# Maria K. Cameron

### University of Maryland, Department of Mathematics

### Introduction to Numerical Analysis
Spring 2020

##### Syllabus

###
Introduction: sources of errors, the absolute and the relative errors.

- 1. Computer numbers: integers, floating-point numbers, computer arithmetics, and the IEEE standard, overflow and underflow.

- 2. Nonlinear equations in 1D: fixed-point iteration, contraction map, basic methods (bisection, Newton’s, secant, hybrid), the order of convergence.

- 3. Linear algebra: vector and matrix norms, inner products, condition numbers, PLU for solving linear systems of algebraic equations, least squares, QR decomposition, SVD decomposition.

- 4. Interpolation and approximation of functions: polynomial interpolation in Lagrange’s and Newton’s form, Chebyshev interpolation, spline interpolation

- 5. Numerical differentiation: basic formulas for f' and f'', derivation of higher-order rules, the Richardson extrapolation, the optimal choice of h.

- 6. Numerical quadrature: basic rules (rectangle, midpoint, trapezoid, Simpson's), composite rules, the Euler-Maclaurin summation formula, the Romberg integration, adaptive quadrature, Gaussian quadrature.

- [1] G. W. Stewart, Afternotes on Numerical Analysis, SIAM, 1996 (freely available online: http://epubs.siam.org.proxy-um.researchport.umd.edu/doi/book/10.1137/1.9781611971491)
(Links to an external site.)
- [2] D. Bindel and J. Goodman, Principles of Scientific Computing

- [3] James Demmel, Applied Numerical Linear Algebra, SIAM, 1997 (freely available online via http://www.lib.umd.edu)

- [4] Maria Cameron, Lecture notes on some topics will be posted on ELMS in ‘Files/ Lecture notes’

- [5] A. Gil, J. Segura, N. Temme, Numerical Methods for Special Functions, Book 2007, Publisher: SIAM, ISBN: 978-0-89871-634-4, available at research gate: https://epubs.siam.org/doi/book/10.1137/1.9780898717822
(links to an external site.)