Math/CS 466/666

Fall 2023 University of Nevada Reno

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

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/

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 Textbook:

Hosking, Joe, Joyce and Turner, First Steps in Numerical Analysis, 2nd Edition, Arnold, 1996.

Supplemental References:

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.

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

Information about Software

• The Julia 1.6 Language, official documentation and software download.
• Thomas Breloff, Plots--Powerful Convenience for Visualization in Julia.

Student Learning Outcomes

• 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

• LAB1 due Sep 8
• LAB2 due Sep 15
• LAB3 due Sep 29
• LAB4 due Oct 20
• LAB5 due Nov 16
• LAB6 due Dec 8

Homework

• HW1 due Sep 20 (solutions)
• HW2 due Oct 2 (solutions)
• HW3 due Oct 25 (solutions)
• HW4 due Nov 20 (solutions)
• HW5 due Dec 16

Lecture Notes

• Lecture 1: Course Outline
• Lecture 2: Types of Errors
• Lecture 3: Relative and Absolute

• Labor Day: no notes
• Lecture 4: Rounding and Addition
• Computer Lab 1: Copy and Paste Hints

• Lecture 5: Rounding and Multiplication
• Lecture 6: Taylor's Theorem
• Computer Lab 2: no notes

• Lecture 7: Evaluating a Polynomial
• Lecture 8: Taylor Series Example
• Lecture 9: Bisection Method

• Lecture 10: False Position and Secant
• Lecture 11: Convergence of Secant Method
• Computer Lab 3: Newton's Method

• Lecture 12: Convergence of Newton's Method
• Theoretical Midterm: no notes
• Lecture 13: Partial Pivoting

• Lecture 14: Backwards Error Analysis
• Lecture 15: The Condition Number
• Lecture 16: Matrix Norms and Singular Values

• Lecture 17: The Gauss-Seidel Method
• Lecture 18: The Power Method
• Computer Lab 4: no notes

• Lecture 19: Finite Differences
• Lecture 20: Polynomials
• Nevada Day: no notes

• Lecture 21: Polynomial Interpolation (table2.jl)
• Lecture 22: Backward Differences (table3.jl)
• Lecture 23: Lagrange Basis Functions

• Lecture 24: Polynomial Interpolation Theorem
• Lecture 25: Newton's Divided Differences (table6.jl)
• Veteran's Day: no notes

• Lecture 26: Linear Least Squares
• Lecture 27: Hausholder Reflectors
• Computer Lab 5: Vandermonde Matrices

• Lecture 27: Hausholder Example
• Lecture 28: Trapezoid Method
• Thanksgiving: no notes

• Lecture 29: Simpson's Method
• Lecture 30: Gaussian Quadrature
• Practical Midterm: no notes

• Lecture 31: Orthogonal Polynomials
• Lecture 32: Taylor Methods for ODEs
• Computer Lab 6: no notes

• Review: Final Exam Review

Announcements

[20-Dec-2023] Final Exam

• Review all checkpoint questions from Steps 1-13, 16-23, 26-27, 29-33.
• 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.
  • How to create a difference table.
  • Use a difference table to find an interpolating polynomial.
• Be able to state Newton's method for solving f(x)=0.
• Know the statement and proof of Taylor's Theorem.
• Know the statement and proof of the polynomial interpolation theorem.
• Know the proof of the rate of convergence of Newton's method.
• State the Gauss-Seidel method for solving Ax=b and explain what kinds of matrices A are appropriate for using this method.
• State the power method for finding the eigenvalue of largest magnitude and a corresponding eigenvector.
• The definition of the Lagrange polynomial basis functions.
• 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 matrix norm ||A||₂ = max{ ||Ax||₂ : ||x||₂ ≤ 1 } corresponding to the Euclidian norm is given by ||A||₂ = max{ λ^1/2 : λ is an eigenvalue of A^TA }.
• Given a matrix A find the Householder reflector for the first step in the QR factorization.
• State Simpson's method for approximating a definite integral.
• Prove the n-point Gauss quadrature method is exact for polynomials of degree 2n-1.
• Use Taylor series to derive the n-th order Taylor method.

[16-Dec-2023] Homework 5

[08-Nov-2023] Computer Lab 6

[01-Dec-2022] Computational Midterm

• Computer labs 1 through 5.
• Homeworks 2 through 4.

[28-Nov-2023] Homework 4 Solutions

[18-Nov-2023] Computer Lab 5

[20-Nov-2023] Homework 4

[29-Oct-2023] Homework 3 Solutions

[20-Oct-2023] Computer Lab 4

[16-Oct-2023] Homework 3

[3-Oct-2023] Homework 2 Solutions

[2-Oct-2023] Summary of Newton's Method

[4-Oct-2023] Theoretical Midterm

• Review all checkpoint questions from Steps 1 to 10.
• Review homework 1 and 2 and be prepared to perform simple calculations and derivations. The computer will not be available for this exam.
• 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.

[29-Sep-2023] Computer Lab 3

[25-Sep-2023] Homework 2

[24-Sep-2023] Homework 1 Solutions

[20-Sep-2023] Homework 1 Due

[15-Sep-2023] Computer Lab 2

[08-Sep-2023] Computer Lab 1

[01-Sep-2023] Homework 1

[28-Aug-2023] Welcome Fall 2023

My lecture notes should complement the notes you take in class. I'd recommend comparing the notes you take with the ones I post after class along with the relevant sections from the text. Then use these three sources of information to create a final version of your notes. In my experience reviewing the lecture in this way is important. Though tempting with an iPad, one should not try make the final version of your lecture notes during class. That takes too much time and omits the comparison and review steps mentioned above.

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

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

Quiz and Exam Schedule

Aug 28 -- Overview of the Text
Aug 30 -- Step 1 Sources of Error
Sep 01 -- Step 2 Approximation to Numbers

Sep 04 ***Labor Day***
Sep 06 -- Step 3 Error Propagation and Generation
Sep 08 -- Lab 1 Quadratic Equations

Sep 11 -- Step 4 Floating Point Arithmetic
Sep 13 -- Step 5 Approximation to Functions
Sep 15 -- Lab 2 Scientific Visualization

Sep 18 -- Step 6 Nonlinear Algebraic and Transcendental Equations
Sep 20 -- Step 7 The Bisection Method
Sep 21 -- Step 8 Method of False Position

Sep 25 -- Step 9 Method of Simple Iteration
Sep 27 -- Step 10 The Newton-Raphson Iterative Method
Sep 29 -- Lab 3 Newton's Method

Oct 02 -- Step 10 Newton's Method (continued)
Oct 04 -- Theoretical Midterm
Oct 06 -- Step 11 Solution by Elimination

Oct 09 -- Step 12 Errors and Ill-conditioning
Oct 11 -- Step 16 Testing for Ill-conditioning
Oct 13 -- Special Topic: Matrix Norms and Singular Values

Oct 16 -- Step 13 The Gauss-Seidel Iterative Method
Oct 18 -- Step 17 The Power Method
Oct 20 -- Lab 4 The Spectral Norm

Oct 23 -- Step 18 Tables
Oct 25 -- Step 19 Forward, Backwards and Central Differences
Oct 27 ***Nevada Day***

Oct 30 -- Step 20 Polynomials
Nov 01 -- Step 21 Linear and Quadratic Interpolation
Nov 03 -- Step 22 Newton Interpolation Formulae

Nov 06 -- Step 23 Lagrange Interpolation Formula
Nov 08 -- Step 26 Least Squares
Nov 10 ***Veteran's Day***

Nov 13 -- Step 27 Least Squares and Linear Equations
Nov 15 -- Step 29 Finite Differences
Nov 17 -- Lab 5 Polynomial Fitting

Nov 20 -- Step 30 The Trapezoidal Rule
Nov 22 -- Step 31 Simpson's Rule
Nov 24 ***Family Day***

Nov 27 -- Step 32 Gaussian Quadrature
Nov 29 -- Step 32 Gaussian Quadrature (continued)
Dec 01 -- Computer Exam

Dec 04 -- Step 33 Single-Step Methods
Dec 06 -- Step 33 Single-Step Methods (continued)
Dec 08 -- Lab 6 Numerical Integration

Dec 11 -- In-class Review
Dec 13 ***Prep Day***

Dec 20 Final Exam from 12:45-2:45pm

Course Policies

Communications Policy

During the epidemic I discovered that Zoom also allowed me to meet individually with students who are sick or can't come to campus just to ask a question. If you wish to set up an appointment for office hours please send me a message through WebCampus.

Late Policy

Plagiarism

Exams and quizzes, unless otherwise noted, will be closed book, closed notes and must reflect your own independent work.

Academic Conduct

If you are unclear as to what constitutes cheating, please consult with me.

Statement on Academic Success Services

Equal Opportunity Statement

Diversity

Statement on Audio and Video Recording

Final Exam