- Gautschi
- Numerical analysis 2nd edition. BirkhĂ¤user/Springer 2012 Walter Gautschi

- 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

Spectral methods, Review of class.

- Readings
- Everything below
- Trefethen, Chapter 6, 7
- Final Review Slides
- Julia
- Spectral differentiation (html) (ipynb)

A-stable methods for ODEs and A-stability and eigenvalues.

- Readings
- Section 5.9
- Trefethen, Chapter 6, 7
- Julia
- Stiff ODEs (html) (ipynb)

Global error analysis and step length control

- Readings
- Section 5.7, 5.8, 5.9
- ODE Error
- Julia
- ODEs and Error Analysis (html) (ipynb)

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

- Readings
- Section 5.3, 5.4, 5.5, 5.6
- ODE Theory

Forward and Backward Euler and systems of equations

- Readings
- Section 5.6.1

Review of Chapter 4 and introduction to ODEs

- Chapter 4
- [Section 5.1][Gaustchi]

Remote lecture on the convergence of the secant method, the method of fixed points for nonlinear equations and Newton's method for nonlinear equations.

Newton's method and fixed points (Guest lecture by Ahmed Sameh)

- Readings
- Section 4.6, 4.7

Rates of convergence, method of false position, secant method.

- Readings
- Section 4.2, 4.4
- Julia
- How quickly sequences converge (html) (ipynb)

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.

Gauss-Hermite quadrature and Richardson extrapolation

- Readings
- Section 3.2.4, 3.2.7

Midterm review

Midterm

Review for the midterm!

- Readings
- Everything below
- Midterm Review Slides

We reviewed the relationship between the zeros of orthogonal polynomials, quadrature, and eigenvalues. Then we discussed quadrature via the method of undeteremined coefficients, which is a handy way to compute these!

- Readings
- Integration and Quadrature, Chapter 12, Spectral Methods in Matlab, Trefethen
- Section 3.2.2, 3.2.3, 3.2.5
- Julia
- Orthogonal polynomials, eigenvalues, and quadrature (html) (ipynb)

- Interpolatory quadrature via integrating polynomials exactly and getting
- better accuracy via the node polynomial. We ended on how orthogonal
- polynomials arise in quadrature.

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

- Readings
- Gautschi Section 3.2 (Trapezoidal and Simpson's rule)

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. There is some julia code to illustrate some non-obvious effects here.

- Readings
- Gautschi Section 3.1
- Julia
- Approximating derivatives (html) (ipynb)

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

- Readings
- Gautschi Section 2.3
- Chapter 11 This has a good treatment of piecewise interpolants from the basis function perspective.

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.

- Readings
- Gautschi Section 2.2
- Julia
- Hermite interpolation (html) (ipynb) (wrong name, it's really Newton)

Guest lecture by Youhan Fang (our TA!) on Barycentric interpolation and Newton interpolation.

- Readings
- Gautschi Section 2.2
- Berrut and Trefethen, Barycentric Lagrange Interpolation
- Julia
- Evaluating Polynomial Interpolants (html) (ipynb)

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

- Readings
- Gautschi Section 2.2

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

- Readings
- Gautschi Section 2.1
- Chapter 6

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

- Readings
- Gautschi Section 2.0
- Trefethen, Numerically computing with functions
- Chapter 1-3
- Links
- chebfun
- ApproxFun
- Julia
- Approximating functions with Julia (html) (ipynb)

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.

- Readings
- Gautschi Section 2.0
- Sample variance computations
- Julia
- Julia In-class

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

- Readings
- Gautschi Section 1.4, 1.5
- Julia
- Julia Intro

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

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

- 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)

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

- 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

We introduced the class and reviewed the syllabus.

- Readings
- Syllabus
- What is numerical analysis, Trefethen
- Fast inverse square root, Wikipedia
- Handouts
- First week survey
- Slides
- Intro slides
- Julia
- Julia Intro