Numerical Computation

Lecture 0: Course IntroductionDr. Weifeng SU

United International CollegeAutumn 2010

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

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%

• 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

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 - Derivatives

Calculus Review - DerivativesDefinition Example

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?

h

xfhxf

0

1

2

3

-3 -2 -1 0 1 2 3

• 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 called integration.

Calculus Review - AntiDerivatives

( )f x dx

Mika Seppälä: Differentiation Rules

Basic Differentiation Rules

D fg D f g fD g

D f g D f g D g

f g f g g

x x x

d xx x

dx

3

4

The Product Rule

The Chain Rule

D 1x 1 The 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 D f g D f D g

Mika Seppälä: Differentiation Rules

Derived Differentiation Rules

-1 1

D fD f

6

2

gD f fD gfD

g g

5 The Quotient Rule. Follows from the

Product Rule.

Inverse Function Rule. Follows from the Chain Rule.

Mika Seppälä: Differentiation Rules

Special Function Rules

sincos

d xx

dx8

9

7 1, r

rdxrx r

dx

cossin

d xx

dx

10 2

tan 1

cos

d x

dx x

11

2

arcsin 1

1

d x

dx x

12 2

arctan 1

1

d x

dx x

13e

ex

xd

dx

14 lnx

xdaa a

dx

15 ln 1d x

dx x