Math 270A (Numerical Linear Algebra)

Course Topics: Numerical Linear Algebra
Instructor: Prof. Michael Holst (5739 AP&M, mholst@math.ucsd.edu; Regular Office Hours: Mon 3-4:30pm)
Term: Fall 2016
Lecture: 2-2:50p MWF, 5402 AP&M
TA: Xuefeng Shen (295E SDSC, xus009@ucsd.edu Office Hours: 3-5p Th)
Discussion: None
Main Class Webpage: http://ccom.ucsd.edu/~mholst/teaching/ucsd/270a_f16/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: 270A. Numerical Linear Algebra (4)
Error analysis of the numerical solution of linear equations and least squares problems for the full rank and rank deficient cases. Error analysis of numerical methods for eigenvalue problems and singular value problems. Iterative methods for large sparse systems of linear equations. Prerequisites: Graduate standing 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 I will lecture on at the beginning of the course (most of which is also in the textbook), and some material on iterative methods at the end (I will post my notes on that material).

Important dates:

 First lecture: FRI 09/23 Last lecture: FRI 12/02 Finals week: MON-FRI, 12/05-12/09 NO LECTURE: FRI 10/21 (Professor giving lecture at Utah) NO LECTURE: MON 11/14 (Professor giving lecture at USC) NO LECTURE: FRI 11/25 (Thanksgiving Break) NO LECTURE: FRI 12/2: (Professor Holst released from hospital Friday, back at work on Monday) Final Exam: WED 12/7: 3-6pm, 5402 APM TA-led Final Prep (1 of 3): MON 12/5: 10am-12pm TA-led Final Prep (2 of 3): TUE 12/6: 10am-12pm TA-led Final Prep (3 of 3): TUE 12/6: 6-8pm Prof. Holst Availability: MON 12/5 and TUE 12/6, 1pm-4pm

Study tips for the Final Exam: Professor Holst strongly encourages you to take advantage of one or both of the Final Exam Preparation sessions that the TA has offered to hold on Monday 12/5 and Tuesday 12/6. The TA will work out the details of any problem you ask him to solve. I suggest you ask him to focus on the sample final exam problems that I am posting below; the final you see Wednesday 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 Wednesday, 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: Review of Vector spaces, Matrix Theory, Multivariate Calculus (Chapters 1 and 2)
Homework Exercises:
• Chapter 1 (1.13): 3, 4, 5, 6, 7, 8, 10, 12, 13, 14, 15, 17
• Chapter 2 (2.6): 1, 2, 3, 4, 5
PART 2 of 4: Direct Methods for the Solution of Linear Systems (Chapter 3)
Homework Exercises:
• Chapter 3 (3.15): 1, 2, 3, 4, 5, 9, 10, 13, 14, 15
PART 3 of 4: Iterative Methods for Solving Linear Systems (Chapter 4)
Homework Exercises: (NOTE: Some of the problems require MATLAB or a similar tool.)
• Chapter 4 (4.8): 1, 2, 3, 6, 7, 8, 11, 12, 13
PART 4 of 4: Approximation of Eigenvalues and Eigenvectors (Chapter 5)
Homework Exercises: (NOTE: Some of the problems require MATLAB or a similar tool.)
• Chapter 5 (5.13): 2, 3, 4, 5, 6, 7, 9, 10, 13, 14
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 1 and 2 homework problems can be found [ here ].
• Solutions to a subset of the Chapters 3 and 4 homework problems can be found [ here ].
• Solutions to a subset of the Chapters 5 homework problems can be found [ here ].