M209Introduction

8/8/2019 M209Introduction

1/31

Math 209: Numerical Analysis

1


2/31

Website

http://courses.gdeyoung.com

DescriptionA study of numerical methods for root finding, interpolation,

approximation, integration, differentiation, divided differences, and

applications, using the computer.

2


3/31

Objectives

For students to obtain an intuitive and working understandingof some numerical methods for the basic problems of numerical

analysis.

For students to gain some appreciation of the concept of error

and the need to analyze and predict it.

For students to develop some experience in the implementation

of numerical methods by using a computer.

3


4/31

Programming

The programming tool that will be used for some assignmentsand illustrations is SciLab.

http://www.scilab.org/

Examples of SciLab equivalents of the Text MatLab programs

can be found on the course web site.

Help for SciLab

SciLab for Dummies, PDF and HTML on website.

SciLabs build in documentation

4


5/31

Tests & Final

Tests: 50% and Final 25% of course grade. No late exams given

Announced at least one week before test date.

Format may vary.

5


6/31

Homework/Programs

25% of course Grade. Announced at end of class

Please keep up. Late homework is significantly penalized

Homeworknonprograming: follow guidelins in syallbusUse

cover Sheet.

6


7/31

Homeworkprograming

Email subject line: Subject: Math 209, Due Date:

yyyy/mm/dd, Section x.x Problem y Short programs (less than 2 pages) you must submit paper

copy in class

Email as attachements include description of your test run

and evidence of successful run.

Programs must include comments.

Programs MUST be syntax error free, that is load with no

errors.

7


8/31

The Main Problem

How to computationally solve mathematical problems?

8


9/31

The Main Problem

How to computationally solve mathematical problems?

What problems?

Mathematical models are becoming ubiquitous

Global Warming

Enviormental Impact Engineering Studies

Mine pit reclamation

Invasive speices

...

9


10/31

Commonality in the problems

Problems include numerical method in solving or exploring the

model.

Approximating functional relationships from data and theroy .

Root Finding f(x) = 0.

Solving linear systems.

Solving DEs

10


11/31

Computers ability

Addition Subtraction

Multiplication

Computer only has a finite set of numbers to work withExact

representation is impossible!What is the error?

11


12/31

SciLab

Adapted from

ftp://ftp.math.uiowa.edu/pub/atkinson/ENA Materials

/Overheads/matlab lect.pdf

12


13/31

About SciLab

SciLab designed for numerical computing as a free alternative

to MatLab. (Another free alternative is PerlPDL).

Strongly oriented towards use of arrays (one dimensional) and

matrices (two dimensional).

Graphics that are easy to use. (Once you get the hang of it)

It can be used interactively or by writing and executing scripts.

It is a procedural language, not an object-oriented language.

SciLab can be installed on Linux, Windows and Mac OS X

from http://www.scilab.org/.

SciLab comes with a SciPad (an editor) and a Scicos

13


14/31

SciLab also has a help feature with a built in broswer to

quickly find help various items.

SciLab is an interactive computer language. For example, toevaluate

y = 6 4x + 7x2 + 3x5 +3

x + 2use

y = 6 - 4*x + 7*x*x - 3*x5 + 3/(x+2);

There are many built-in functions, e.g.

exp(x), cos(x), sqrt(x), log(x)

The default arithmetic used in SciLab is double precision and

real. However, complex arithmetic appears automatically when

needed. sqrt(-4) results in an answer of 2i.

14


15/31

SciLab works very efficiently with arrays, and many tasks are

best done with arrays. For example, plot sin(x) and cos(x) on

the interval [0, 10].t = [0:.1:10];

x = cos(t);

y = sin(t);

plot2d(t,[x,y])

15


16/31

The statement

t = [a:h:b]

with h > 0 creates a row vector of the form

t = [ a ; a + h ; a + 2 h ; : : : ]

giving all values a + jh that are less than b. When h isomitted, it is assumed to be 1. Thus

n = 1:5

creates the row vector

n = [1; 2; 3; 4; 5]

16


17/31

ARRAYS

b = [1, 2, 3]

creates a row vector of length 3.

A = [1 2 3; 4 5 6; 7 8 9]

creates the square matrix

1 2 3

4 5 6

7 8 9

Spaces or commas can be used as delimiters in giving the

components of an array; and a semicolon will separate the variousrows of a matrix.

17


18/31

For a column vector,

b = [1 3 -6]

results in the column vector

1

3

6

18


19/31

ARRAY OPERATIONS

Addition: Do componentwise addition.

A = [1, 2; 3, -2; -6, 1];

B = [2, 3; -3, 2; 2, -2];

C = A + B;

results in the answer

3 50 0

4 1

19


20/31

ARRAY OPERATIONS

Multiplication by a constant: Multiply the constant times each

component of the array.

D = 2*A;


2 4

6 4

12 2

20


21/31

ARRAY OPERATIONSMatrix multiplication: This has the standard meaning.

E = [1, -2; 2, -1; -3, 2];

F = [2, -1, 3; -1, 2, 3];

G = E*F;


G =

1 2

2 1

3 2

2 1 31 2 3

=

4 5 3

5 4 3

8 7 3

A nonstandard notation: H = 3 + F; results in the computation

H = 3

1 1 1

1 1 1

+2 1 31 2 3

=

5 2 6

2 5 6

21


22/31

COMPONENTWISE OPERATIONS

SciLab also has component-wise operations for multiplication,

division and exponentiation. These three operations are denoted byusing a period to precede the usual symbol for the operation. With

a = [ 1 2 3 ] ; b = [ 2 - 1 4 ] ;

we have

a.*b = [2 -2 12]

a./b = [0.5 -2.0 0.75]

a.^3 = [1 8 27]

2.^a = [2 4 8]

b.^a = [2 1 64]

22


23/31

The expression

y = 6 - 4*x + 7*x*x - 3*x5 + 3/(x+2);

can be evaluated at all of the elements of an array x using thecommand

y = 6 - 4*x + 7*x.*x - 3*x.^5 + 3./(x+2);

The output y is then an array of the same size as x.

23


24/31

SPECIAL ARRAYS

A = zeros(2,3)

produces an array with 2 rows and 3 columns, with all components

set to zero, 0 0 0

0 0 0

24


25/31

SPECIAL ARRAYS

B = ones(2,3)

produces an array with 2 rows and 3 columns, with all components

set to 1, 1 1 1

1 1 1

25


26/31

SPECIAL ARRAYS

eye(3,3) results in the 3 3 identity matrix,

1 0 0

0 1 0

0 0 1

Inzeros

,ones

andeye

of the above the argument may be replacedwith a matrix, the size of the result is the size of the matrix.

26


27/31

ARRAY FUNCTIONS

There are many SciLab commands that operate on arrays, we

include only a very few here. For a vector x, row or column, oflength n , we have the following functions.

max(x) = maximum component of x

min(x) = minimum component of x

abs(x) = vector of absolute values of components of x

sum(x) = sum of the components of x

Use the help browser to find many other functions.

27


28/31

OTHER COMMANDS

clear: To remove the current variables from use.

clc: To clear the output screen.clf: To clear the graphics screen.

help command name

: Brief description of command name.

28


29/31

SCRIPT FILES

A list of interactive commands can be stored as a script file. For

example, store

t = 0:.1:10;

x = cos(t);

y = sin(t);

plot(t,x,t,y)

with the file name plot trig.sci. Then to loaded in to SciLab using

the file menu or loading it from SciPad.

29


30/31

FUNCTIONS

To create a function, we proceed similarly, but now there are inputand output parameters. Consider a function for evaluating the

polynomial

p(x) = a1 + a2x + a3x2 + +anx

n1

SciLab does not allow zero subscripts for arrays. The following

function would be stored under the name polyeval.sci. no Thecoeffcients {aj} are given to the function in the array named coef,

and the polynomial is to be evaluated at all of the components of

the array x.

30


31/31

function [value] = polyeval(x,coef);

//

// function value = polyeval(x,coef)

//

// Evaluate a polynomial at the points given

// in x. The coeficients are to be given in

// coef. The constant term in the polynomial

// is coef(1).

n = length(coef)

value = coef(n)*ones(x);

for i = n-1:-1:1

value = coef(i) + x.*value;

end

endfunction

31

Date post:	10-Apr-2018
Category:	Documents
Upload:	kish-nvs
View:	218 times
Download:	0 times

M209Introduction

Documents