Math 170B (Introduction to Numerical Analysis: Approximation and Nonlinear Equations)

Course Topics: Numerical methods for solving nonlinear equations and for approximation
Instructor: Prof. Michael Holst (5739 AP&M,; Normal Office Hours: Mon 2-4pm; Finals: Tue (6/12/12) 1-3pm)
Term: Spring 2012
Lecture: 12:00p-12:50p MWF, AP&M B412
TA: Shi (Fox) Cheng (5760 AP&M,; Office Hours: 3:30p-4:30p Tu-Fri)
Discussion: 5:00p-5:50p, B412 AP&M
Main Class Webpage:
Textbook(s): D. Kincaid and W. Cheney, Numerical Analysis: Mathematics of Scientific Computing, Third Edition.
Printable Syllabus: Can be found [ here ].

CATALOG DESCRIPTION: 170B. Introduction to Numerical Analysis: Approximation and Nonlinear Equations (4)
Rounding and discretization errors. Calculation of roots of polynomials and nonlinear equations. Interpolation. Approximation of functions. Three lectures, one recitation. Knowledge of programming recommended. Prerequisites: Math 170A.

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. The Math 170ABC series at UCSD provides an introduction to the exciting field of numerical analysis, which is also sometimes referred to as computational mathematics or scientific computing. Professor Holst has a passion for this particular area of mathematics, and much of his published research is in this area, and in particular in the topics covered in 170B. Math 170B deals primarily with the development and analysis of algorithms (or, numerical methods) for solving systems of nonlinear equations, and for doing approximation (of data, functions, or even solutions to differential or integral equations).

GRADES, HOMEWORKS, EXAMS, AND IMPORTANT DATES: Course information, such as homework assignments, due dates, and exam dates, will be maintained on the class webpage. Note that I sometimes make minor changes to the homework assignments as the quarter progresses, based on how much I am able to cover in the lectures. Therefore, CHECK THE WEBPAGE FREQUENTLY. The course will be graded on the homework assignments, two midterm examinations and a final examination, according to the following guidelines:

Written and Computer HW (five homeworks): 20% of grade
Midterm #1 (In class week 4, FRI 4/27, AP&M B412): 20% of grade
Midterm #2 (In class week 8, FRI 5/25, AP&M B412): 20% of grade
Final (11:30a-2:29p WED 6/13, AP&M B412) 40% of grade

Here are some other important dates:

First lecture: MON 4/2
Last lecture: FRI 6/8
Finals week: MON-FRI, 6/11-6/15
Holiday: MON 5/28 (Memorial Day -- NO LECTURE)

There will be five homework assignments throughout the quarter. The first midterm will be based on homeworks 1 and 2. The second midterm will be based on homeworks 3 and 4. The final will be cummulative and based on homeworks 1-4, as well as a small amount of new material from homework 5. The following policies regarding homeworks and exams will be applied:
  1. I normally do not accept late homeworks so that I can post solutions to the homeworks in a timely way for the class.
  2. The default plan is to have all HW assignments count towards the final grade in the class, unless I make an announcement to the contrary later in the quarter.
  3. In order to receive credit on a homework, you must at least attempt the computer parts of the homework assignments (if there are any).
  4. There will be no make-up exams. If you miss a midterm with an excused absence (i.e., illness with a note from a doctor), the other midterm and the final exam will be weighted accordingly.
  5. You are not allowed (and will not need) to use a calculator on midterms or finals.
  6. You are allowed to bring a single 8x11 sheet of paper containing notes on both sides (formulas, whatever you find useful) to each midterm and to the final. My view is that this allows you to focus on learning how to do the problems and understanding the material, rather than on memorizing formulas.
  7. Hint for Midterms and Final: The questions on all three exams should look very familiar. I will make most of the problems on all three exams look very much like the homework problems; in some cases, they will be exactly the same as some of the homework problems, and in other cases, they will be minor variations of homeworks. (I will put at least one slightly more challenging problem on each exam, which is not just a variation of a homework problem; this ensures that everyone will have some challenge on the exam.)

LECTURES: The lectures will follow the textbook quite closely. Homework assignments will be a combination of theoretical and computer problems; this will require some computer programming using MATLAB. The TA will be able to assist you in accessing your computer accounts as well as MATLAB.

Week Topics Covered
Week 1

Topics: Review of Linear Algebra and Calculus in both R and Rn.

Week 2

Topics: Taylor Expansion; Bisection method, Newton's method, and fixed-point methods for f:R->R.
Homework 1 due WED 4/18.

Week 3

Topics: Newton's Method for F:Rn->Rn.

Week 4

Topics: Unconstrained Optimization and Midterm.
Homework 2 due WED 4/25.
Midterm 1 given in class on Friday 4/27. Covers: Homeworks 1 and 2.

Week 5

Topics: Unconstrained Optimization, Polynomial Interpolation.

Week 6

Topics: Polynomial Interpolation, Divided Differences, Hermite Interpolation.
Homework 3 due FRI 5/11.

Week 7

Topics: Splines, B-Splines, Taylor Series again.

Week 8

Topics: Best Approximation and Midterm.
Homework 4 due WED 5/23.
Midterm 2 given in class on Friday 5/25. Covers: Homeworks 3 and 4.

Week 9

Topics: Least Squares Approximation, Orthogonal Polynomials, Connection to Iterative Methods for Linear Systems.

Week 10

Topics: Any Final Topics and Review for Final.
Homework 5 due FRI 6/8.

SAT 6/9
AP&M B402

Review Session: The TA (Fox Cheng) will be giving a special 1-hour review for the final.
WED 6/13
AP&M B412

Final Exam: Covers Homeworks 1-5.