Math 467/667

Spring 2021 University of Nevada Reno

467/667 NUMERICAL METHODS II (3+0) 3 credits

Instructor  Course                             Time            Room
Eric Olson  Math 467/667 Numerical Methods II  TR 3:00-4:15pm  Remote

Course Information

Eric Olson
ejolson at unr dot edu
Through Zoom by appointment.

Required Texts and Equipment

  1. Arieh Iserles, A First Course in the Numerical Analysis of Differential Equations, Second Edition, 2008. Note that an online copy of this book is available from the UNR library.
  2. Home computer running Windows, Linux or MacOS and a suitable web camera. Note that it is possible to provision certain mobile phones as web cameras for use on a computer.

Supplemental Texts on Numerical Methods

  1. Anthony Ralston and Philip Rabinowitz, A First Course in Numerical Analysis, Second Edition, Dover, 1978.
  2. Justin Solomon, Numerical Algorithms: Methods for Computer Vision, Machine Learning and Graphics, CRC Press, 2015.
  3. David Kincaid and Ward Cheney, Numerical Analysis: Mathematics of Scientific Computing, 3rd Revised Edition, Pure and Applied Undergraduate Texts, American Mathematical Society, 2002.
  4. R.J. Hosking, S. Joe, D.C. Joyce, J.C. Turner, First Steps in Numerical Analysis, 2nd Edition, Hodder Education Publishers, 1998.
  5. R.W. Hamming, Numerical Methods for Scientists and Engineers, Second Edition.
  6. Kendall Atkinson, An Introduction to Numerical Analysis, Second Edition, Wiley, 1989.
  7. Eugene Isaacson, Analysis of Numerical Methods, Revised Edition, Dover Books on Mathematics, 1993.

Information about Software

Student Learning Outcomes

Upon completion of this course students will be able to
  1. Use Runge-Kutta and Multistep methods to solve IVP's for ODE's.
  2. Use energy-conserving IRK methods to solve conservative ODE's.
  3. Use techniques to solve elliptic, parabolic and hyperbolic PDE's.

Lecture Notes

Here are lecture notes from the distance learning classes to help people catch up who may have experienced technical difficulties during the lecture. Attendance is mandatory and will be taken starting with the second week of class. Don't forget to check WebCampus for graded discussions, pending homework assignments, quizzes and the schedule of Zoom lectures.


[15-May-2021] Project 2 Solutions

I have made solutions to project 2. Please let me know if you see any errors in my work.

[12-May-2021] Final Exam

The final exam is scheduled for Wednesday, May 12 from 2:30-4:30pm through alternative remote. Please make sure you have a web camera available for the final exam.

In preparing for the final exam please consider the following topics, techniques and questions:

  1. There were five video assignments and one reading assignment. Of these six participation activites, which one did you find the most meaningful. Explain why, what inspired you and something you learned.
  2. Explain in a mathematical way how the logarithm of the shift operator is connected with the derivative? Provide an example based on the finite-difference method of how this connection can be used to construct a numerical approximation.
  3. Compare and contrast the advantages and disadvantages of explicit schemes versus implicit schemes for approximating the solutions to ordinary differential equations.
  4. Define the terms truncation error, convergence and stability. Explain the significance of each concept. List one other property a numerical scheme might possess and explain the importance of that property.
  5. What's the difference difference between a Runge-Kutta method and a multistep method? Discuss how these two kinds of methods are used and when one might be preferred over the other.
  6. Given a particular numerical scheme be able to find the truncation error and linear stability domain.
  7. Given a multistep method be able to use the root condition to determine whether the method is convergent.
  8. Be able to convert an RK tableau into computer code that approximates a given initial value problem.
In the review on Tuesday we will discuss the above items, details about how the final will be administered and what to do if there are technical difficulties. Please let me know if you have any questions or concerns.

[13-May-2021] Programming Project 2 Due

The final report in pdf format for Programming Project 2 is due on WebCampus by May 13, 2021. Please upload your report sooner if possible. It may be updated multiple times up until the final deadline.

[01-May-2021] Solutions to Homework 2

I have made solutions to homework 2. Please let me know if you see any errors in my work.

[29-Apr-2021] In-Class Exam (written part)

There will be a closed-book in-class exam on Thursday administered through Zoom covering the following topics:
  1. The proof that Euler's method converges as in the lecture notes or Theorem 1.1 in the textbook.
  2. The proof that the n-point Gaussian Quadrature formula is exact for polynomials of degree 2n-1 as in the lecture notes or Theorem 3.3 in the textbook.
  3. The definition of linear stability domain, A-stability and how these notions effect the maximum step-size that can be used when approximating the solution to a differential equation.
  4. What it means intuitively for a differential equation to be stiff.
  5. What it means for a differential equation y'=f(t,y) to preserve a quadratic invariant of the form y(t)·Sy(t) where S is a symmetric matrix.
  6. [Extra Credit and for Math 667] The definition of the famous matrix M and why M=0 implies the corresponding RK method exactly preserves all quadratic invariants up to rounding error as in Theorem 5.4.

[07-May-2021] Video Participation 6 Due

Please watch the video Thinking Machines Corporation the CM-1 and CM-2, read the Wikipedia page Connection Machine and then answer the following questions.
  1. Who created the Connection Machine?
  2. What is a data parallel computer?
  3. Are GPUs also data parallel? Explain.
  4. Give an example of a computational task that can be sped up using data parallel techniques?
Your answers are due on WebCampus by May 7, 2021 and will be counted as part of your participation grade. Note that many of these questions are open ended.

[18-Apr-2021] Project 1 Solutions

I have made solutions to project 1. Please let me know if you see any errors in my work.

[16-Apr-2021] Homework 2

Homework 2 is due on April 16 and consists of the following problems from Chapter 3 and Chapter 4. Please scan and upload your work to WebCampus. Try to upload your assignment ahead of time and let me know if you have any difficulties.

[09-Apr-2021] Video Participation 5 Due

Please watch the video Can We Make Quantum Technology Work? and answer the following questions.
  1. When Leo Kouwenhoven went to university, what subject was his first choice of study?
  2. List some possible applications for quantum computing.
  3. What is a qubit? In theory how much faster is a 32-qubit quantum computer compared to a 16-qubit quantum computer?
  4. Predict when, if ever, a quantum computer will be available for student use at UNR. Explain your reasoning.
Your answers are due on WebCampus by April 9, 2021 and will be counted as part of your participation grade. Note that many of these questions are open ended.

[30-Mar-2021] Solutions to Homework 1

I have made solutions to homework 1. Please let me know if you see any errors in my work.

[26-Mar-2021] Reading Participation 4 Due

The article Why Julia is Turning Heads in 2021 by Nicole Hemsoth interviews Keno Fischer a long-time developer of Julia and co-founder of Julia Computing. The article explains that While this is obviously difficult to do, I chose Julia for this course not only because of its growing popularity, but because of how it combines interactive convenience with the execution speed of a compiled language. I particularly liked where Fischer said I know Julia is not perfect, especially when it displays a stack trace with too many error messages for a simple mistake. Hopefully you've not been struggling with this too much.

Please read the article linked above and answer the following questions:

Your answers are due on WebCampus by March 26, 2021 and will be counted as part of your participation grade.

[12-Mar-2021] Midterm Exam (computer part)

The midterm will be given through Proctorio and available throughout the weekend starting Friday morning March 12 through Monday night March 15. Please make sure you have a working webcamera and microphone. Do not wait for the last day as there may be technical problems that need to be solved before the deadline.

In preparing for the midterm please know the following topics and techniques:

  1. Given a numerical scheme for approximating an ODE, be able to compute the truncation error of the scheme.
  2. Under the assumption that the method converges, explain how the order of the truncation error is related to the expected order of the resulting ODE solver.
  3. Convert between the explicit form of a multistep method and the polynomials rho(w) and sigma(w).
  4. Use the root condition to prove a multistep method is convergent.
  5. Given a polynomial rho(w) the satisfies the root condition be able to solve for sigma(w) to find a multistep method with a specified order.
  6. [Extra Credit] If one is given sigma(w) is it possible to find a function rho(w) that constructs a multistep method? If having a high order of convergence is not necessary is it always possible to find such a rho(w) that satisfies the root condition?
  7. Be able to rescale the [-1,1] interval of a Gaussian quadrature method to approximate an integral in an arbitrary interval [a,b].
  8. Be able to translate an RK tableaux into the actual equations used to compute a numerical scheme and vice versa.
  9. Be familiar enough with Julia and Mathematica to perform some calculations using Euler's method and/or Gaussian quadrature.
In the review on Thursday we will discuss the above items, details about how the exam will be administered and what to do if there are technical difficulties. Please let me know if you have any questions or concerns.

[12-Mar-2021] Homework 1

Homework 1 is due on March 12 and consists of the following problems from Chapter 1 and Chapter 2. Please scan and upload your work to WebCampus. Try to upload your assignment ahead of time and let me know if you have any difficulties.

[19-Mar-2021] Programming Project 1 Due

The final report in pdf format for Programming Project 1 is due on WebCampus by March 19, 2021. Please upload your report sooner if possible. It may be updated multiple times up until the final deadline.

[05-Mar-2021] Video Participation 3 Due

Please watch the video The Citadel Campus and answer the following questions.
  1. Where is the Citadel Campus located?
  2. What three energy sources are used to generate the electricity used at the Citadel Campus?
  3. A Tier 4 datacenter guarantees 99.995% availability. How many minutes could a datacenter be unavailable per year and still meet these requirements?
  4. What is the advertised network latency from Reno to Las Vegas? How does this compare to the theoretical limits based on the speed of light found in Video Participation 2?
Your answers are due on WebCampus by March 5, 2021 and will be counted as part of your participation grade. Note that many of these questions are open ended.

[19-Feb-2021] Video Participation 2 Due

Please watch the video Grace Hopper on Letterman and answer the following questions.
  1. Who is Grace Hopper?
  2. How many nanoseconds does it take to travel at the speed of light from Reno to Las Vegas?
  3. How many picoseconds does each clock cycle of a CPU running at 2.8 Ghz take?
  4. Research other sources of information about Grace Hopper and relate something you found interesting which doesn't appear in the video.
Your answers are due on WebCampus by February 19, 2021 and will be counted as part of your participation grade. Note that many of these questions are open ended.

[05-Feb-2021] Video Participation 1 Due

Please watch the video Introducing Aurora, Argonne's Intel-Cray Exascale Supercomputer and answer the following questions.
  1. Where is Argonne National Laboratory?
  2. List some planned applications for the Aurora Supercomputer.
  3. What is an exascale computer and what does exascale mean?
  4. What are the possible benefits of mixing numeric simulation with artifical intelligence?
Your answers are due on WebCampus by February 5, 2021 and will be counted as part of your participation grade. Note that many of these questions are open ended.

[04-Feb-2021] Mathematica

We will occasionally use the computer algebra system Mathematica to derive numeric schemes and analyze their convergence properties. It is available to all students at the UNR DataWorks Remote Desktop. Mathematica also comes free with a Raspberry Pi, the $35 credit-card sized computer developed in the United Kingdom for teaching computer science.

[26-Jan-2021] First Day of Class

We will meet over Zoom at 3:00pm. Please see WebCampus for the meeting link.

[21-Jan-2021] Mathematics and Statistics Book Club

You are invited to join the Department of Mathematics and Statistics Book Club! This club will create opportunities for everyone to meet, discuss and learn from one another in a positive, supportive environment. Faculty and students are all welcome!

The first book we will read and discuss is "Mathematics for Human Flourishing" by Francis Su. The e-book is available for free at the UNR Knowledge Center. Here is a brief introduction where the author explains his view of human flourishing.

We will meet via Zoom on Tuesdays from 1:30 to 2:30pm starting in Spring 2021. The schedule is

Please email Diana Moss or Amit Saini in the Department of Mathematics and Statistics for the Zoom link if you plan to attend or have questions.

I'll also put a link on our WebCampus page for your convenience. Note that this is not a mandatory assignment but just an announcement of an activity you might find fun while sheltering at home to escape the epidemic.

[20-Jan-2021] Zoom for Students

Information on Zoom for Students is available here. If you sign into Zoom ahead of time using your UNR student account at the UNR Zoom website, you can enter the online class lectures directly and bypass the waiting room.

[20-Jan-2021] Zoom

I will be giving online interactive lectures through the Zoom Video Conferencing system integrated into WebCampus. If possible, please install and test this software before the first day of class. Note that the university has sponsored Zoom accounts for every student. Accounts may be activated by visiting You do not need to pay Zoom any money to use this software on your home computer. My understanding is that study rooms may be reserved in Mathewson-IGT Knowledge Center and equipment checked out from the @One Digital Media and Technology Center by students who need a suitable location to attend lectures delivered over Zoom.

[20-Jan-2021] WebCampus

This course will be delivered through the UNR WebCampus a customized version of the Canvas learning managment platform. According to the documentation Canvas supports access from Windows, MacOS and Linux using current and first previous major releases of the Chrome, Firefox, Edge and Safari browsers. If you are having trouble accessing WebCampus from home or on campus, please contact the UNR OIT Helpdesk.

[20-Jan-2021] Julia

Julia is a free open-source software designed at MIT for performing matrix and vector computations similar to Matlab. This language is quickly becoming popular in science, technology, engineering and mathematics because it is easy to use and generally performs faster than Matlab. Click and install versions can be downloaded for Windows, macOS and Linux from the official Julia language website. If you try to download it over summer and encounter difficulties, please let me know.

[20-Jan-2021] Alternative Remote

This course was originally scheduled to be delivered in-person, but has moved to entirely online due to social distancing and capacity limitations. We will be using a combination of Zoom, WebCampus and other Internet resources which will be announced later. Luckly, this course will not include the additional $34 per credit online fee; however, please make sure you have a computer, suitable web camera and the Internet connection needed for online learning. More information is available at the UNR Coronavirus Information for Students webpage.


     Midterm (written part)    50 points
     Midterm (computer part)   50 points
     2 Homework Assignments    40 points each
     2 Programming Projects    50 points each
     Attendence/Participation  20 points
     Final Exam               100 points
                              400 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 he believes they are warranted.

Course Schedule

The University will be implementing four reading days and an additional no-instruction Day during the Spring 2021 semester. These five days will replace Spring Break. Classes (including this one) will not be in session on these days, although depending on course goals, some laboratory and studio classes will still be scheduled.
Jan 25-Jan 29   Week 1: 1.1-1.2 Euler's Method
Feb 01-Feb 05   Week 2: 1.3-1.4 Theta Method
Feb 08-Feb 12   Week 3: 2.1-2.2 Adams Bashforth

*** President's Day Monday Feb 15

Feb 16-Feb 19   Week 4: 2.3 Backwards Difference Formulas
Feb 22-Feb 24   Week 5: 3.1 Gaussian Quadrature

*** Reading Day Thursday Feb 25

Mar 01-Mar 05   Week 6: 3.2-3.3 Runge Kutta
Mar 11-Mar 12   Week 7: Review, Midterm

*** Reading Day Tuesday Mar 9
*** Reading Day Wednesday Mar 10

Mar 15-Mar 19   Week 8: 3.4-4.1 Implicit RK and Stability

*** No-Instruction Day Wednesday Mar 24

Mar 22-Mar 26   Week 9: 4.2-4.3 Linear and A Stability

*** Spring Break Cancelled

Mar 29-Apr 02   Week 10: Chapter 5 or 6 or other
Apr 05-Apr 09   Week 11: optional topics
Apr 12-Apr 16   Week 12: 8.1-8.2
Apr 19-Apr 23   Week 13: 8.3

*** Reading Day Wednesday Apr 21

Apr 26-Apr 30   Week 14: 9.1-9.2
May 03-May 04   Week 15: Review

*** Prep Day May 5
*** Final exam Wednesday, May 12 from 2:30 to 4:30am

Course Policies

Communications Policy

Lectures and classroom activities will be held online through Zoom at the scheduled meeting time listed in MyNevada for this course. Please check the canvas page for the Meeting ID and Join URL under the Zoom tab. To promote an open communication through this interactive environment, video attendance will be mandatory and count as participation in your final grade. If you wish to set up an appointment for office hours please send me a message through WebCampus and ask through chat after one of the online lectures.

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. Homework may be turned in late--with a possible deduction of points depending on the circumstances--as long as I have not already graded the assignment. When attending a Zoom lecture for the course, it's always better to be late than never.


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 as well as personal communications. Exams and quizzes, unless otherwise noted, will be closed book, closed notes and must reflect your own independent work. Please consult the section on academic conduct below for additional information.


A web camera will be required for this course in order to comply with university requirements for identity verification. Bring your student ID to all online quizzes and Zoom lectures as if attending class on campus. At the beginning of each class please send a quick hello through chat and a quick goodbye at the end. This will indicate to me that you are ready and also count towards your attendance and participation score.


This course is designed to comply with but not satisfy the UNR Core Objective 10 requirement on diversity and equity. More information about the core curriculum may be found in the UNR Catalog here.

COVID-19 Policies

Statement on COVID-19 Training Policies

Students must complete and follow all guidelines as stated in the Student COVID-19 Training modules, or any other trainings or directives provided by the University.

Statement on COVID-19 Face Coverings

In response to COVID-19, and in alignment with State of Nevada Governor Executive Orders, Roadmap to Recovery for Nevada plans, Nevada System of Higher Education directives, the University of Nevada President directives, and local, state, and national health official guidelines face coverings are required at all times while on campus, except when alone in a private office. This includes the classroom, laboratory, studio, creative space, or any type of in-person instructional activity, and public spaces. A "face covering" is defined as a covering that fully covers a person's nose and mouth, including without limitation, cloth face mask, surgical mask, towels, scarves, and bandanas (State of Nevada Emergency Directive 024). Students that cannot wear a face covering due to a medical condition or disability, or who are unable to remove a mask without assistance may seek an accommodation through the Disability Resource Center.

Statement on COVID-19 Social Distancing

Face coverings are not a substitute for social distancing. Students shall observe current social distancing guidelines where possible in accordance with the Phase we are in while in the classroom, laboratory, studio, creative space (hereafter referred to as instructional space) setting and in public spaces. Students should avoid congregating around instructional space entrances before or after class sessions. If the instructional space has designated entrance and exit doors students are required to use them. Students should exit the instructional space immediately after the end of instruction to help ensure social distancing and allow for the persons attending the next scheduled class session to enter.

Statement on COVID-19 Disinfecting Your Learning Space

Disinfecting supplies are provided for you to disinfect your learning space. You may also use your own disinfecting supplies.

Contact with Someone Testing Positive for COVID-19

Students must conduct daily health checks in accordance with CDC guidelines. Students testing positive for COVID-19, exhibiting COVID-19 symptoms or who have been in direct contact with someone testing positive for COVID-19 will not be allowed to attend in-person instructional activities and must leave the venue immediately. Students should contact the Student Health Center or their health care provider to receive care and who can provide the latest direction on quarantine and self-isolation. Contact your instructor immediately to make instructional and learning arrangements.

Local, State and Federal COVID-19 Information

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 Mathematics Department 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.

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.

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 over Zoom or in the classroom 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.

Final Exam

The final exam is scheduled for Wednesday, May 12 from 2:30-4:30pm through alternative remote. Please make sure you have a web camera available for the final exam.
Last Updated: Mon Jan 18 14:41:57 PST 2021