Numerical Methods

Textbook• Chapra, “Applied Numerical Methods with

MATLAB for Engineers and Scientists, 3nd Edition.”

• Publisher: McGraw-Hill• Class Note: www.ntut.edu.tw/~juiching

Grading Policy• 15 Weekly programming home work:

75%.• Programming capability evaluation:

25%.

Teaching Hours• Lecture: 1:10 pm ~ 2:00 pm, Wednesday.• Lab: 2:10 pm ~ 4:00 pm, Wednesday.• Office hour: 4:10 pm~6:00 pm, Mon., Wed.

and Thur..

Goal• Learn numerical methods. • Learn MATLAB.

What is “Numerical Methods”?

• Solving mathematical equations by computer programs numerically.

Why learn “Numerical Methods”?

• Equations in engineering mathematics– the simplest.– solutions in functional form.

• Problems in real life– usually not solvable by the technique in engineering

mathematics.– solutions not necessarily in functional form.

Why learn “Numerical Methods”? (cont.)

• Example: the volume of liquid V in a hollow horizontal cylinder of radius r and length L is related to the depth of the liquid h by

• If r is fixed, can youdetermine h for a given V?

V' r 2cos&1 r&hr

&(r&h) 2rh&h 2 LV' r 2cos&1 r&hr

&(r&h) 2rh&h 2 L

Lh

Symbolical v.s. Numerical

• Symbolical– Solve equations by derivative technique.– Mimic the way human brains solve problems.– Solutions are in function form.– Specific but not general.– Efficient but limited. Many are not solvable.– Software: Mathemetica, Maple

Symbolical v.s. Numerical (cont.)

• Numerical– Utilize the advantages of computers: fast, large and

repetitive.– Very different from the way human brains solve

problems.– Solutions are in numerical form.– General.– Not efficient but not limited.– Software: MATLAB and any programming languages.

Symbolical v.s. Numerical (cont.)

• Symbolical– Example:

&d 2f(x)dx 2

'1%4x 2, f(0)'f(1)'0

Y f(x)'5x6&

x 2

2&

x 4

3

&d 2f(x)dx 2

'1%4x 2, f(0)'f(1)'0

Y f(x)'5x6&

x 2

2&

x 4

3

Symbolical v.s. Numerical (cont.)

• Numerical– Example: same as previous.

f )(x)' f(x%Δ x)& f(x&Δx)2Δx

f ))(x)' f(x%Δ x)&2 f(x)% f(x&Δ x)Δx 2

&fi% 1& 2 fi% fi& 1

Δ x 2' 1% 4x 2

i

Δx'0 .1

1Δx2

2 &1 0 0 0 0 0 0 0&1 2 &1 0 0 0 0 0 00 &1 2 &1 0 0 0 0 00 0 &1 2 &1 0 0 0 00 0 0 &1 2 &1 0 0 00 0 0 0 &1 2 &1 0 00 0 0 0 0 &1 2 &1 00 0 0 0 0 0 &1 2 &10 0 0 0 0 0 0 &1 2

f1f2!

f9

'

1%4x 21

1%4x 22

!

1%4x 29

1Δx2

2 &1 0 0 0 0 0 0 0&1 2 &1 0 0 0 0 0 00 &1 2 &1 0 0 0 0 00 0 &1 2 &1 0 0 0 00 0 0 &1 2 &1 0 0 00 0 0 0 &1 2 &1 0 00 0 0 0 0 &1 2 &1 00 0 0 0 0 0 &1 2 &10 0 0 0 0 0 0 &1 2

f1f2!

f9

'

1%4x 21

1%4x 22

!

1%4x 29

Symbolical v.s. Numerical (cont.)

• Comparison

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8 1

f(x)

x

exactfinite difference

Symbolical v.s. Numerical (cont.)

DescriptionOf Problem

Formulate

SolveSymbolically

SolveNumerically

NumericalData

Contents

• Roots of Equations• Linear Algebraic Equations and Matrices

– Direct solver– Iterative solver

• Curve Fitting• Numerical Integration Formulas• ODEs• Eigenvalues

What is MATLAB?• Short for MATrix LABoratory• A numerical computing environment and

programming language.• Merits:

– Interactive– Easy matrix manipulation– Plotting of functions and data– Creation of user interfaces– Interface with programs in other languages– Many libraries of useful functions

Importance of MATLAB• A must-have tool for engineers.• A common software engineering students

must learn in the United States.• At least, a good replacement of calculators.• Example: computing the following formula

V' r 2cos&1 r&hr

&(r&h) 2rh&h 2 LV' r 2cos&1 r&hr

&(r&h) 2rh&h 2 L

A Simple Demo of MATLAB• Every variable is a matrix. Case sensitive.• Assigning values to variables

– Real number– Complex number– Matrix

• Mathmatical operations– Matrix operation– Element-by-element operation

A Simple Demo of MATLAB(cont.)

• Built-in functions• Graphics• Script files• Function files• Conditional statement “if”• Loops

– “for”– “while”

Free Compatible MATLAB Packages

• Octave• FreeMAT• SciLAB

Milestones of AI

• Autonomous Cars• Robots• Chess• Quiz

Autonomous Cars

• DARPA Grand Challenge 2004– Route: Mojave Desert, 150 miles.– Prize: 1 million.– Result: no team succeeded.

• DARPA Grand Challenge 2005– Result: 5 teams succeeded.

• DARPA Grand Challenge 2007– Route: urban area in California, 60 miles.– Prize: 3.5 million total.– Result: 6 teams succeeded.

Robots

• DARPA Robotics Challenge 2015– Semi-autonomous ground robots that could do

complex tasks in dangerous, degraded, human-engineered environments.

– Complete 8 tasks simulating disaster response.– Prize: 3.5 millions total.– Winner: Korea KAIST (Korea Advanced

Institute of Science and Technology).

Chess

• IBM Deep Blue chess computer defeated World Champion Kasparov in 1997.

• In 2016, Google Alpha Go defeated Lee Sedol, the second-highest in Go international Championship ranking.

Quiz

• In 2011, IBM Watson defeated on quiz show Jeopardy! former winners Brad Rutterand Ken Jennings and received the first place prize of $1 million.