Math/CS 466/666

Fall 2025 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
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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 1:00-1:50pm in DMS 238 and through Zoom by appointment
Homepage:: http://fractal.math.unr.edu/~ejolson/466/
Course Textbook:: T.A. Driscoll and R.J. Braun, Fundamentals of Numerical Computation, Julia Edition, SIAM, 2022.; Online web version Julia edition of the textbook.; Unified Julia, MATLAB and Python edition of the textbook.
Supplemental References:: Hosking, Joe, Joyce and Turner, First Steps in Numerical Analysis, 2nd Edition, Arnold, 1996.; Germund Dahlquist and Ake Bjorck, Numerical Methods, Dover, 2003.; Clemens Heitzinger, Algorithms with JULIA, Springer, 2022.
Other Books:: 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.; Endre Suli, David F. Mayers, An Introduction to Numerical Analysis, 1st Edition, Cambridge University Press, 2003.; Jeffery Leader, Numerical Analysis and Scientific Computation, Pearson, 2004.

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

The Julia 1.10 Language, official documentation and software download.
Support Materials in Julia, Python and Matlab for the Textbook.
Thomas Breloff, Plots--Powerful Convenience for Visualization in Julia.

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

Lab 1 due Sep 12
Lab 2 due Sep 26
Lab 3 due Oct 10
Lab 4 due Nov 7

Homework

Homework 1: Exercises 1.1.1(a,b,c), 1.1.3(a,b); Exercises 1.2.1, 1.2.2, 1.2.3, 1.2.4; Exercises 1.3.3; Exercises 1.4.2(a,b,c,d), 1.4.3(a,b,c); Exercises 2.1.1(a,b,c); Exercises 2.2.4; Exercises 2.3.1; Exercises 2.4.1(a), 2.4.5, 2.4.6; Exercises 2.5.1(a,b,c,d), 2.5.2(a,b,c,d), 2.5.5, 2.5.7; Exercises 2.6.1(a); Exercises 2.7.2(a,b), 2.7.6(a,b,c), 2.7.9(a,b); Exercises 2.8.6(a,b,c), 2.8.7(a,b)
Homework 2: Exercises 3.1.2(a,b), 3.1.5(a,b); Exercises 3.2.4, 3.2.8(a-c); Exercises 3.3.1, 3.3.4, 3.3.5, 3.3.8(a-c); Exercises 3.4.6; Exercises 4.1.1(a-c); Exercises 4.2.1(a-c), 4.2.2(a-c), 4.2.3(a-d), 4.2.5; Exercises 4.3.1(a-e), 4.3.2(a-e), 4.3.3(a-e), 4.3.5, 4.3.6; Exercises 4.4.1(a-e), 4.4.2(a-e), 4.4.3(a-e), 4.4.5, 4.4.7; Exercises 4.5.1, 4.5.2, 4.4.4(a-d), 4.4.5(a-c)

Lecture Notes

Lecture 1: Floating Point
Lecture 2: Double Precision
Lecture 3: Arithmetic
Lecture 4: Conditioning
Lecture 5: Vandermonde Matrix
Lecture 6: Matrix Operations
Lecture 7: Outer Products
Lab 1: Quadratic Equations
Lecture 8: LU Factorization (mylu.jl)
Lecture 9: Asymptotic Efficiency
Lecture 10: Row Pivoting
Lecture 11: Matrix Norms (myplu.jl)
Lecture 12: The Matrix 1-Norm
Lecture 13: Matrix Conditioning
Lecture 14: Least Squares
Lecture 15: Conditioning of Normal Equations
Lecture 16: Householder Reflectors
Lecture 17: QR Factorization
Lecture 18: Programming Considerations
Lecture 19: Bisection Method (bisect.js, bisect2.jl)
Lecture 20: Fixed Point Iteration
Lecture 21: Newton's Method
Lecture 22: Inverse Interpolation
Lecture 23: Polynomial Interpolation Theorem
Lecture 24: Inverse Interpolation Example
Lecture 25: Inverse Interpolation Convergence Rate
Lecture 26: Newton for Systems
Lecture 27: Symbolic Computation
Lecture 28: Difference Quotients
Lecture 29: Optimal Value of h

Announcements

[29-Oct-2025] Theoretical Midterm

There will be a theoretical midterm in class covering the theory problems from the homework along with select topics from the lectures.

Theoretical Homework

1.1.1, 1.2.1(abcd), 1.2.2(abc), 1.2.3(abc), 1.2.4, 1.4.2(a)
2.1.1(ab), 2.3.1, 2.5.1(abcd), 2.5.2(abcd), 2.5.7, 2.7.2(ab)
2.7.6(abc), 2.7.9(ab), 2.8.7(ab)

Lecture Notes

Definition of machine epsilon (Lecture 1).
Why computer arithemetic + and * are non-associative (Lecture 3).
Relative condition number κ_f(x)=|xf'(x)|/|f(x)| (Lecture 4).
Definition of induced matrix norm ||A|| (Lecture 11).
Compute matrix-1 and matrix-∞ norms (Lecture 12).
Definition of matrix condition number κ=||A^-1|| ||A|| (Lecture 13).
Proof that ||x^*-x||/||x|| ≤ κ||b^*-b||/||b|| (Lecture 13).
Here x^* is an approximation of the solution to Ax=b and b^*=Ax^*.
Definition of Householder reflector H (Lecture 16).
Proof that H^TH=I (Lecture 16).
Explanation of the bisection method algorithm (Lecture 19).
Statement of Newton's method (Lecture 21).
Prove quadratic convergence |ε_n+1| = ε_n²|f''(η_n)|/|2f'(x_n)| for Newton's method (Lecture 21).
Here ε_n=x_n-r and η_n is some point between r and x_n.

Extra Credit and for Graduate Students

Proof of the polynomial interpolation theorem (Lecture 23).

[23-Oct-2025] Homework 1 Solutions

Homework solutions are now posted here please look at the solutions and note any questions you may have regarding them. We will discuss these solutions on Monday October 27 as part of the review for the midterm.

[20-Oct-2025] Homework 1 Due

Please make sure to turn in the three homework problems assigned to you before October 20. This is important to give everyone time to study for the Midterm on October 29.

[02-Oct-2025] Homework 1

Each person in the course has been randomly assigned to turn in solutions for three homework problems selected from chapters 1 and 2. The list of which problems you should work is now available. This homework is due in a couple weeks and should be turned in electronically through WebCampus. Do not write your name on the pages you turn in. After that I will post your work for the class to read online.

[22-Sep-2025] Extra Credit

There were two problems for graduate students and extra credit mentioned in class today.

Show that lim_p→∞ ||x||_p = max{ |x₁|, |x₂|, ..., |x_n| }.
Read the paper Estimating the Matrix p-norm by Higham, Numerische Mathematik, Vol. 62, 1992, pp. 539-555 and create a program in Julia to compute ||A||_p for values of p>1.

[10-Sep-2025] Student Discount

The student discount code for the textbook is available but password protected here. Since this discount is available only to enrolled students, the username and password needed for access are given in WebCampus. I will also explain how to obtain the discount code in class.

[03-Sep-2025] Student Discount

I have requested a 20 percent student discount for the printed version of our textbook. Please wait until I hear from the publisher. I am expecting to receive a code that you can use when ordering.

[23-Aug-2025] Setting up Julia

We will be using the Julia edition of Fundamentals of Numerical Computation by Driscoll and Braun. I will set up all necessary software including the Julia programming language and support files for the text in the computing lab reserved for the course. Since the software is free and open source you may wish to set it up at home or on a laptop computer. Instructions how to set up your computing environment are here.

Note that the textbook in available online in a Julia only version as well as a combined Julia, Matlab and Python version. I will focus on Julia in the course. If you are an expert with Matlab or Python you may choose to use those instead; however, I will be unable to provide as much help with those languages.

[25-Aug-2025] Welcome Fall 2025

I am looking forward to seeing you August 25 starting the first week of class.

There will be a number of in-class computing labs and quizzes, a theoretical exam, a practical exam and a final exam. In person attendance is mandatory for all computing labs, quizzes, exams and the final.

Grading

     Theoretical Midterm       50 points
     Practical exam            50 points
     5 Computing Labs          10 points each
     3 Homework Assignments    10 points each
     Final                    100 points
    ------------------------------------------
                              280 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.

Calendar

Aug 25 -- Chapter 1.1: Floating-Point Numbers
Aug 27 -- Chapter 1.2: Problems and Conditioning
Aug 29 -- Chapter 1.3: Algorithms

Sep 01 ***Labor Day***
Sep 03 -- Chapter 1.4: Stability
Sep 05 -- Chapter 2.1: Polynomial Interpolation

Sep 08 -- Chapter 2.2: Computing with Matrices
Sep 10 -- Chapter 2.3: Linear Systems
Sep 12 -- Computing Lab 1

Sep 15 -- Chapter 2.4: LU factorization
Sep 17 -- Chapter 2.5: Efficiency of Matrix Computations
Sep 19 -- Chapter 2.6: Row Pivoting

Sep 22 -- Chapter 2.7: Vector and Matrix Norms
Sep 24 -- Chapter 2.8: Conditioning of Linear Systems
Sep 26 -- Computing Lab 2

Sep 29 -- Chapter 2.9: Exploiting Matrix Structure
Oct 01 -- Chapter 3.1: Fitting Functions to Data
Oct 03 -- Chapter 3.2: The Normal Equations

Oct 06 -- Chapter 3.3: The QR Factorization
Oct 08 -- Chapter 3.4: Computing QR Factorizations (optional)
Oct 10 -- Computing Lab 3

Oct 13 -- Chapter 4.1: The Rootfinding Problem
Oct 15 -- Chapter 4.2: Fixed-point Iteration
Oct 17 -- Chapter 4.3: Newton's Method

Oct 20 -- Chapter 4.4: Interpolation-based Methods
Oct 22 -- Chapter 4.5: Newton for Nonlinear Systems
Oct 24 -- Chapter 4.6: Quasi-Newton Methods

Oct 27 -- Chapter 4.7: Nonlinear Least Squares (optional)
Oct 29 -- Theoretical Midterm
Oct 31 ***Nevada Day***

Nov 03 -- Chapter 5.1: The Interpolation Problem
Nov 05 -- Chapter 5.2: Piecewise Linear Interpolation
Nov 07 -- Computing Lab 4

Nov 10 -- Chapter 5.4: Finite Differences
Nov 12 -- Chapter 5.5: Convergence of Finite Differences
Nov 14 -- Chapter 5.6: Numerical Integration

Nov 17 -- Chapter 6.1: Basics of Initial-Value Problems
Nov 19 -- Chapter 6.2: Euler's Method
Nov 21 -- Computing Lab 5

Nov 24 -- Chapter 6.3: IVP Systems
Nov 26 -- Chapter 6.4: Runge-Kutta Methods
Nov 28 ***Family Day***

Dec 01 -- Chapter 6.5: Adaptive Runge-Kutta
Dec 03 -- Chapter 6.6: Multistep Methods
Dec 05 -- Practical Exam

Dec 08 -- Chapter 6.7: Implementation of Multistep Methods
Dec 10 ***Prep Day***

Dec 17 ***Final Exam at 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.

AI Policy

In this course you are welcome to use generative artificial intelligence/large language model tools (such as ChatGPT, Claude, Gemini, Grok, Perplexity, etc.). Using these tools aligns with the course learning outcomes/student goals for an in-depth understanding of calculus.

Please be aware that many AI companies collect and store personal information. Please do not enter your confidential information as part of a prompt.

Also, please note that some of these large language models may make up or hallucinate information. These tools may reflect misconceptions and biases of specific data. Students are responsible for checking facts, finding reliable sources for, and making a critical examination of any work that is submitted.

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.

Plagiarism

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.

Diversity

This course is designed to comply with the UNR Core Objective 10 requirement on diversity and equity. More information about the core curriculum may be found in the UNR Catalog.

Statement on Academic Success Services

Your student fees cover usage of the University Math Center, (775) 784-4433; University Tutoring Center, (775) 784-6801; and University Writing & Speaking Center, (775) 784-6030. 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.

Statement of Disability Services

Any student with a disability needing academic adjustments or accommodations is requested to speak with me or the Disability Resource Center (Pennington Achievement Center Suite 230) as soon as possible to arrange for appropriate accommodations. This course may leverage 3rd party web/multimedia content, if you experience any issues accessing this content, please notify your instructor

Mental Health Support Statement

There are times when you may experience difficulties in life, and you may benefit from seeking help. Mental health services are available to you as a student at no additional cost through Counseling Services at the Pennington Student Achievement Center. This includes same-day in-person and tele mental health initial consultations, brief individual counseling, and group counseling sessions. Limited same-day appointments can be scheduled online via Counseling Services or by calling 775-784-4648. Additional brief drop-in "Let's Talk" student consultations are also available in the Counseling Services Annex located at the southwest corner of Great Basin Hall.

Veteran Statement

Veterans, Reservists, National Guard and military connected family members may wish to check the office of Veteran Services for benefits and support. Besides processing VA educational benefits, the department offers a variety of programs year-round to support student academic and personal success while transitioning to higher education and throughout your educational experience. They welcome inquiries regarding VA benefits and assist in navigating resources, the campus, and in the Reno community.

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 17, 2025 from 12:45-2:45pm in DMS106.

Last Updated: Mon Aug 18 10:26:44 AM PDT 2025