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Welcome to Numerical Analysis Math 448/548 Cpt S 430/530

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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Welcome to Numerical Analysis Math 448/548 Cpt S 430/530

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  1. Welcome to Numerical Analysis Math 448/548 Cpt S 430/530

  2. 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

  3. 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

  4. 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.

  5. 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/

  6. 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.

  7. How do we get this result? Non-linear problem reduced to sequence of linear problems

  8. Generalize to xn and xn+1

  9. From “Essentials of MATLAB” p150

  10. Graphical solution provides initial guess

  11. Working in the command window after newtfun has been written and saved What does .^ mean?

  12. No more roots?

  13. 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.

  14. Modify newtfun to get output needed for assignment 1 Note: logre is a vector Note: log(x) is natural log

  15. 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

  16. Y axis is log10(re) -5 corresponds to re = 10-5

  17. Derive expected convergence rate of Newton’s method What does this mean?

  18. Convergence problems with Newton’s method Example: 3.2-19 text p102

  19. Roots with multiplicity greater than one

  20. Bisection method for continuous functions What does this imply? Why require continuity? Which root will bisection method converge to in the example shown?

  21. MatLab implementation What is missing here? Test f(a)f(b)<0

  22. How do you solve for n?

  23. How does this explain the bisection result in Newton vs bisection (slide 16)?

  24. How is this method like Newton’s method (slide 7)?

  25. 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?

  26. 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?

  27. function[x,f,conv,logre]=secant(fh,xnm1,xn) Pseudo code from text p113

  28. Note similarity to Newton’s method (slide 17) en+1 = [ ] en2

  29. See text p115

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