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Numerical Computation

Lecture 0: Course Introduction

Dr. Weifeng SU

United International College

Autumn 2010

Course Contacts

- Instructor: Dr. Weifeng SU
- Email: [email protected], mobile phone: 13411565789
- Office: E408, Room 7
- Contact me: Email - any time; Phone – during office hours;

- TA: Ms. Yanyan Ji
- Email: [email protected]
- 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 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.

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%

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.

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

Mathematical Background

- Differential Calculus, Taylor’s Theorem
- Integral Calculus
- Linear Algebra
- Differential Equations

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?

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)

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

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.

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)

Calculus Review - AntiDerivatives

- Class Practice: Find anti-derivatives for
- x13
- x-5
- √x
- 1/x3
- sin(x) + e2x

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.

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

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

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