
Least squares approximation of sign(x) on [1,1] with polynomials of degree 10,20,40

Course information
SYLLABUS: Information about exams, homeworks, grades
 Since Fall 2015 the course AMSC 666 has a new syllabus, combining the most important topics of the former AMSC666, AMSC667. The goal was to make this course more attractive:
 students who only want to take one graduate numerical analysis course get to see more relevant topics
 students who are interested in topics like numerical PDE methods can more quickly take
courses like AMSC714, AMSC715
 The AMSC qualifying exam rules were updated accordingly: The sequences
AMSC666, AMSC714 and AMSC666, AMSC715
were added to the list of permitted 2course sequences
 AMSC/CMSC 466 (Introduction to Numerical Analysis) is a prerequisite for this course.
You should be familiar with the basic topics of numerical methods:
 The idea of AMSC/CMSC666 is to build on these results and cover more advanced techniques and theory. I will give a brief review of the AMSC466 results which we use in this class.
 I recommend the following book:
E. Süli: "An Introduction to Numerical Analysis",
Cambridge University Press, 2003
Chapters 1,2,6,7 in this book contain the prerequisites for this course.
Chapters 8,9,10,12,13,14 are related to the material I want to cover in this course.
Additional Course Material
Topics
 Approximation Theory (3 weeks, [1,2,3])
 Vector, Matrix and Functional Norms
 Least Squares, QR, SVD
 Orthogonal Polynomials
 Chebyshev Expansions
 Gaussian Quadrature
 Numerical Solution of InitialValue Problems (3 weeks, [4,5,6])
 Consistency, Stability, and Convergence Analysis
 OneStep Methods, RungeKutta Methods
 Multistep Methods
 Methods for Stiff Problems
 Error Estimation and Adaptivity
 Iterative Methods for Linear Algebraic Systems (3 weeks, [7,8,9,10])
 Motivation: BoundaryValue Problems for Elliptic PDEs
 Classic Iterative Methods: Jacobi, GaussSeidel, SOR methods
 Conjugate Gradient Method and Preconditioning
 GMRES
 Optimization (3 weeks, [9,10,11])
 Steepest Descent, Newton and QuasiNewton Methods
 Line Search Methods and Trust Region Methods
 Rates of Convergence
 Nonlinear Conjugate Gradient Method
 Nonlinear Least Squares Problem
Literature
 Lloyd N. Trefethen, Approximation Theory and Approximation Practice, SIAM 2012
first six chapters,
Errata
 James Demmel, Applied Numerical Linear Algebra, SIAM, 1997
freely available online via UMD Library
 Rivlin, T. J., an Introduction to the Approximation of Functions, Dover, 1969
 Gil, A., Segura, J., Temme, N., Numerical Methods for Special Functions, SIAM, 2007
chapter on Chebyshev Expansions is freely available online
 Hairer, E., Norsett, S.P., Wanner, G., Solving Ordinary Differential Equations I. Nonstiff Problems, Second Revised Edition, Springer 1993
 Hairer, E., Wanner, G., Solving Ordinary Differential Equations II. Stiff and DifferentialAlgebraic Problems, Second Revised Edition, Springer 1993
 Deuflhard, P., Bornemann, F., Scientific Computing with Ordinary Differential Equations, Springer, 2002
 Elman, H., Silvester, D., Wathen, A., Finite Elements and Fast Iterative Solvers, Second Edition, Oxford Science Publications, 2014
 Morton, K. W., Mayers, D.F., Numerical Solutions of Partial Differential Equations, Second Edition, Cambridge, 2005
 Nocedal, J., Wright, S., Numerical Optimization, Second Edition, Springer 2006
freely available online via UMD Library
 Kelley, C. T., Iterative Methods for Linear and Nonlinear Equations, SIAM 1995
 Kelley, C. T., Iterative Methods for Optimization, SIAM 1999
freely available online via UMD Library
Matlab programming
We will use Matlab to see how various algorithms perform (or fail).
Matlab can be downloaded for free from TERPware.
How to hand in Matlab results for homeworks:
 You have to write an mfile for each problem. (Typing commands at the >> prompt and printing this out is not allowed)
 You have to hand in the mfile together with the output and graphics.
The easiest way is to use the publish command in Matlab
(if you call additional mfiles, you have to print them out separately).
 How to use the publish command in Matlab:
Example 1: Using publish with the mfile sample.m
generates sample.html (which you can print out and hand in).
Example 2: this mfile shows how to format each problem. This is
the generated output.
More information about publishing markup