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

Course Contacts Instructor: Dr. Weifeng SU – mobile phone: – Office: E408, Room 7 – Contact me: - any time; Phone – during office hours; TA: Ms. Yanyan Ji – – Office: E408

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.

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

Class Resources Textbooks: – Numerical Methods Course Notes, Version 0.11, University of California San Diego, Steven E. Pav, October – 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.

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

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

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

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. Let’s Start!!

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 Application Areas

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

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? Calculus Review - Derivatives

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

Class Practice: Find f ’ (x) for – f(x) = 2x 3 – f(x) = x -1 – f(x) = sin(x) – Derivative Rules : Look at any Calculus website Calculus Review - Derivatives

DefinitionExample Differentiable: A function f is differentiable at x if approaches some number as h approaches zero. The function f (x) = |x| is differentiable for all values of x except x = 0. Why?

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. 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) = x 2 /2 If f ’ (x) = x n then f(x) = x (n+1) /(n+1) (for n not equal to -1) Calculus Review - AntiDerivatives

Class Practice: Find anti-derivatives for x 13 x -5 √x 1/x 3 sin(x) + e 2x 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 called integration. Calculus Review - AntiDerivatives

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

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

Mika Seppälä: Differentiation Rules Special Function Rules