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, mholst@math.ucsd.edu; Regular Office Hours: Mon 1-2pm)
Term: Spring 2014
Lecture: 10:00a-10:50a MWF, B412 AP&M
TA: Adam Mihalik (5768 AP&M, amihalik@math.ucsd.edu; Office Hours: Tue 10-11am)
Discussion: 5:00p-5:50p W, B402A AP&M
Main Class Webpage: http://ccom.ucsd.edu/~mholst/teaching/ucsd/170b_s14/index.html
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): 20% of grade Midterm #2 (In class week 8): 20% of grade Final Exam (Monday 6/9/14, 8am-10:50am, APM B412) 40% of grade

Here are some other important dates:

 First lecture: MON 3/31 Last lecture: FRI 6/6 Finals week: MON-FRI, 6/9-6/13 Holiday: MON 5/26 (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.

THE "DEAL" 06/02/2014: As announced in class, if we can hit a target of 60% of the class giving a CAPE response, then in return for your help with getting a good sample size for the CAPES, I will drop your single lowest homework score for the quarter. Your grade for the homework in the class will then be the average of just your four best scores.

UPDATE 06/06/2014: The CAPE response rate has now exceeded our target of 60% (we hit 72% last night). As promised, in return for your help with getting a good sample size for the CAPES, I will now drop your single lowest homework score. Your grade for the homework will then be the average of just your four best scores. Thanks again to all of you for helping get a good CAPE response; I read every single comment, and I make changes in the way I teach every year based on what you write.

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; however, I will not cover all sections in the book. My lectures may sometimes expand a bit on a particular topic beyond what is in that section of the book, if I think it is particularly important or useful. If I expect you to understand the material in a section and to be able to work problems in that section, then I will give you at least one homework problem from that section. This means that you should have a look at the upcoming homework assignment for the next two weeks to know what parts of the book to read prior to the lectures.

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(3/31-4/4) Topics: Review of Linear Algebra and Calculus in both R and Rn. Week 2(4/7-4/11) Topics: Taylor Expansion; Bisection method, Newton's method, and fixed-point methods for f:R->R. Homework 1 due FRI 4/11 (put in TA box by midnight). Week 3(4/14-4/18) Topics: Newton's Method for F:Rn->Rn. Week 4(4/21-4/25) Topics: Intro to Unconstrained Optimization and Midterm. Homework 2 due WED 4/23 (put in TA box by midnight). Midterm 1 given in class on FRI 4/26. Covers: Homeworks 1 and 2. Midterm 1 solutions are posted [ here ]. Week 5(4/28-5/2) Topics: Unconstrained Optimization. Week 6(5/5-5/9) Topics: Intro to Approximation of Functions; Best Approximation, Orthogonal Systems, Orthogonal Polynomials, Polynomial Interpolation. Homework 3 due FRI 5/9 (put in TA box by midnight). Week 7(5/12-5/16) Topics: Divided Differences, Error in Interpolation, Hermite Interpolation. Week 8(5/19-5/23) Topics: Splines, Taylor Series as Hermite Interpolation, Related topics, and Midterm. Homework 4 due WED 5/21 (put in TA box by midnight). Midterm 2 given in class on FRI 5/23. Covers: Homeworks 3 and 4. Midterm 2 solutions are posted [ here ]. Week 9(5/26-5/30) Topics: Additional topics in Approximation of functions; Approximation of Derivatives. Week 10(6/2-6/6) Topics: Approximation of Integrals and Review for Final. Homework 5 due FRI 6/6 (put in TA box by midnight). Final Exam Final Exam: (Monday 6/9/14, 8am-10:50am, APM B412) Covers Homeworks 1-5.

HOMEWORKS ASSIGNMENTS: The following are the five homework assignments. Each homework consists of exercises listed below.

Homework 1 Exercises (Background and Bisection Method; Midterm 1 material!):
• Problems 1.1: 5, 9, 25
• Problems 1.2: 6, 14, 20, 32, 36
• Problems 3.1: 2, 7 (skip "Marc-32" part)
• Computer Problems 3.1: 1, 3
• Homework 1 solutions are posted [ here ].

Homework 2 Exercises (Newton's and other Fixed-Point Methods; Midterm 1 material!):
• Problems 3.2: 6, 8, 13, 15, 18, 23
• Computer Problems 3.2: 3, 4
• Problems 3.3: 5, 7
• Problems 3.4: 1, 2, 3
• Homework 2 solutions are posted [ here ].

Homework 3 Exercises (Optimization; Midterm 2 material!):
• NOTE: The book is very weak on optimization, and so HW3 is based on my lecture material and not on the book. (The other four homeworks are assigned from the book.) Any midterm questions on optimization will be (VERY) similar to HW3, so just work on being able to do the homework problems!
• The homework on optimization can be found [ here ].
• Homework 3 solutions are posted [ here ].

Homework 4 Exercises (Best Approximation, Orthogonal Systems, Polynomial Interpolation, Hermite Interpolation; Midterm 2 material!):
• Problems 6.8: 4, 16, 18, 21, 22
• Problems 6.1: 2, 4, 5, 6, 21, 22
• Problems 6.2: 5, 8, 24
• Problems 6.3: 1
• Homework 4 solutions are posted [ here ].

Homework 5 Exercises (Splines, Taylor Series, Numerical Differentiation, Quadrature; Final Exam material!):
• Problems 6.4: 21, 22
• Problems 6.7: 16
• Problems 7.1: 6, 12, 14, 16
• Problems 7.2: 10, 11, 16, 18, 31
• Homework 5 solutions are posted [ here ].