Course Website: Additional course material can be found on Desire to Learn
Math 373 - Introduction to Numerical Analysis
Prerequisite: MATH 321 and CSC 150 or CSC 170 or CSC 111 or permission of instructor.
This course is an introduction to numerical methods. Topics include elementary discussion of errors, polynomial interpolation, quadrature, non-linear
equations, and systems of linear equations. The algorithmic approach and efficient use of the computer will be emphasized. Additional topics may include: calculation of eigenvalues and eigenvectors, numerical differentiation and integration, numerical solution of differential equations.
Homework and Course Outline
An update on the sequence of topics and supplementary materials will be available
here. All students are responsible for keeping up with the course
.
Prep work for class
Go through the welcome video that is on YouTube playlist (link for the playlist is on D2L)
Watch the introduction to LaTeX on the YouTube playlist (playlist link is on D2L course page)
Setup your LaTeX (you can setup LaTeX on your own computer, or you can use Overleaf, or you can do anything else that generates LaTeX documents as long as you use the templates provided for this class
Assignment for the week- get Matlab on your machine and also get access to LaTeX (either by download of MikTex, or by starting and account in Overleaf). We will start working with these programs on Friday (1/17). Download homework 1 from the D2L course page.
Download the lecture notes and review the notes for lecture 1 and lecture 2
Strongly encourage students to work through the MATLAB Onramp, which is a free course on the Mathworks website https://www.mathworks.com
Lecture 1 is broken up into five parts: lecture 1.1 (where to find stuff for MATH 373), lecture 1.2 (Getting MATLAB), lecture 1.3 (Introduction to MATLAB), lecture 1.4 (Getting Started in LaTeX), and lecture 1.5 (A few things to start with in the course)
(8/21) --We will start with lecture 2 in the class notes and then circle back to lecture 1 in the class notes. Please see lecture 2.1 and lecture 2.2 on the YouTube playlist (playlist link can be found on the D2L course page).
(8/24) --Introduction to Matlab and LaTeX. Check out the short videos on the YouTube playlist on D2L that give introductions to Matlab and LaTeX. In addition, lecture 1 in the notes provides information. If time allows, we will discuss the program template file and how to manage the program reports.
Recall that lecture 1.3 is a quick introduction to Matlab and lecture 1.4 is a getting started video for LaTeX, both can be found on the YouTube playlist (link for the YouTube playlist is on the D2L course page).
The YouTube playlist will have part a and part b videos covering the MATLAB for section 1.1. Please review both videos.
The YouTube playlist has a video for going over the sample programming assignment program 0 and the LaTeX report for program 0. You can find all the materials in the content module for programs under the submodule for program 0 on the D2L course page. The sample program of progz123456.m is also in the same submodule and is sample of m-file for programming assignment of program 0. Program 0 is just and example and is not assignment you have to complete, it should provide good reference for what is required for the programming assignments.
(8/26) Read section 1.1, 1.2, and 1.3 in LB16. Measuring error, sources of error, Taylor's Polynomial, and convergence.
The material to read is sections 1.5 and 1.6 out of the class notes along with lecture 3 out of the class notes. An optional resource you can consult is Chapter 1 from KK09.
The YouTube playlist has three videos for lecture 3
Video for lecture 3.1 is about Taylor's Polynomial and calculating error bounds using the remainder term
Video for lecture 3.2 is on machine arithmetic and sources of error
Video for lecture 3.3 is about convergence, numerically estimating order of convergence, and how to analyze convergence
There is also a bonus (older) video on order of convergence that is on the YouTube playlist for extra reference
(8/31) Lecture 4 in the notes covers an introduction to Interpolation. We will revisit interpolation again, but lecture 4 will provide a brief introduction. There is a video for lecture 4.1 on the YouTube Playlist.
Read lecture 4 in the notes
Video for lecture 4.1 is on linear interpolation
Video for lecture 4.2 is on Newton's Divided Difference
Programs newdivdiff.m and newdivdiff2.m are part of lecture 4.2
(9/2) Newton's Divided Difference is covered in the class notes in lecture 4.2 and there is a video for 4.2 on the YouTube playlist. A good reference on the application of Newton's Divided Difference is section 5.3 of KK09. The LB16 book does present the same topic in section 3.3, but the notation is different from the presentation in the notes and KK09. We introduce Newton's Divided Difference to provide a nice challenge in your Matlab programming, if you can program Newton's Divided Difference then you are definitely ready for the programming we will have in this course.
(9/4) Getting to know your Matlab
Homework 1 in problems 3.A, 3.B, and 3.C all involve simple Matlab programming. Along with questions in part 4 that involve using Matlab like a calculator.
Homework 2 has several parts that involve programming in Matlab.
You should also be in position to start work on Program 1
(9/7) Root Finding with Bisection
Review the videos on bisection that are on the YouTube playlist
Bisection is covered in 2.1 in LB16 and section 3.3 in KK09
Bisection is also mentioned in lecture 5 in the class notes.
(9/16) Solving Systems of Nonlinear equations is the subject of lecture 6 from the class notes.
There is a wrap up on the numerical solvers for a nonlinear equation that is given in the notes at the end of lecture 5
Lecture 6 in the notes explains the extension of how to apply these same methods to solve a system of nonlinear equations, please see the notes for details.
(9/18) Return of interpolation. Program 1 due
Lecture 7 in the class notes provides the overview of what we are going to cover along with the listing of the specific sections of LB16 and KK09 that would be of interest.
The YouTube playlist has two videos (lecture 7 part 1, and the second part that is labeled Cubic Splines). In addition, at the end of lecture 7 are additional videos that you can review if you wish.
The material also utilizes directpoly.m, which is an m-file in the D2L MATLAB module that will be helpful with the lecture and the homework.
Lecture 8 from the notes on Numerical Differentiation. Background can also be found in Chapter 2 of KK09 and the first part of 4.1 in LB16.
Check out the YouTube playlist for lecture 8 part 1 on an introduction to numerical differentiation and part 2 on how to prove order of convergence. The end of lecture 8 in the notes also has links to additional videos for reference.
(9/23) Numerical Integration
Lecture 9 in the class notes and see Chapter 7 in KK09
Video Lecture 9 part 1 is a brief introduction to numerical integration on the YouTube playlist
At the end of Lecture 9 in the notes there is a list of other videos you can watch to see more presentations on the topic.
KK09 does cover the fundamentals of integration in Chapter 7
LB16 does cover integration in chapter 4, but it is heavy on the derivation using interpolation, which is not an emphasis we will use in this course. We do reference Table 4.5 in LB16 and you should review that table.
(9/25) Get ready for exam 1 and work on program 3. Quiz 2 on D2L due
(9/28)
(9/30)
(10/2) Exam 1
(10/5) Simpson's Methods
Simpson's 1/3 rule is covered in Chapter 7 section 3 of KK09
Section 9.3 in lecture 9 of the class notes does go over the formulas
The video of Lecture 9 part 2 on the YouTube playlist does go over the formulas and discusses the application of composite methods
(10/7) Order of Convergence and Adaptive Methods are covered in Lecture 9 part 3 video on the YouTube playlist
(10/9) Gaussian Quadrature Program 2 due
KK09 does cover Gaussian Quadrature in Chapter 7 seciton 5
The video, Lecture 9 part 4, on the YouTube playlist also covers Gaussian Quadrature
The LB16 book also presents Gaussian Quadrature at the end of section 4.3
(10/12) We start the large module on solving differential equations and boundary value problems.
We will start by reviewing separation of variables and integration factors, which are techniques you should know from differential equations.
The kick off for Ordinary Differential Equations in LB16 is in section 6.1
KK09 has a nice condense summary of analytical methods in Chapter 8 section 1 and we will make reference to many of the methods mentioned in chapter 8 section 1 of KK09. However, you will only need to know how to implement separation of variables and integration factors for the work we will be doing in this class.
See lectures 10.1, 10.2, and 10.3 on the YouTube playlist
(10/14) Review of ODEs see Lecture 10 in the notes,
The review also includes a review of the analytical methods you will need to use in this class: Separation of Variables, and Integration Factors. You can find videos on the YouTube playlist the give examples of both these methods. We do expect that you have mastery of both methods from Differential Equations.
The review also includes the videos at the end of lecture 10 as additional resources.
(10/16) Euler's Method and Taylor's Methods (see section 6.2 in LB16 and lecture 11 in the class notes ) Quiz 3 on D2L due
Please see the video on the YouTube playlist for Euler's Method and Taylor Methods
A derivation of Euler's Method can also be found in Chapter 8 section 2 of KK09
Lecture 13 in the class notes also contains work on finite differences
The YouTube playlist has two videos on finite differences with one being an introduction and another covering how to use the technique with derivative boundary conditions. The link for the YouTube playlist is on D2L
(11/18) Start Linear Systems (lecture 14 in the notes)
Solving singular and nonsingular systems, along with Gaussian Elmination (covered in Chapter 4, section 6 in KK09).
Check out the video about solving singular systems on the YouTube playlist
(11/20) Solving Linear Systems using Gaussian Elminiation with partial pivoting Quiz 4 on D2L due
(11/23) Condition number and ill-conditioned systems plus Gauss-Seidel (Chap 4 section 8 in KK09)