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Welcome to Numerical Analysis Math 448/548 Cpt S 430/530. CptS 430/Math448 Numerical Analysis Spring 2014 Instructor: John Miller, West 134E jhmiller@tricity.wsu.edu Class web page can be found at http://www.tricity.wsu.edu/~jhmiller. Objectives:

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slide1

Welcome to Numerical Analysis

Math 448/548

Cpt S 430/530

slide2

CptS 430/Math448

Numerical Analysis

Spring 2014

Instructor: John Miller, West 134E

jhmiller@tricity.wsu.edu

Class web page can be found at

http://www.tricity.wsu.edu/~jhmiller

slide3

Objectives:

To present the mathematical basis for methods of numerical analysis

To introduce students to MATLAB as a software package for

numerical analysis

Textbook: “Numerical Mathematics and Computing” 6th Edition

by Cheney and Kincaid

Course content:

Chapter 3: Locating roots of an equation

Chapters 5&6: Numerical integration

Chapter 10&11: Systems of ordinary differential equations

Midterm exam

Chapter 2: Floating-point representation and errors

Chapters 7&8: Systems of linear equations

Chapter 12: Smoothing data and method of least squares

Final exam

slide4

Optional supplemental textbook:

“Essential MATLAB for Scientist and Engineers” 2nd Edition by Hahn

Required assignments:

Students will be assigned problems to be solved by writing MATLAB codes

Prior approval is required for late submission.

Criteria for student evaluation:

homework 25%, quizzes 25%, midterm exam 25% final exam 25%.

No partial credit on homework. Full credit for corrected homework

500-level students must define a project (subject to approval by the instructor) that involves numerical analysis and write a report that describes the numerical method, code developed, and results obtained. Project reports are due on the on the last class period.

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Academic integrity:

Academic integrity will be strongly enforced in this course. Any student caught cheating on any assignment will be given an F grade for the course and will be reported to the Office Student Standards and Accountability. Cheating is defined in the Standards for Student Conduct WAC 504-26-010 (3). It is strongly suggested that you read and understand these definitions:

http://www.studentmediagroup.com/planners/palouse2012/#/18/

I encourage you to work with classmates on assignments. However, each student must turn in original work. No copying will be accepted. Students who violate WSU’s Standards of Conduct for Students will receive an F as a final grade in this course, will not have the option to withdraw from the course and will be reported to the Office Student Standards and Accountability. Cheating is defined in the Standards for Student Conduct WAC 504-26-010 (3). It is strongly suggested that you read and understand these definitions:

http://www.studentmediagroup.com/planners/palouse2012/#/18/

Academic integrity is the cornerstone of the university. Any student who attempts to gain an unfair advantage over other students by cheating, will fail the assignment and be reported to the Office Student Standards and Accountability. Cheating is defined in the Standards for Student Conduct WAC 504-26-010 (3).

http://www.studentmediagroup.com/planners/palouse2012/#/18/

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Campus Safety Statement

WSU Tri-Cities is committed to maintaining a safe environment for its faculty, staff and students. The Campus Safety Plan can be found at http://www.tricity.wsu.edu/safetyplan/

See also the WSU Office of Emergency Management site at http://oem.wsu.edu/emergencies

Up-to-date WSU emergency alerts are available at http://alert.wsu.edu/

Disability Services Reasonable Accommodations Statement: Reasonable accommodations are available for students who have a documented disability. Classroom accommodation forms are available through the Disability Services Office. If you have a documented disability, even if it’s temporary, make an appointment as soon as possible with the Disability Services Coordinator, Cherish Tijerina Pearson, West Building, Room 269J, at 372-7352 or ctijerina@tricity.wsu.edu.You will need to provide your instructor with the appropriate classroom accommodation form. The form should be completed and submitted during the first week of class. Late notification can delay your accommodations or cause them to be unavailable. All accommodations for disabilities must be approved through the Disability Services Coordinator.

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How do we get this result?

Non-linear problem reduced to sequence of linear problems

slide13

Assignment 1, Due 1/23/14

f(x) = exp(x) - 3x2 has a root in the interval [0.5, 1.5].

Write MATLAB codes to find this root by the Bisection, Newton,

and Secant methods.

Use format long to compare your results to the value returned by

MATLAB’s fzero function.

Define convergence by Dn = |(xn – xn-1)/xn|

On a semi-log plot, compare the rates of convergence of the 3 methods.

slide14

Modify newtfun to get output needed for assignment 1

Note: logre is a vector

Note: log(x) is natural log

slide15

MatLab code to compare bisection and newton methods

Separate m-files for function and its derivative

Note @ before file name in function calls

Must be in the same folder

Inline function definitions (slide 11) could be used

When vector is single argument of plot, (k, ak) pairs are plotted

slide16

Y axis is log10(re)

-5 corresponds to re = 10-5

slide20

Bisection method for continuous functions

What does this imply?

Why require continuity?

Which root will bisection method converge to

in the example shown?

slide21

MatLab implementation

What is missing here? Test f(a)f(b)<0

slide23

How does this explain the bisection result in

Newton vs bisection (slide 16)?

slide25

Convergence may not be uniform

Always use best estimate of root as xn in each cycle

If starting values are 1 and -1,

what do we expect to happen?

slide26

Initial xn+1 is not as good as xn, (how do we know this?)

Use best estimate of root as xn (what value is this?)

Use bad first estimate as xn-1

Did the second attempt work?

How do we avoid this type of convergence?

slide30

Note similarity to

Newton’s method (slide 17)

en+1 = [ ] en2