Math/CS 466/666

466/666 NUMERICAL METHODS I (3+0) 3 credits

Instructor  Course Section                     Time              Room
------------------------------------------------------------------------
Eric Olson  Math 466/666 Numerical Methods I   MWF noon-12:50pm  DMS106
                                               due to COVID the first
                                               week is on Zoom

Course Information

Instructor:

Eric Olson

email:

Please contact me through WebCampus

Office:

DMS 238 and through Zoom (preferred) by appointment

Homepage:

http://fractal.math.unr.edu/~ejolson/466/

Live Stream:

If you can't come to class due to sickness, quarantine or other reasons, please join via the Zoom link in WebCampus.

Grader:

to be determined (contact through WebCampus)

Course 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.

Supplemental References:

Justin Solomon, Numerical Algorithms: Methods for Computer Vision, Machine Learning and Graphics, CRC Press, 2015.

Endre Suli, David F. Mayers, An Introduction to Numerical Analysis, 1st Edition, Cambridge University Press, 2003.

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

Computer Labs

• Computer Lab 1 (Due Sept 19)
• Computer Lab 2 (Due Sept 30)
• Computer Lab 3 (Due Oct 7)
• Computer Lab 4 (Due Oct 14)
• Computer Lab 5 (Due Nov 9)
• Computer Lab 6 (Due Nov 18)
• Computer Lab 7 (Due Dec 9)

Homework

• HW1 from First Steps in Numerical Analysis (solutions)
  • Step 1 Exercise 1
  • Step 2 Exercise 2abcd
  • Step 3 Exercises 1, 3, 4
  • Step 4 Exercises 1abef, 2abef
  • Step 5 Exercises 1abc, 2abc, 3, 5

• HW2 from First Steps in Numerical Analysis (Due Oct 14) (solutions)
  • Step 6 Exercise 2abc
  • Step 7 Checkpoint 3, Exercise 3 (except to 4D)
  • Step 8 Exercise 2abc
  • Step 9 Exercises 1, 3
  • Step 10 Exercises 2, 5

• HW3 from Numerical Analysis and Scientific Computation (Due Nov 16) (solutions)

  Please use Julia on problems where it is appropriate and do the rest using pencil and paper. Include all input and output from any computations you make on a computer. Sorry for the confusion earlier about where I put the with Julia hint. It is fine but not necessary to do your handwritten work with an iPad or other annotation software. If you decide to present your work in typed form, make sure to include all intermediate steps from your pencil and paper calculations so I can follow.

  • Chapter 2.1 Problems 7abc, 8abc
  • Chapter 2.2 Problems 5ab
  • Chapter 2.3 Problem 15
  • Chapter 2.4 Problem 6ab, 10abc
  • Chapter 2.5 Problem 2ab, 10ab

• HW4 from Numerical Analysis and Scientific Computation (Due Dec 2) (solutions)
  • Chapter 2.6 Problem 4abcde, 14
  • Chapter 2.7 Problem 2, 10ab

• HW5 from Numerical Linear Algebra (Due Dec 12) (solutions)
  • The Spectral Theorem

Lecture Notes

• Lecture 1: Types of Errors
• Lecture 2: Propagation of Error
• Lecture 3: Generated Error

• Lecture 4: Taylor's Theorem
• Lecture 5: Lagrange Form of the Remainder

• Lecture 6: Booting the Linux Image
• Lecture 7: Loss of Precision

• Lecture 8: Bisection Method
• Lecture 9: Loops and Conditionals (bisect.jl)
• Lecture 10: Method of False Position

• Lecture 11: Convergence Rate of Secant Method
• Lecture 12: Fixed-point Methods
• Lecture 13: Derivation of Newton's Method

• Lecture 14: When Newton Converges
• Lecture 15: How Fast Newton Converges
• Lecture 16: More about Newton

• Lecture 17: Numerical Convergence
• Lecture 18: LU Factorization
• Lecture 19: Partial Pivoting with LU

• Lecture 20: Cholesky Decomposition
• Lecture 21: Speed of Cholesky versus LU

• Lecture 22: Backward Error Analysis
• Lecture 23: Matrix Norms

• Lecture 24: The Spectral Norm
• Lecture 25: The Condition Number
• Lecture 26: Reduced QR Decompostion

• Lecture 27: Householder Reflectors

• Lecture 28: Householder Example
• Lecture 29: Power Iteration

• Lecture 30: More Power Iteration
• Lecture 31: Spectral Mapping

• Lecture 32: Rayleigh Quotient Iteration
• Lecture 33: Cubic Convergence

• Lecture 34: Hessenberg Form
• Lecture 35: Shifted QR Iteration

• Lecture 36 Remarks on HW 5

Information about Software

• The Julia 1.6 Language, official documentation and software download.
• Thomas Breloff, Plots--Powerful Convenience for Visualization in Julia.
• Jupyter Lab, A Comprehensive List of Links for the Juptyer Project.

Announcements

[16-Dec-2022] Final Exam

• Review all checkpoint questions from Steps 1 to 10.
• Review homeworks 1 to 5 and be prepared to perform simple calculations related to
  • Significant digits, relative error, rounding error.
  • Propagated error, generated error.
  • Bisection, secant and Newton methods.
  • Matrix norms ||A||_∞ and ||A||₁.
  • The Cholesky factorization of a 2x2 positive definite matrix.
  • Show that ||A||_s=||A^T||_s.
• Be able to state Newton's method for solving f(x)=0.
• Know the proof of the rate of convergence of Newton's method.
• The definition of the condition number of a matrix.
• Know how to use the condition number for backward error analysis.
• Know how to prove that the spectral norm ||A||_s = ρ(A^T A)^1/2.
• The Rayleigh quotient iteration (Algorithm 27.3) for the eigenvector problem.
• Given a matrix A find the Householder reflector for the first step in
  • The QR factorization.
  • Making the Hessenberg form.

[10-Dec-2022] Homework 4 Solutions

[9-Dec-2022] In-class Computer Lab 7

[2-Dec-2022] Computational Midterm

• Computer labs 1 through 6.
• Homework 2.

[21-Nov-2022] Homework 3 Solutions

[18-Nov-2022] In-class Computer Lab 6

[9-Nov-2022] In-class Computer Lab 5

[21-Oct-2022] Theoretical Midterm

• Review all checkpoint questions from Steps 1 to 10.
• Review the homework and be prepared to perform simple calculations with a calculator related to error estimates and root finding. The computer will not be used for this exam.
• Know the proof of Taylor's theorem.
• Know the proof of the rate of convergence of Newton's method.

[17-Oct-2022] Homework 2 Solutions

[14-Oct-2022] In-class Computer Lab 4

[14-Oct-2022] Homework 2 Due

[7-Oct-2022] In-class Computer Lab 3

[30-Sep-2022] In-class Computer Lab 2

[21-Sep-2022] Homework 1 Solutions

For an experiment I used an iPad with pencil, folio keyboard and the Notability app to make these solutions. Although Julia doesn't run on an iPad, it was possible to log in to a Linux computer via secure shell and access it with the Terminus app. After that most of my work was done split screen with Terminus on one side and Notability on the other. The iPad experienced a hard crash at one point and had to be power cycled, but otherwise things worked reasonably well.

As far as I can tell secure shell hangs up in about 15 seconds from an iPad if you swipe it into the background. Does anyone know how to open a secure shell session that remains connected as one swipes between other apps?

Update: Apparently the way to keep a task running in the background on an iPad is to register it to receive location data from the GPS. It turns out that Terminus has the option to request location data and after enabling that my connection to the server doesn't hangup anymore.

[16-Sep-2022] Homework 1 Due

[28-Aug-2022] Welcome Fall 2022

Do not come to class if you are sick--even if it's something other than COVID-19. If you are subject to quarantine because of exposure to the disease, please stay home. If you are already sick or in quarantine (like I am) and can't come on the first day of class, check the WebCampus page for this course later today for the Zoom link and other information.

While this section of Math 330 is not high-flex, in anticipation of increased absences due to the epidemic I will live-stream our class meetings each day at a link available in WebCampus and maintain an online archive of course materials including lecture notes, assignments and other announcements. Unless there is a change in policy, in-person attendance will be required for all exams and the final.

Last year I discovered that Zoom worked well for office hours as it provided greater flexibility to meet with students: Our meeting times can be arranged around individual schedules and there is no need to go to campus just to ask a question. Please send me a message on WebCampus to schedule all office hours.

As in previous semesters, students should avoid congregating around instructional space entrances before or after class sessions and 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. Note that students who cannot wear a face covering due to a medical condition or disability, or who are unable to remove a mask without assistance may seek accommodation through the Disability Resource Center.

Grading

     Theoretical Midterm       75 points
     Practical Midterm         75 points
     Homework                  50 points
     Computer Labs             50 points
     Final                    100 points
    ------------------------------------------
                              350 points total

    Grade        Minimum Percentage
      A                 90 %
      B                 80 %
      C                 70 %
      D                 60 %

Quiz and Exam Schedule

Course Policies

Communications Policy

Late Policy

Plagiarism

Diversity

COVID-19 Policies

Statement on COVID-19 Training Policies

Statement on COVID-19 Social Distancing

Statement on COVID-19 Disinfecting Your Learning Space

Contact with Someone Testing Positive for COVID-19

Local, State and Federal COVID-19 Information

• Return of the Pack
• Reno/Sparks/Washoe Regional Information Center
• State of Nevada Heath Response
• CDC: Coronavirus

Statement on Academic Success Services

Equal Opportunity Statement

Statement on Audio and Video Recording

Academic Conduct

Final Exam