Numerical Computation

1. Numerical Computation Lecture 0: Course Introduction Dr. Weifeng SU United International College Autumn 2010

2. Course Contacts • Instructor: Dr. Weifeng SU • Email: wfsu@uic.edu.hk, mobile phone: 13411565789 • Office: E408, Room 7 • Contact me: Email - any time; Phone – during office hours; • TA: Ms. Yanyan Ji • Email: yyji@uic.edu.hk • Office: E408

3. Class Lectures • Lectures are on: • Monday, 10:00-10:50pm, C306 • Thursday, 15:00-16:50am, D407 • Attendance is required – at lectures and tutorials • Lectures cover main points of course • But, NOT ALL MATERIAL WILL BE ON SLIDES • Some essential material may be covered only in the lecture period.

4. Class Tutorials/Labs • Tutorials (Labs) are Critical for success in this class! • Tutorials will be scheduled starting next week • Tutorials will be scheduled for one hour each week. They will include work on: • Homework Exercises • Programming Exercises • Review of Lecture Material

5. Class Resources • Textbooks: • Numerical Methods Course Notes, Version 0.11, University of California San Diego, Steven E. Pav, October 2005. • Numerical Computing with Matlab, C. Moler (on-line text) • Both of these texts are on-line. They can be accessed through the Links section of the course page.

6. Learning Objectives • Understand the mathematical algorithms used in computational science • Understand error analysis and error propagation in numerical algorithms • Understand how computational science is used in modeling scientific applications • Understand the underlying mathematics of calculus and linear algebra needed for computational science • Develop programming skill at implementing numerical algorithms • Develop confidence in creating computational solutions to scientific applications

7. 10 minute review • Each students is require to give a ten minute review based on the content last week. • Purpose: • To learn if you are understanding what I am saying. • Practice presentation

8. Assessment • Attendance and Class Participation 5% • Periodic Quizzes/Homework: 10% • Programming Assignments: 20% • Midterm Examination: 15% • Final Examination: 50%

9. Let’s Start!! • We will study Numerical Computation a subfield of Computer Science. • What is Numerical Computation? • Given a scientific or mathematical problem. • Create a mathematical model. • Create an algorithm to numerically find a solution to the model. • Implement the algorithm in a program. • Analyze the robustness (accuracy, speed) of the algorithm. Adjust the algorithm, if needed.

10. Application Areas • CAD – Computer-Aided Design • CAM - Computer-Aided Manufacturing • Fluid Flow – Weather models, airplanes • Optimization – business, government, labs • Prototyping – Virtual Models in Car Design • Econometrics – financial models • Signal Processing – Video, Wireless algorithms

11. Mathematical Background • Differential Calculus, Taylor’s Theorem • Integral Calculus • Linear Algebra • Differential Equations

12. Calculus Review - Derivatives • The derivative of a function f(x) at a point x measures how fast the function is changing at that point. (Rate of change) • It also can be thought of as the slope of the tangent line to the curve at the point (x, f(x)). • How do we calculate a derivative?

13. Calculus Review - Derivatives

14. Calculus Review - Derivatives • Example: Let f(x) = 4x2– 2x +3. • Find the limit as h  0 of [f(x + h) – f(x)]/h • The difference quotient is • {[4(x+h)2– 2(x+h)+3] – [4x2– 2x +3]}/h = [4x2 + 8xh +4h2– 2x –2h +3 - 4x2 + 2x -3]/h = (8xh +4h2–2h)/h = 8x +4h – 2 • So, limit as h  0 of the difference quotient is 8x –2 = f’(x)

15. Calculus Review - Derivatives • Class Practice: Find f’(x) for • f(x) = 2x3 • f(x) = x-1 • f(x) = sin(x) • Derivative Rules : Look at any Calculus website

16. Calculus Review - Derivatives

17. Calculus Review - AntiDerivatives • Is it possible, knowing the derivative of a function, to work backwards and determine the function? • This process of converting a derivative back to the original function is called finding the anti-derivative, or anti-differentiation.

18. Calculus Review - AntiDerivatives • Definition: The anti-derivative of f(x) is the function F(x) such that F’(x) = f(x). • Examples: • If f’(x) = 0 then f(x) = c (constant) • If f’(x) = c (a constant) then f(x) = cx (linear) • If f’(x) = x then f(x) = x2/2 • If f’(x) = xn then f(x) = x(n+1)/(n+1) (for n not equal to -1)

19. Calculus Review - AntiDerivatives • Class Practice: Find anti-derivatives for • x13 • x-5 • √x • 1/x3 • sin(x) + e2x

20. Calculus Review - AntiDerivatives • The symbol used for finding an anti-derivative is called the integral and is denoted as • The process of evaluating an integral is calledintegration.

21. Basic Differentiation Rules 1 The derivative of the function f(x)=x is 1. 2 3 The Product Rule The Chain Rule 4 These are the basic differentiation rules which imply all other differentiation rules for rational algebraic expressions. Mika Seppälä: Differentiation Rules

22. Derived Differentiation Rules 5 The Quotient Rule. Follows from the Product Rule. Inverse Function Rule. Follows from the Chain Rule. 6 Mika Seppälä: Differentiation Rules

23. Special Function Rules 12 7 8 13 9 14 10 15 11 Mika Seppälä: Differentiation Rules