Math/CS 466/666
Fall 2024 University of Nevada Reno
Math 466/666 NUMERICAL METHODS I (3+0) 3 credits
Numerical solution of linear systems, including linear programming;
iterative solutions of non-linear equations; computation of eigenvalues
and eigenvectors, matrix diagonalization. Prerequisite(s): MATH 330.
Instructor Course Section Time Room
------------------------------------------------------------------------
Eric Olson Math 466/666 Numerical Methods I MWF noon-12:50pm DMSC106
Course Information
- Instructor:
- Eric Olson
- email:
- Please contact me through WebCampus
- Office:
- MWF 11:00-11:50PM in DMS 238 and through Zoom by appointment
- Homepage:
- http://fractal.math.unr.edu/~ejolson/466/
- Grader:
- to be determined (contact through
WebCampus)
- Course Textbook:
- Endre Suli, David F. Mayers,
An Introduction to Numerical Analysis, 1st Edition,
Cambridge University Press, 2003.
- Supplemental References:
- Hosking,
Joe, Joyce and Turner, First Steps in Numerical Analysis,
2nd Edition, Arnold, 1996.
- Jeffery Leader, Numerical Analysis and Scientific
Computation, Pearson, 2004.
- Lloyd Trefethen, Numerical Linear Algebra, Siam 1997.
- Justin Solomon,
Numerical Algorithms: Methods for Computer
Vision, Machine Learning and Graphics, CRC Press, 2015.
- Anthony Ralston and Philip Rabinowitz, A First Course in Numerical
Analysis, Second Edition, Dover, 1978.
- Richard Hamming, Numerical Methods for Scientists and Engineers,
Second Edition, Dover, 1986.
Class Handouts
Course materials specific for this section of Math 466 are available
by clicking on this link. Details for how to
access these files may be found on our course page in WebCampus.
Information about Software
Student Learning Outcomes
Upon completion of this course, students will be able to
- The effects of rouding error in numerical computation.
- Newton's method, interpolation and numerical Linear Algebra.
- Practical use of the computer to solve numerical problems.
In-Class Computer Labs
Homework
- HW1 problems 1.1, 1.4 and 1.6 due Sept 23
(solutions).
- HW2 problems 2.1, 2.8i,ii due Nov 8
(solutions).
- HW3 problems 2.9, 2.10 and 2.12 due Nov 22
(solutions).
- HW4 problems 4.1 and 4.2 due Dec 6
(solutions).
- HW5 problems click here due Dec 18.
Lecture Notes
Announcements
[17-Dec-2024] Sample Final Exam
Here is a sample final to help you study
for the exam on Wednesday.
[18-Oct-2024] Homework 5
Homework 5 is now available and is due the
night of the final exam.
[18-Dec-2024] Final Exam
The final exam is Wednesday, December 18, 2024
from 12:45-2:45pm in DMS106.
Here is a list of ideas to help you review:
- Review HW1-HW4 questions and answers.
- Know the proof of Taylor's theorem.
- State Newton's, the secant and bisection methods.
- Show that Newton's method is quadratically convergent.
- Use A=PLU factorization to find det(A).
- Explain the purpose of partial pivoting.
- Definition of the p vector norms ||x||p.
- Proof of Cauchy's inequality and Holder's inequality.
- Definition of the matrix norm ||A||p
induced by ||x||p.
- Compute 1, 2 and ∞ vector norms.
- Compute 1, 2 and ∞ matrix norms.
- Be able to state the Contraction Mapping Theorem.
- Explain the idea behind Jacobi's Eigenvalue Algorithm.
- Definition of the Gerschgorin disks Di.
- Use of Gerschgorin Theorems to estimate eigenvalues.
- Definition of a Householder reflector.
- Use of QR factorization to minimize ||Ax-b||2.
This is a closed-book closed-notes only scientific-calculator-allowed
in-class exam. Efforts have been made to keep the arithmetic simple.
If it turns out to be complicated, that's either because I made a mistake
or you did. In either case, do the best you can and check your work
where possible. While getting the right answer is nice, this is not an
arithmetic test. It's more important to clearly explain what you did
and what you know.
[02-Dec-2024] Computational Midterm
The computational midterm will be given in class on December 2.
[27-Nov-2024] Example Question
The computational midterm will consist of number of computational
tasks along with incomplete code solutions to be completed in class
using the lab computer or your personal laptop.
This will be an
open book and open computer exam; however, you must work independently.
The questions are written in such a way that it is
possible to use an independently created program—even in a different
programming language—to obtain full credit. I would
recommend sticking with Julia.
Each incomplete code solution explains where to add more code and
further includes a routine to check that the solution is correct.
Here is a sample
question to give you an idea what to expect on the exam:
- Find 5 iterations of the bisection method with I0=[1,4]
to approximate the solution to f(x)=0 where f(x)=log(x)-x+2.
The incomplete code solution you need to fix is here.
To further prepare I
suggest looking through the programs you wrote for the computer
labs and add comments explaining what the code does.
[22-Nov-2024] Lab Activity 7
On November 22 we will have a computing lab in class.
Please make sure to attend.
[15-Nov-2024] Lab Activity 6
On November 15 we will have a computing lab in class.
Please make sure to attend.
[08-Nov-2024] Lab Activity 5
On November 8 we will have a computing lab in class.
Please make sure to attend.
[01-Nov-2024] Lab Activity 4
On November 1 we will have a computing lab in class.
Please make sure to attend.
[23-Oct-2024] Lab Activity 3
On October 23 we will have a computing lab in class.
Please make sure to attend.
[07-Oct-2024] Extra Credit
For extra credit please read
- Nicholas Higham, Estimating the matrix p-norm, Numer. Math.,
vol 62, 1992, pp. 539-555.
and implement the algorithm to find the Matrix p-norm in Julia.
Though it may not help much, there is a
C++ implementation of this
algorithm for GNU Octave.
I do not know of any native Julia code which does the same.
[02-Oct-2024] Homework 1 Solutions
I have posted my solutions to Homework 1
to help you study.
[02-Oct-2024] Theoretical Midterm
The theoretical midterm will be given in class on October 2.
Here is a list of ideas to help you review:
- Review HW1 questions and answers.
- Know the proof of Taylor's theorem.
- Be able to state Newton's method, the secant method and
the bisection method.
- Show that Newton's method is quadratically convergent.
This is a closed-book closed-notes only scientific-calculator-allowed
in-class exam. Efforts have been made to keep the arithmetic simple.
If it turns out to be complicated, that's either because I made a mistake
or you did. In either case, do the best you can and check your work
where possible. While getting the right answer is nice, this is not an
arithmetic test. It's more important to clearly explain what you did
and what you know.
[20-Sep-2024] Lab Activity 2
On September 20 we will have a computing lab in class.
Please make sure to attend.
[26-Aug-2024] Welcome Fall 2024
I am looking forward to seeing you August 26 starting the first
week of class.
We will be using WebCampus to turn in written homework--either
scanned from pencil and paper or prepared digitally using an
iPad or similar device.
There will also be a number of in-class computing labs.
In addition to the computing labs and written homework, there will
be two exams and a final exam. In person attendance
is mandatory for all exams and the final.
Grading
Theoretical Midterm 50 points
Practical Midterm 50 points
Homework 50 points
Computer Labs 50 points
Final 100 points
------------------------------------------
300 points total
Exams and quizzes will be interpreted according to the following
grading scale:
Grade Minimum Percentage
A 90 %
B 80 %
C 70 %
D 60 %
The instructor reserves the right to give plus or minus grades and
higher grades
than shown on the scale if it is believed they are warranted.
Quiz and Exam Schedule
There will be two midterms one covering your knowledge of the theory
and another covering your ability to use a computer to perform
practical computations. The final exam will test both. In
person attendance is mandatory for all exams and the final.
Aug 26 -- Newton's Method
Aug 28 -- Taylor's Theorem
Aug 30 -- Fixed-point Methods
Sep 02 ***Labor Day***
Sep 04 -- Newton with Julia
Sep 06 -- Lab 1 Quadratic Equations
Sep 09 -- More Fixed-point Methods
Sep 11 -- The Secant Method
Sep 13 -- The Bisection Method
Sep 16 -- Review of Newton's Method
Sep 18 -- Secant with with Julia
Sep 20 -- Lab 2 Newton's Method
Sep 23 -- Solution by Elimination
Sep 25 -- Gaussian Elimination and Pivoting
Sep 27 -- LU Factorization with Julia
Sep 30 -- Vector and Matrix Norms
Oct 02 -- Theoretical Midterm
Oct 04 -- Ill-conditioning
Oct 07 -- Minkowsky Inequality
Oct 09 -- The Matrix p Norm
Oct 11 -- The Condition Number
Oct 14 -- Double Precision Arithmetic
Oct 16 -- Interpretation of Condition Number
Oct 18 -- Power Method for the Matrix 2 Norm
Oct 21 -- Finding the Norm of Inv(A)
Oct 23 -- Lab 3 FLOPS when Solving Ax=b
Oct 25 ***Nevada Day***
Oct 28 -- Solution of Nonlinear Systems
Oct 30 -- The Contraction Mapping Theorem
Nov 01 -- Lab 4 The Spectral Norm
Nov 04 -- Derivative Condition for Contraction
Nov 06 -- Relaxation and Newton for Systems
Nov 08 -- Lab 5 Nonlinear Systems
Nov 11 ***Veteran's Day***
Nov 13 -- Quadratic Convergence of Newton for Systems
Nov 15 -- Lab 6 Jacobi's Eigenvalue Algorithm
Nov 18 -- Convergence of Jacobi's Eigenvalue Algorithm
Nov 20 -- Gerschgorin Theorems
Nov 22 -- Lab 7 Plotting, Gerschgorin Theorems
Nov 25 -- Householder Reflectors
Nov 27 -- QR Factorization
Nov 29 ***Family Day***
Dec 02 -- Computer Exam
Dec 04 -- Comparison of Householder with Gram-Schmidt
Dec 06 -- Least Squares with QR Factorization
Dec 09 -- Least Squares Example
Dec 11 ***Prep Day***
Dec 18 Final Exam in DMSC106 from 12:45-2:45pm
Course Policies
Communications Policy
Lectures and classroom activities will held in person.
If you wish to set up an appointment for office hours
please send me a message through
WebCampus.
Late Policy
Students must have an approved university excuse to be eligible for a
make-up exam. If you know that you will miss a scheduled exam please
let me know as soon as possible.
Plagiarism
Students are encouraged to work in groups and consult resources outside
of the required textbook when doing the homework for this class. Please
cite any sources you used to complete your work including Wikipedia, other
books, online discussion groups, generative AI such as ChatGPT as well
as personal communications. Note that answers obtained from any source
should be verified and fully understood for homework to have a positive
learning outcome. In all cases your sources need to be cited.
Exams and quizzes, unless otherwise noted, will be closed book, closed
notes and must reflect your own independent work.
Academic Conduct
Bring your student identification to all exams.
Work independently on all exams and quizzes.
Behaviors inappropriate to test taking may disturb other
students and will be considered cheating.
Don't send electronic messages, talk or pass notes with other
students during a quiz or exam.
Homework may be discussed freely.
When taking a quiz or exam
don't read notes or books unless explicitly permitted.
Sanctions for violations are specified in the
University Academic Standards Policy.
If you are unclear as to what constitutes cheating,
please consult with me.
Statement on Academic Success Services
Your student fees cover usage of the University Math Center, University
Tutoring Center, and University Writing and Speaking Center. These
centers support your classroom learning; it is your responsibility to
take advantage of their services. Keep in mind that seeking help outside
of class is the sign of a responsible and successful student.
Equal Opportunity Statement
The University of Nevada Department of Mathematics and Statistics
is committed to equal opportunity in education
for all students, including those with documented physical disabilities
or documented learning disabilities. University policy states that it is
the responsibility of students with documented disabilities to speak
with the Disability Resource
Center during the first week of each semester to discuss appropriate
accommodations to ensure equity in grading, classroom experiences and
outside assignments.
For assistance with accessibility, or to report an issue,
please use the
Accessibility
Help Form. The form is set up to automatically route your request
to the appropriate office that can best assist you.
Diversity
This course is designed to comply with UNR Core Objective 10.
More information about the core curriculum may be found in the
UNR Catalog
here.
Statement on Audio and Video Recording
Surreptitious or covert video-taping of class or unauthorized audio
recording of class is prohibited by law and by Board of Regents
policy. This class may be videotaped or audio recorded only with the
written permission of the instructor. In order to accommodate students
with disabilities, some students may be given permission to record class
lectures and discussions. Therefore, students should understand that
their comments during class may be recorded.
Final Exam
The final exams will be held in person at the time listed in
the standard schedule of final exams for this section. Namely,
the final exam is Wednesday, December 18, 2024
from 12:45-2:45pm in DMS106.
Last Updated:
Mon Aug 26 11:42:57 AM PDT 2024