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

Instructor Course Time Room ---------------------------------------------------------------------- Eric Olson Math 466/666 Numerical Methods I TR 3:00-4:15pm Remote

- Instructor:
- Eric Olson
- email:
- ejolson at unr dot edu
- Office:
- Through Zoom by appointment.
- Homepage:
- http://fractal.math.unr.edu/~ejolson/466/

- Anthony Ralston and Philip Rabinowitz, A First Course in Numerical Analysis, Second Edition, Dover, 1978.
- 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.

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

- The Julia Project,
*The Julia 1.5 Language*, official documentation. - Thomas Breloff,
*Plots--Powerful Convenience for Visualization in Julia*, official documentation. - Jupyter Lab,
*A Comprehensive List of Links for the Juptyer Project*, official documentation.

- Students will be able to implement a numerical method to solve a nonlinear equation using the bisection method and Newton's method.
- Students will be able to solve linear systems using direct and iterative methods.
- Students will be able to construct interpolating functions.

- August 25 -- Course Outline and Section 1.2 Types of Errors
- August 27 -- Sections 1.4 and 1.5 Rounding Errors
- September 1 -- Sections 1.4, 1.5 and 2.2 Floating Point Precision
- Bisection Example JupyterLab

- September 3 -- Section 3.1 Lagrange Polynomial in Julia
- Lagrange Interpolation JupyterLab

- September 8 -- Section 3.2 Theorem on Error in Interpolating Polynomials
- September 10 -- Section 1.3 Matrix Norms
- Matrix Norms JupyterLab

- September 15 -- Section 3.3 Binomial Theorem
- September 17 -- Section 3.3 Newton Polynomial
- September 22 -- Section 3.3 Newton Polynomial in Julia
- Finite Differences JupyterLab (pdf)

- September 29 -- Section 3.6 Inverse Interpolation
- Inverse Interpolation JupyterLab

- October 1 -- Section 8.3 The Secant Method
- October 6 -- Section 8.3 Order of Convergence
- Secant Method JupyterLab

- October 8 -- Section 8.4 One step Methods
- October 13 -- Section 8.7 Aitkin's δ
^{2}Process- Linear Convergence JupyterLab (pdf)

- October 15 -- Aitkin's δ
^{2}Process in Julia- Accelerated Convergence JupyterLab (pdf)

- October 20 -- Multiple Roots and Gaussian Elimination
- October 22 -- Carefully Counting FLOPS
- October 27 -- Section 9.4 Condition Number
- November 3 -- Section 9.6 Matrix Iterative Methods
- Simple Iteration JupyterLab

- November 5 -- Section 9.7 Matrix Iterative Methods
- Jacobi Iteration JupyterLab

- November 10 -- Section 9.7.2 Gauss-Seidel Method
- Positive Definite Matrices JupyterLab

- November 12 -- Proof of Theorem 9.3
- November 17 -- QR and Hausholder Reflectors
- November 19 -- The Matrix 2-norm of A
- Hausholder Reflector Example JupyterLab

- November 24 -- The Power Method
- Computing the Matrix 2-norm JupyterLab

- December 1 -- The Spectral Mapping Theorem
- December 3 -- The Inverse Power Method
- Inverse Power Example JupyterLab (pdf)

- December 8 -- Review for the Final Exam
- Marked up version of the Sample Final

You must complete two out of three specified computations. For each computation I will provide a program written in Julia that is missing some code. Your task is to fill in the missing lines so the program runs correctly. You could also create a new program of your own using C, Fortran, Matlab, Python or a different language. In preparation for the quiz, I would recommend that you look over all of the programs written this semester to make sure you understand how they work. In particular, please know how to perform the following tasks using a computer:

- Use the bisection method to search for a critical value where a system, process or function changes state or behavior.
- Construct an interpolating polynomial
p(x) of degree n-1 passing through the points
(x
_{i},f(x_{i})) where i=1,...,n and then evaluate it at a specified value x=α. - Use the Secant Method to approximate x such that f(x)=0.
- Use Aitkin's δ
^{2}process to accelerate a linearly convergent sequence.

Quiz 1 will cover the following topics and tasks:

- Definition and computation of the vector p-norms.
- Definition of the induced or natural matrix p-norm.
- Given a matrix A compute the matrix 1-norm and the ∞-norm.
- Definition of absolute, relative error and significant digits.
- Given an approximation ξ of the exact value x, be able to compute the absolute and relative error in ξ.
- How to contruct the Lagrange interpolating polynomial of
degree n−1 passing
through specified points for
(x
_{i},y_{i}) for i=1,...,n. - Exact statement of the theorem on the error in interpolating polynomials (see the end of the lecture notes from September 8 and also Homework Assignment 1).
- Definitions of the difference operators E, Δ, ∇ and δ.
- Statement of Newton's binomial theorem.
- If f(x) is a polynomial of degree n, be able to prove that
Δ
^{n}f_{j}and ∇^{n}f_{j}are constants.

Here are some worked example problems related to the topics on Quiz 1 which will be given Thursday. This quiz will be closed book and closed notes. Don't forget to study the theorems proofs and definitions as well!

- Commentary with mathematical notation.
- Functioning Julia code.
- Scanned-in pencil-and-paper work.

There was an unfortunate typo in the Julia code used to approximate
the limit which appears in the video. The line which read
`n=2&j` should have been `n=2^j`.

Except for getting the wrong answer, this does not affect the subsequent discussion on how to create a pdf version of the notebook for upload into WebCampus. For reference, the corrected files appear below

Once people have installed Julia and verified it is working, I'll further describe how to install the JupyterLab notebook interface.

COVID-19 Training Quiz 5 points Written Quiz 30 points Computer Quiz 30 points Midterm 50 points 2 Homework Assignments 20 points each 2 Programming Projects 30 points each In-class Lab Work 20 points Final Exam 70 points ------------------------------------------ 305 points totalExams 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.

Aug 24-Aug 28 Sections 1.1-1.3 Error Definitions Sections 1.4-1.6 Computer Arithmetic Aug 31-Sep 04 Section 2.1 Weierstrass Approximation Section 2.2 Bisection Method *** Labor Day Sep 07 Sep 07-Sep 11 Section 3.2 Interpolation Theorem Section 1.3 Matrix Norms Section 3.5 Divided Differences Sep 14-Sep 18 Section 2.3 Peano Kernel Theorem Section 2.4 Undetermined Coefficients Sep 21-Sep 25 Section 3.6 Inverse Interpolation Section 3.7 Hermite Interpolation *** Written Quiz covering 1.1 through 3.2 Sep 24 Sep 28-Oct 02 Section 8.1-8.2 Functional Iteration Oct 05-Oct 09 Section 8.3 The Secant Method Oct 12-Oct 16 Section 8.4-8.5 Newton's Method Oct 19-Oct 23 Section 8.6 Multiple Roots Oct 26-Oct 30 Section 8.7 delta^2 Acceleration *** Computer Quiz covering 8.1 through 8.7 Oct 29 *** Nevada Day Oct 30 Nov 02-Nov 06 Section 9.1-9.3 Gaussian Elimination Nov 09-Nov 13 Section 9.6 Jacobi Iteration Section 9.7 Successive Overrelaxation *** Midterm covering 1.1 through 9.7 Nov 10 *** Veteran's Day Nov 11 Nov 16-Nov 20 Section 9.9 Overdetermined Systems Section 9.10 The Simplex Method Nov 23-Nov 27 Section 10.1 Eigenvectors and Eigenvalues *** Thanksgiving Nov 26 *** Family Day Nov 27 Nov 30-Dec 04 Section 10.2 The Power Method Dec 07-Dec 08 *** Prep Day Dec 9 *** Final exam Dec 16 from 2:30 to 4:30pm

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Last Updated: Sun Aug 2 20:59:53 PDT 2020