Numerical computation
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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

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Numerical computation

Numerical Computation

Lecture 0: Course Introduction

Dr. Weifeng SU

United International College

Autumn 2010


Course contacts

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

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

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

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

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

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

Assessment

  • Attendance and Class Participation 5%

  • Periodic Quizzes/Homework: 10%

  • Programming Assignments: 20%

  • Midterm Examination: 15%

  • Final Examination: 50%


Let s start

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

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

Mathematical Background

  • Differential Calculus, Taylor’s Theorem

  • Integral Calculus

  • Linear Algebra

  • Differential Equations


Calculus review derivatives

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 derivatives1

Calculus Review - Derivatives


Calculus review derivatives2

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 derivatives3

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 derivatives4

Calculus Review - Derivatives


Numerical computation

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.


Numerical computation

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)


Numerical computation

Calculus Review - AntiDerivatives

  • Class Practice: Find anti-derivatives for

  • x13

  • x-5

  • √x

  • 1/x3

  • sin(x) + e2x


Numerical computation

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

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

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


Special function rules

Special Function Rules

12

7

8

13

9

14

10

15

11

Mika Seppälä: Differentiation Rules


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