Phys 331
Introduction to Numerical Techniques in Physics
Spring 2016
Course information
Instructor: Joaquín E. Drut.
Email: drut at email.unc.edu.
Office: Phillips 296
Where and When:
Class: Phillips 265 - Mo-We-Fr, 12:20pm-1:10pm
Mo-We : Lectures
Fr : Review / Q&A session / Exams (see below)
Lab 1: Phillips 265 - Mo, 5:45pm-7:45pm
TAs: Ryan Tanner (rjtanner .at. physics.unc.edu)
Philip Wulfken (wulfken .at. email.unc.edu)
Lab 2: Phillips 265 - We, 4:40pm-6:40pm
TAs: Ryan Tanner (rjtanner .at. physics.unc.edu)
Andrew Loheac (loheac .at. live.unc.edu)
Office hours: By appointment only.
To obtain an appointment, email me directly. Your subject heading must begin with Phys331.
Website: http://user.physics.unc.edu/~drut/public_html_UNC/physics-331.html
Bibliography:
- A. Gilat & V. Subramaniam, Numerical Methods for Engineers and Scientists.
- Press, Teukolsky, Vetterling, Flannery, Numerical recipes in C (2nd Edition, 1992).
- M.L. Boas, Mathematical Methods in the Physical Sciences.
Software: MATLAB
Midterm 1: Friday, February 12th (in class).
Midterm 2: Friday, March 11th (in class).
Final Exam: Saturday, April 30th, 12pm, Phillips 265
Other remarks:
- We have a total of 42 lectures (29 Mo-We Lectures + 13 Fr Review/Q&A sessions/Exams).
- The holidays affecting this course are:
- Monday, January 18th (MLK day).
- Spring break: March 11th (Fri) @ 5pm - March 21st (Mon) @ 8am.
- Friday, March 25th (Good Friday).
- Last lecture: Wednesday, April 27th.
Introduction
The goal of this course is to learn how to solve an arbitrary problem using a numerical approach.
These skills are useful regardless of whether you continue your studies in graduate school or go
into industry. Usually problems involve one or more of the following: nonlinear equations, linear
algebra, differentiation, integration, ordinary differential equations.
The basic algorithms are given in the lecture notes and most of them have been implemented in C
or Fortran and can be downloaded, or are directly available within MATLAB. Hence, the essential
part of numerical analysis is to know what can go wrong and how to prevent it.
Specific objectives
(1) Programming and basic numerical methods.
(a) programming concepts, structure, constructs, syntax
(b) floating point representation, definition and limitations
(c) plotting
(d) polynomials
(e) interpolation
(f) non-linear equations and root-finding (concept of iterative solutions)
(2) Linear algebra and linear systems. The students are introduced to basic
concepts of linear algebra (vector spaces, complete bases, linear operators), and they will
apply the concepts to develop methods for solving linear systems.
(a) vectors, matrices, vector spaces, bases
(b) linear operators (rotation), determinants, eigenvalues & vectors
(c) linear systems (over/under-determined, solutions)
(d) direct methods: Gauss elimination, LU decomposition
(e) iterative methods: Jacobi and Gauss-Seidel
(3) Vector calculus. Introduction to basic concepts needed for multi-dimensional root-finding and
optimization.
(a) Gradient, divergence, curl, Stokes & Gauss.
(b) multi-dimensional root-finding (Newton-Raphson)
(c) multi-dimensional optimization I: conjugate gradient methods
(4) Fourier transforms I. Introduction to time-series analysis and Fourier techniques
(a) function spaces, orthogonal basis functions
(b) Fourier series and integrals (analytical and numerical)
(5) Ordinary differential equations I. Numerical integration and differentiation.
(a) Numerical differentiation and integration. Gaussian quadrature.
(b) Euler step, simple numerical integration of 1st order ODEs.
(c) non-stiff coupled 1st order ODEs.
(6) Miscellaneous topics: Numerical evaluation of polynomials; preconditioners; further FFT; path
integrals in QM.
Grading
A linear scale between 0-100 will be used. There will be no “grading on the curve”.
A: 95.000-100.000; A-:
90.000-94.999;
B+: 85.000-89.999;
B:
80.000-84.999; B-: 75.000-79.999;
C+: 70.000-74.999;
C:
65.000-69.999; C-:
60.000-64.999;
D: 50.000-59.999; F: 0.000-49.999 Activity
Homework sets
Midterm 1
Midterm 2
Final
Percentage of grade
45%
15%
15%
25%
You will need to pass each activity with at least 50%. Thus, you need at least 50% in the two
midterms (each of them), at least 50% in the final, and at least 50% for the homework.
Prerequisites
You will need to know calculus and have had some exposure to differential equations for this
course. Linear algebra is also useful, but I will review necessary concepts in class.
You will need to have MATLAB installed on your computer. It is possible to use the university
license. Instructions on how to use that license to download the program will be sent to you by
email. If you have questions about this, please contact the TAs.
Attendance policy
Homework, class and lab attendance are mandatory for everyone, except the Friday review
sessions (unless otherwise indicated by the instructor, e.g. for a make-up lecture). If you hand the homework in before the corresponding lab session starts, you are excused from
the lab (you will still need to attend class). If you know in advance that you will miss a class or lab
session and you have a legitimate reason, you need to contact me prior to the class hour by
email, telephone or in person. If you miss because of an acute onset of illness, and you are still
alive, you need to contact me as soon as possible.
Classroom activities
We will work out many of the basic concepts in the classroom, partially in lecture format, but
mostly in form of interactive exercises. Thus, you need to bring your laptop to class.
Lectures
No cell phones, blackberries, open laptop computers (except for classroom exercises) or other
electronic devices are allowed during class. You are not allowed to make audio or video recordings
in class without instructor permission.
Homework
Homework assignments will be posted online and the due date will be indicated at the top of the
first page. Programs need to be uploaded to Sakai as indicated in the first day of class. The
homework will be graded within a week and returned to you on the next lab session.
For the first two homework sets, late homework will be penalized by 10% per day, up to a
maximum of 30%. Homework handed in later than 3 days will not be accepted. Starting with
homework #3, no late homework will be accepted.
I expect the solutions to be the results of your individual work. This means that copying
someone else's solutions (this includes verbatim copying of MATLAB code published on
the web or in the help files, or verbatim copies of available solutions) is a serious
violation of the Honor Code.
Course calendar and tentative schedule of lectures.
Lecture
#
1
Date
Lecture
Week 1: Jan 11
2
Jan 13
3
Jan 15
4
Week 2: Jan 20
5
Jan 22
Introduction, motivation, brief
math review. Course logistics.
Number representation,
sources of error.
Programming elements;
Using MATLAB.
Programming elements;
Using MATLAB. Root-finding.
SNOW DAY
6
Week 3: Jan 25
7
Jan 27
8
Jan 29
9
Week 4: Feb 1
10
Feb 3
11
Feb 5
12
Week 5: Feb 8
13
Feb 10
14
Feb 12
15
Week 6: Feb 15
SNOW DAY
16
Feb 17
17
Feb 19
Bilinear interpolation. Elements of
linear algebra
Review. Linear operators.
18
Week 7: Feb 22
19
Feb 24
Operators, matrices and linear
equations.
Gaussian elimination I
20
Feb 26
Gaussian elimination II
21
Week 8: Feb 29
LU decomposition.
22
Mar 2
LU decomposition. Linear algebra
in real life: Software and
Hardware.
Notes
Root-finding algorithms:
bisection
Root-finding algorithms:
secant
Review/Q&A
Root-finding: Newton-Raphson,
fixed-point, practice
Root-finding: convergence,
practice
Review/Q&A
Polynomial interpolation: Lagrange
Polynomial interpolation:
Splines
Midterm 1:
Everything up to
and including Week 4.
23
Mar 4
Review. Norms. Iterative solvers.
24
Week 9: Mar 7
Iterative solvers
25
Mar 9
Function spaces.
26
Mar 11
27
Week 10: Mar 21
Exam review
28
Mar 23
Fourier transforms I
29
Week 11: Mar 28
Fourier transforms II
30
Mar 30
31
Apr 1
32
Week 12:Apr 4
33
Apr 6
34
Apr 8
35
Week 13: Apr 11
36
Apr 13
37
Apr 15
38
Week 14: Apr 18
39
Apr 20
40
Apr 22
41
Week 15: Apr 25
42
Apr 27
**
April 30th
Midterm 2:
Everything from
Feb 8th up to and
including Week 8.
Review/Q&A
ODEs
Review/Q&A
Review/Q&A
FINAL EXAM
(covers everything)
Disclaimer
The instructor reserves the right to make changes to the syllabus, including due dates and test
dates (excluding the officially scheduled final examination), when unforeseen circumstances occur.
These changes will be announced as early as possible so that students can adjust their schedules.