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

- 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.

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

- 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.

- 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 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

- 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^{T}A }. - 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.

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

- 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.

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.

Theoretical Midterm 50 points Practical Midterm 50 points Homework 50 points Computer Labs 50 points Final 100 points ------------------------------------------ 300 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 it is believed they are warranted.

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

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.

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

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

Last Updated: Sat Aug 26 10:48:12 AM PDT 2023