Math 174/274 (Numerical Methods for Physical Modeling)

Course Topics: Numerical for Physical Modeling
Instructor: Prof. Michael Holst (5739 AP&M, mholst@math.ucsd.edu; Regular Office Hours: Mon 3-4:30pm)
Term: Fall 2014
Lecture: 2:00p-2:50p MWF, 5402 AP&M
TA: Shi (Fox) Cheng (5768 AP&M, scheng@math.ucsd.edu; Office Hours: Wed 4-5pm, Fri 3-4pm)
Discussion: 2:00p-2:50p Th, 2402 AP&M
Main Class Webpage: http://ccom.ucsd.edu/~mholst/teaching/ucsd/174_f14/index.html
Textbook(s): T. Sauer, Numerical Analysis, Second Edition.
Printable Syllabus: Can be found [ here ].


CATALOG DESCRIPTION: 174/274. Numerical Methods for Physical Modeling (4)
Floating point arithmetic, direct and iterative solution of linear equations, iterative solution of nonlinear equations, optimization, approximation theory, interpolation, quadrature, numerical methods for initial and boundary value problems in ordinary differential equations. (Students may not receive credit for both Math 174 and PHYS 105, AMES 153 or 154. Students may not receive credit for Math 174 if Math 170A, B, or C has already been taken.) Graduate students will do an extra assignment/exam. Prerequisites: Math 20D or Math 21D, and either Math 20F or Math 31AH, 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. The Math 174/274 at UCSD provides a quick but thorough 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. Math 174/274 deals primarily with the development and analysis of algorithms (or, numerical methods) for solving systems of linear systems of equations, nonlinear systems of equations, optimization problems, and ordinary differential equations, and for approximating functions, their derivatives, and their integrals.


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, end of week 3): 20% of grade
Midterm #2 (In class, end of week 7): 20% of grade
Final Exam (Wednesday 12/17/14, 3pm-5:59pm, APM 5402) 40% of grade

Here are some other important dates:

First lecture: FRI 10/03
Last lecture: FRI 12/12
Finals week: MON-FRI, 12/15-12/19
Holiday: FRI 11/28 (Thanksgiving Break -- NO LECTURE)
Final Exam: WED 12/17, 3pm-5:59pm, APM 5402

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. However:

    THE "DEAL" 12/08/2014: As announced in class, if we can hit a target of 70% 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 12/17/2014: The CAPE response rate has now exceeded our target of 70% (we hit 79% 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 help you with both the theoretical and computer problems in both the discussions sections and by appointment.

Week Topics Covered
Week 0
(10/3)
Topics: Overview of Class, Some Background Material

Week 1
(10/6-10/10)
Topics: Solving Single Equations (Chapter 1)

Week 2
(10/13-10/17)
Topics: Solving Systems of Linear and Nonlinear Equations (Chapter 2)
Homework 1 due MON 10/13 (put in box outside TA office by 6pm-ish).

Week 3
(10/20-10/24)
Topics: Solving Systems of Linear and Nonlinear Equations (Chapter 2; continued) and Midterm
Homework 2 due Thu 10/23 (put in box outside TA office by 6pm-ish).
Midterm 1 given in class on FRI 10/24.
Covers: Homeworks 1 and 2 up through Section 2.5; Sections 2.6-2.7 WILL NOT be on Midterm 1.
Midterm 1 solutions are posted [ here ].
Week 4
(10/27-10/31)
Topics: Interpolation (of data and/or functions) using Polynomials (Chapter 3)

Week 5
(11/3-11/7)
Topics: Interpolation (of data and/or functions) using Polynomials (Chapter 3; continued)

Week 6
(11/10-11/14)
Topics: Numerical Differentiation and Integration (Chapter 5)
Homework 3 due MON 11/10 (put in box outside TA office by 6pm-ish).

Week 7
(11/17-11/21)
Topics: Numerical Differentiation and Integration (Chapter 5; continued) and Midterm.
Homework 4 due THU 11/20 (put in box outside TA office by 6pm-ish).
Midterm 2 given in class on FRI 11/21.
Covers: Homeworks 3 and 4, AND ALSO Sections 2.6-2.7 that we did not cover on Midterm 1.
Midterm 2 solutions are posted [ here ].

Week 8
(11/24-11/26)
Topics: Numerical Methods for Ordinary Differential Equations (Chapters 6 and 7)

Week 9
(12/1-12/5)
Topics: Numerical Methods for Ordinary Differential Equations (Chapter 6 and 7; continued)

Week 10
(12/8-12/12)
Topics: A Few Special Topics and Review for Final
Homework 5 due FRI 12/12 (put in box outside TA office by 6pm-ish).

Final Exam
Final Exam: WED 12/17, 3pm-5:59pm, APM 5402. 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 Chapter 1; Midterm 1 material!):
  • 0.5 Exercises: 1, 2, 3, 4
  • 1.1 Exercises: 1, 3, 5
  • 1.1 Computer Problems: 1, 3
  • 1.2 Exercises: 1, 7, 11
  • 1.2 Computer Problems: 1, 3
  • 1.4 Exercises: 1, 3, 10
  • 1.4 Computer Problems: 1, 10
  • Homework 1 solutions are posted [ here ].

Homework 2 Exercises (Chapter 2; Midterm 1 material!):
  • 2.1 Problems: 1, 3
  • 2.3 Problems: 2, 8
  • 2.5 Problems: 1
  • 2.5 Computer Problems: 1, 3, 5(a)
  • 2.6 Problems: 1, 13(a)
  • 2.6 Computer Problems: 1, 3
  • 2.7 Problems: 1, 3, 4
  • 2.7 Computer Problems: 1, 3
  • 2.2 Problems: 1 (Guest Lecture covering Sections 2.2 and 2.4, Wed Oct 22)
  • 2.4 Problems: 3 (Guest Lecture covering Sections 2.2 and 2.4, Wed Oct 22)
  • Homework 2 solutions are posted [ here ].

Homework 3 Exercises (Chapter 3; Midterm 2 material!):
  • 3.1 Problems: 1, 2, 3, 11
  • 3.1 Computer Problems: 1
  • 3.2 Problems: 1, 3, 5
  • 3.2 Computer Problems: 1, 3, 4
  • 3.3 Problems: 1, 2
  • 3.4 Problems: 1, 2, 4
  • Homework 3 solutions are posted [ here ].

Homework 4 Exercises (Section 4.1 and Chapter 5; Midterm 2 material!):
  • 4.1 Problems: 1, 5, 6
  • 5.1 Problems: 1, 5, 11, 21
  • 5.2 Problems: 1, 3, 9, 12
  • 5.3 Problems: 1, 3
  • 5.5 Problems: 1, 7
  • Homework 4 solutions are posted [ here ].

Homework 5 Exercises (Chapters 6 and 7; Final Exam material!):
  • 6.1 Problems: 2(a), 5, 9, 11, 13
  • 6.1 Computer Problems: 1
  • 6.2 Problems: 1
  • 6.2 Computer Problems: 1, 3
  • 6.3 Problems: 1(a), 3(a)
  • 6.3 Computer Problems: 1(a)
  • 6.4 Problems: 1(a)
  • 6.6 Problems: 1, 3
  • 7.1 Problems: 1(a), 3
  • 7.2 Computer Problems: 1(a)
  • Homework 5 solutions are posted [ here ].