## AMCS/MATH 603: Introduction
to Numerical Analysis I

Spring 2021

**Instructor: **Charles L. Epstein
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### The Course

Numerical Analysis is a study of the interface between the idealized world
of continuum mathematics and the realities of finite, discrete numerical
computation. The principal motivation for most numerical analysis was to
develop methods for approximately solving equations, ODEs and PDEs. There
are a variety of topics that mirror those occuring in continuum mathematics.
Most of the course will focus on the 1d-case, though of course most
current research in the field focuses on higher dimensional problems.
Indeed, "Machine Learning" is largely devoted to the problem of
approximating functions of many variables. We will take material from
a variety of sources, among them:

- Rivlin, T.,
*An Introduction to the Approximation of Functions*,
Dover Press **Rivlin**
- Stoer and Bullirsch,
*Introduction to Numerical Analysis, *Springer **
S & B**
- Trefethen, L.N.,
*Approximation Theory and Approximation Practice*,
SIAM **ATAP**
- Trefethen, L.N.,
*Spectral Methods in MATLAB*,
SIAM
**SMM**

A problem set will
be assigned every other week on Tuesday, which will
be due two
weeks hence. The problem sets appear at the bottom of this
web-page. Everything will be posted on the AMCS 603 Canvas page.

- The class meets on TuTh from 12:00-1:30 online!

- My office hour will be Mondays 3:00-4:00PM. Contact me by e-mail for
an appointment if you can not come during this time.
- My office in the Math Department is 4E7 DRL, tel. 8-8476. But I'm very
unlikely to be there!
- email: cle@math.upenn.edu.
Send
e-mail if you have a question or need to contact me.

# Syllabus

- Representing functions (Rivlin Chapters 1,2, 4, and ATAP
Chapters 6-10, for practical aspects of the subject)
- Notions of optimal approximation, normed linear spaces.
- Best polynomial/trigonometric approximant.
- Rates of convergence and smoothness of the function.
- L
^{2}-approximation and orthogonal polynomials.
- 3-term recurrence relations, and practical computation of
polynomials approximants.
- Rational approximation. (if time permits)

- Interpolation (S &B Chapter 1, ATAP 1-5, 10-16)
- Polynomial interpolation.
- Some algorithms.
- Effects of the choice of interpolation points, Lebesgue constants,
the Runge phenomenon.
- Trigonometric interpolation, the FFT.
- Chebyshev points.
- Splines (if time premits).

- Calculus: (S&B Chapter 3 and 5, ATAP Chapters 18, 19, 21, SMM
Chapters 1-6, 12)
- Solving equations.
- Bisection.
- Newton's method.
- Gradient descent.

- Integration.
- Newton-Cotes.
- Euler-MacClurin summation formula.
- Gaussian quadrature.

- Differentiation
- Finite differences.
- Spectral methods.

- Applications to ODEs and PDEs (S & B Chapter 7, SMM
7-10)(As time permits)
- Initial value problems for ODEs.
- Multistep methods (Runge-Kutta, predictor-corrector).
- Boundary value problems for ODEs.
- Some PDEs.

### Announcements

- This class will have its first meeting on Thursday, January 21, 2021.

### Problem sets

### Online resources

Return to cle's home page.