Math 270B (Numerical Linear Algebra)

Course Topics: Numerical Linear Algebra
Instructor: Prof. Michael Holst (5739 AP&M, mholst@math.ucsd.edu; Regular Office Hours: Mon & Wed 11-11:50am)
Term: Winter 2017
Lecture: 10-10:50a MWF, B412 AP&M
TA: Xuefeng Shen (295E SDSC, xus009@ucsd.edu; Regular Office Hours: Thu 4-5:50pm)
Discussion: None
Main Class Webpage: http://ccom.ucsd.edu/~mholst/teaching/ucsd/270b_w17/index.html
Textbook(s): A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Second Edition, Springer-Verlag, 2000.
Printable Syllabus: Can be found [ here ].

CATALOG DESCRIPTION: 270B. Numerical Approximation and Nonlinear Equations (4)
Iterative methods for nonlinear systems of equations, Newtonâ€™s method. Unconstrained and constrained optimization. The Weierstrass theorem, best uniform approximation, least-squares approximation, orthogonal polynomials. Polynomial interpolation, piecewise polynomial interpolation, piecewise uniform approximation. Numerical differentiation: divided differences, degree of precision. Numerical quadrature: interpolature quadrature, Richardson extrapolation, Romberg Integration, Gaussian quadrature, singular integrals, adaptive quadrature. Prerequisites: Math. 270A or consent of instructor.

COURSE INFORMATION: Many of the advances of modern science have been made possible only through the sophisticated use of computer modeling. The mathematical foundation of the computer modeling techniques now used in all areas of mathematics, engineering, and science is known as numerical analysis (sometimes referred to as computational mathematics, numerical mathematics, or scientific computing). The Math 270ABC series at UCSD provides a graduate level overview of some of the foundation topics in numerical analysis. The Math 270ABC sequence covers all of the material that appears on one of our core written qualifying examinations for the mathematics doctoral program at UCSD.

Math 270A deals with various aspects of numerical linear algebra, including direct methods for solving systems of equations involving dense and sparse matrices, iterative methods for solving linear systems, and methods for solving eigenvalue problems. Math 270B focuses on numerical methods for solving nonlinear equations and optimization problems, and on classical and modern approximation theory as needed for analyzing numerical methods. Math 270C concentrates on bringing the tools from 270A and 270B together to develop numerical methods for solving initial value problems (IVPs) and boundary value problems (BVPs) in ordinary differential equations (ODEs). The topics in 270A and the first half of 270B may be viewed as preparation for Math 271ABC (numerical optimization), whereas the topics in the entire series, with emphasis on the topics in the second half of 270B and the topics in 270C, may be viewed as preparation for 272ABC (numerical PDE) and 273ABC (advanced techniques in computational and applied mathematics).

GRADES, HOMEWORKS, EXAMS, AND IMPORTANT DATES: Course information, such as any homework assignments given out, exam dates, and so forth, will be maintained on this course webpage. Note that I sometimes make changes to the lecture schedule and homework assignments as the quarter progresses, depending on how far I get each week in the lectures. Therefore, check this webpage regularly. I will periodically (about every 2-4 weeks) give out homeworks to help you prepare for the final exam. The last week I will finish any remaining material, and then focus mostly in the last lecture on reviewing the material from the class for the final exam. The final exam will be based on the material I cover in class (a subset of the textbook), supplemented with some background material.

Important dates:

 First lecture: MON 01/09 Last lecture: FRI 03/17 Finals week: MON-FRI, 03/20-03/24 NO LECTURE: FRI 01/13 (Professor in Hospital) NO LECTURE: MON 01/17 (Campus Holiday) 1ST FINAL REVIEW: TUE 3/21/17, 2-4pm, 7421 APM (TA-led session) 2ND FINAL REVIEW: THU 3/23/17, 5-6pm, 295E SDSC (TA-led session) Final Exam: FRI 03/24, 8-11am, B412 APM

Study tips for the Final Exam: Professor Holst strongly encourages you to take advantage of Final Exam Preparation sessions that the TA has offered to during finals week (date/time/location posted above and below). He is also available by appointment during finals week. The TA will work out the details of any problem you ask him to solve. I suggest you ask him to focus on the problems I have highlighted in red from the homework problems, and also sample final exam problems that I am posting below; the final you see on Friday will look a lot (i.e., a lot) like the sample exam below.

PRACTICE/SAMPLE FINAL EXAM is posted [ here ]

NOTE: This is a little longer than the final exam you will see on Friday, but I wanted to give you a sample problem on each of the main topics of the class. (Some of these are a bit challenging, but most should be doable.)

Homeworks: I will give out some homework problems that will be very similar to problems that will appear on the final exam (and later the qual exam). The TA will be available to help you work the problems in the weekly discussion sessions, and by appointment as needed. The TA will collect and mark some of the homework problems to give you feedback, but your grade for the course will be based only on your final examination at the end of the quarter. (The idea is to give you a practice run at 1/3 of the 270ABC qual exam.)

Turning in the homeworks: Please do not give all of your homeworks to the TA toward the end of the quarter; he will not have time to give you any feedback before the final. To make best use of his help, try to do the homeworks within a couple of weeks of their posting, meeting with the TA during his office hours as needed, or by appointment. As noted above, the TA will give you feedback on your homework, both during his office hours, and also by providing some mark ups of the homeworks if you ask for it. He will not generally grade all questions, but he will try to give you some feedback on most of the questions if you ask him. The best way to make use of the TA in the course is to try working the problems, and then ask him questions in person if you get stuck.

Working Together: You are permitted, and very much encouraged, to work on the homeworks together, including writing up your solutions to the homeworks if you would like feedback from the TA on your solutions. However, keep in mind that your final exam is done individually, so you need to make sure you can solve the homework problems, and also write them up on your own, since you will need to be able do this at the end of the quarter. (In my experience, it is easy to convince yourself that you understand how to solve a problem, but unless you actually write it all out carefully, many times you find that you do not completely understand things without writing it out.)

PART 1 of 4: Rootfinding for Nonlinear Equations (Chapter 6)
Homework Exercises (red are targets for final exam):
• Chapter 6 (6.8): 1, 2, 5, 7, 8
PART 2 of 4: Nonlinear Systems and Numerical Optimization (Chapter 7)
Homework Exercises (all problems are targets for final):
• Chapter 7: Some notes that supplement the book Chapter 7 can be found [ here ]
• Chapter 7: The homework covering Chapter 7 can be found [ here ]
PART 3 of 4: Polynomial Interpolation (Chapter 8)
Homework Exercises (red are targets for final exam):
• Chapter 8 (8.10): 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13
PART 4 of 4: Numerical Integration (Chapter 9)
Homework Exercises (red are targets for final exam):
• Chapter 9 (9.11): 1, 2, 3, 4, 5, 6, 7, 8, 10
HOMEWORK SOLUTION SETS: The TA has provided worked out solutions for some of the homework problems from each chapter. He skipped problems that were either particularly easy or had solutions already written out in the book.
• Solutions to a subset of the Chapters 6 and 7 homework problems can be found [ here ].
• Solutions to a subset of the Chapters 8 and 9 homework problems can be found [ here ].
1ST REVIEW FOR FINAL: Our excellent TA, Xuefeng Shen, has again offered to do a long 2-hour review session for the final exam. Below is the day/time/location for the first review session.
• 2-Hour Review for Final: TUE 3/21/17, 2-4pm, 7421 APM
2ND REVIEW FOR FINAL: Xuefeng Shen has offered to do a second 1-hour review session for the final exam; for this second review, he will focus entirely on the practice exam. Below is the day/time/location for this second review session.
• 1-Hour Review for Final: THU 3/21/17, 5-6pm, 295E SDSC