23. Example 1 m=68.1kg, c=12.5 kg/s Numerical Methods,
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Prof. Jinbo Bi CSE, UConn

24. Example 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

25. Euler’s Method
Use the finite divided difference approximation of the derivative
The approximation becomes exact as Δt → 0
Now, we want to re-formulate the problem so it can approximately solved using numerical steps, or arithmetic operations.
Numerical Methods,
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Prof. Jinbo Bi CSE, UConn

26. Euler’s Method
Using Euler’s method, we can approximate the velocity curve
Numerical Methods,
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Prof. Jinbo Bi CSE, UConn

27. Euler’s Method …… Assume Δt=2 Numerical Methods,
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Prof. Jinbo Bi CSE, UConn

28. Euler’s method
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Prof. Jinbo Bi CSE, UConn

29. Euler’s Method 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?
Numerical Methods,
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Prof. Jinbo Bi CSE, UConn

30. Overview of the problems
Numerical Methods,
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Prof. Jinbo Bi CSE, UConn

31. Overview of the problems
Numerical Methods,
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Prof. Jinbo Bi CSE, UConn

32. Next class Programming and Software Read Chapters 1 & 2
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Prof. Jinbo Bi CSE, UConn