Readings and topics

Main reference

Gautschi: Numerical analysis 2nd edition. Birkhäuser/Springer 2012 Walter Gautschi

Other references

ATAP: Lloyd N. Trefethen. Approximation Theory and Approximation Practice, SIAM
First course: Uri M. Ascher and Chen Greif. A first course in numerical methods, SIAM

Last office hour

Video: Zoom-2021-05-06

Lecture 28

A-stable methods for ODEs and A-stability and eigenvalues. Spectral methods, Review of class.

Readings: Everything below; Section 5.9; Trefethen, Chapter 6, 7; Final Review Slides
Julia: Spectral differentiation (html) (ipynb)
Notes: Lecture 28
Video: Zoom-2021-04-29

Lecture 27

Global error analysis and step length control

Readings: Section 5.7, 5.8, 5.9; ODE Theory
Video: Zoom-2021-04-27
Notes: Lecture 27
Julia: ODEs and Error Analysis (html) (ipynb)

Lecture 26

One-step methods for ODEs and an intro to error analysis.

Readings: Section 5.3, 5.4, 5.5, 5.6, 5.8, 5.9; ODE Theory; ODE Theory
Video: Zoom-2021-04-22
Notes: Lecture 26
Julia: ODEs and Error Analysis (html) (ipynb)

Lecture 25

One-step methods for ODEs and an intro to error analysis.

Readings: Section 5.3, 5.4, 5.5, 5.6; ODE Theory
Video: Zoom-2021-04-20
Notes: Lecture 25

Lecture 24

Intro to ODEs and Forward Euler.

Readings: [Section 5.1][Gaustchi]; Section 5.6.1
Video: Zoom-2021-04-15
Notes: Lecture 24

Lecture 23

Exam 2

Lecture 22

Newton's method and fixed points and exam 2 review.

Readings: Section 4.6, 4.7
Notes: Lecture 22 [Missing a piece because the app crashed!]
Video: Zoom-2021-04-06
Handouts: Exam 2 Review

Lecture 21

I decided to re-use the materials from a previous class recording (2016) on the convergence of the secant method, the method of fixed points for nonlinear equations and Newton's method for nonlinear equations. There is a quick new (2021) intro video to give an overview of these ideas. But then we jump into them!

Video: Lecture-21-intro; Lecture-21-Secant-analysis; Lecture-21-Nonlinear-systems-2016
Julia: Secant Convergence Rates (html) (ipynb); 2d Bisection (html) (ipynb)
Notes: Lecture 21
Readings: Section 4.4, 4.9
Julia: Secant Convergence Rates (html) (ipynb); 2d Bisection (html) (ipynb)

Lecture 20

Rates of convergence, method of false position.

Readings: Section 4.2, 4.4
Notes: Lecture 20
Julia: How quickly sequences converge (html) (ipynb)
Video: Zoom-2021-03-30

Lecture 19

Intro to nonlinear equation solving, example problems and the method of bisection.

Readings: Section 4.1, 4.3; Bisecting roots in Julia Some interesting details that arise with bisection in floating point.
Notes: Lecture 19
Video: Zoom-2021-03-25

Lecture 18

We continued with orthogonal polynomials and Gaussian quadrature and finished up with how to compute eigenvalues of the Jacobi matrix to get nodes of the Gauss Quadrature formula! And a bit about Gauss-Hermite quadrature.

Reading: Integration and Quadrature, Chapter 12, Spectral Methods in Matlab, Trefethen; Section 3.2.2, 3.2.3, 3.2.5; Section 3.2.4, 3.2.7 (Not Richardson though)
Notes: Lecture 18
Video: Zoom-2021-03-23
Julia: Orthogonal polynomials, eigenvalues, and quadrature (html) (ipynb)

Lecture 17

Interpolatory quadrature via integrating polynomials exactly and getting better accuracy via the node polynomial. We ended on how orthogonal polynomials arise in quadrature. Then we discussed quadrature via the method of undeteremined coefficients, which is a handy way to compute these!

Reading: Section 3.2.2; Section 3.2.5 Method of undetermined coefficients
Notes: Lecture 17
Video: Zoom-2021-03-16

Lecture 16

We saw the introduction of numerical integration with the Trapezoidal and Simpsons rule and started thinking about interpolatory methods

Reading: Gautschi Section 3.2 (Trapezoidal and Simpson's rule); Section 3.2.2
Notes: Lecture 16
Video: Zoom-2021-03-11

Lecture 15

We saw a number of ways to approximate derivatives starting from using polynomial interpolation and then going through Taylor series and finite difference methods before reviewing Gaustchi's perspective in the textbook.

Readings: Gautschi Section 3.1
Notes: Lecture 15
Video: Zoom-2021-03-09

Lecture 14

We had the first exam.

Lecture 13

We finished up splines and then started into derivates. There is some julia code to illustrate some non-obvious effects here, including a good choice for the value of h in a finite difference approximation. We also reviewed for the first exam on Thursday.

Notes: Lecture 13
Video: Zoom-2021-03-02
Julia: Approximating derivatives (html) (ipynb)
Handouts: Exam 1 Review

Lecture 12

Today was all about splines and piecewise interpolants. We saw how piecewise interpolants are arbitrarily accurate and it was easy to get a good approximation of the best possible piecewise interpolant.

Notes: Lecture 12
Video: Zoom-2021-02-25
Readings: Gautschi Section 2.3; Chapter 11 This has a good treatment of piecewise interpolants from the basis function perspective.

Lecture 11

We reviewed Newton interpolation and used it to derive Hermite interpolation by studying what it would mean to take divided differences at the same point.

Notes: Lecture 11
Video: Zoom-2021-02-23
Readings: Gautschi Section 2.2
Julia: Newton interpolation (html) (ipynb) (wrong name, it's really Newton)
Handouts: Hermite Interpolation (Not finished)

Lecture 10

Barycentric interpolation and Newton interpolation.

Notes: Lecture 10
Video: Zoom-2021-02-18
Readings: Gautschi Section 2.2; Berrut and Trefethen, Barycentric Lagrange Interpolation
Julia: Evaluating Polynomial Interpolants (html) (ipynb)

Lecture 9

Due to the snow emergency, and the daycare closure, we had to improvise a bit. I had grand visions of recording a fun lecture on the Lagrange interpolation formula -- or what I think of as the pointwise cancellation form of the interpolating polynomials -- along with Chebyshev nodes and Chebyshev polynomials. However, my first attempt doing this quickly failed. So in the interest of getting you material as quickly as possible, I'm going to use an old lecture from Ahmed Sameh on this same topic. I'm so sorry I didn't have time to do more for you but I want to make sure you are equipped for the homework.

Guest lecture by Ahmed Sameh on Lagrange interpolation, the error formula, and Chebyshev nodes.

Video: Pointwise and Least Squares Error 2021 Video on a helpful relationship between pointwise error and least squares error; CS514-2016-Lecture-9 (2016 Guest Lecture by Ahmed Sameh)
Readings: Gautschi Section 2.2

Lecture 8

We covered the normal equations and the Weierstrauss approximation theorem, and orthogonal functions.

Notes: Lecture 8
Video: Zoom-2021-02-11
Readings: Gautschi Section 2.1; Chapter 6
Handouts: Intro to Approx Theory

Lecture 7

We saw how to formalize the best approximation problem in terms of function norms. We got through integrals with a discrete measure, then did a quick demo with Julia

Notes: Lecture 7
Video: Zoom 2021-02-09
Handouts: Intro to Approx Theory
Readings: Gautschi Section 2.0; Trefethen, Numerically computing with functions; Chapter 1-3
Links: chebfun; ApproxFun
Julia: Approximating functions with Julia (html) (ipynb)

Lecture 6

We did more Julia, derived the condition number of a matrix, and saw some issues with the variance computation, then started into chapter 2 on functions.

Notes: Lecture 6
Readings: Gautschi Section 2.0; Sample variance computations
Julia: Julia In-class
Video: Zoom 2021-02-04

Lecture 5

Today we completed our study of the overall floating point error computation using the condition number of an algorithm. We saw a few examples too and then had a demo of Julia

Handouts: Computer Error
Notes: Lecture 5
Readings: Gautschi Section 1.4, 1.5
Julia: Julia Intro (2021)
Video: Zoom 2021-02-02; Zoom 2021-02-02-Julia Extra little bit of lecture on Julia

Lecture 4

We introduced condition numbers for a function and studied two types: a sharp estimate based on the condition number of all the constituent gradients and a weaker condition number based on the norm of the Jacobian.

Readings: Gautschi Section 1.3
Notes: Lecture 4
Handouts: Computer Error
Video: Zoom 2021-01-28

Lecture 3

We got through the analysis of the fundamental floating point operations with respect to input errors.

Notes: Lecture 3
Readings: Gautschi Section 1.2; Pentium FDIV Bug
Fun reading that mentions cancellation as a plot point: Once dead by Richard Phillips (See intro to Chapter 55, page 208)
Video: Zoom 2021-01-26; Syllabus

Helpful Julia Videos

These are from another class. They are more introductory than our type of material,
but ought to be helpful. Some of the details are outdated, but most should still work.: Julia-01 - Introduction -- getting started. But see this instead
for the modern equivalent -- https://techytok.com/julia-vscode/ Everything else
should be adaptable.: Julia-02 - Types; Julia-03 - Plots; Julia-04 - Features; Julia-05 - Control and Scope; Julia-06 - Problems; Julia-07 - Birthday Paradox; Julia-08 - Julia Set; Julia-09 - Julia Set P2

Lecture 2

We reviewed sources for numerical errors in computers and got through the format of IEEE Floating Point Arithmetic and how the numbers are stored.

Notes: Lecture 2
Video: Floating Point Accuracy; Zoom 2021-01-21
Readings: Gautschi Section 1.1; Constructions of the real numbers; Decimal arithmetic; Integers and floating point numbers in Julia Most of the things described here for Float64 will also work in Matlab, except Julia has some nicer functions.; IEEE Floating point; Michael Overton, Numerical Computing with IEEE Floating Point Arithmetic - a great textbook on IEEE arithmetic!
Julia: IEEE Floats in Julia

Lecture 1

We introduced the class and reviewed the syllabus.

Readings: Syllabus; What is numerical analysis, Trefethen; Fast inverse square root, Wikipedia
Notes or Slides: Lecture 1
Video: Zoom-2021-01-19
Julia: Julia Intro

Numerical Analysis

David Gleich

Purdue University

Spring 2021

Course number CS-51400, MATH-51400

Online due to COVID-19 Pandemic

Social distance, wear a mask.

Zoom, Tuesday and Thursday, 10:30-11:45