Numerical Analysis I
AMSC 666, Fall 2003
Basic Information
Course Description

Ten steps of Approximation Theory
 General overview
 On the choice of norm: L^{2} vs. L^{¥}
 Weierstrass' density theorem
 Bernstein polynomials
 Least Squares Approximations I. A general overview
 Gramm mass matrix
 illconditioning of monomials in L^{2}
 Least Squares Approximations II. (Generalized) Fourier expansions
 Bessel, Parseval, ...
 Orthogonal polynomials: Legendre, Chebyshev, Sturm's sequence
Assignment #1 [ pdf file ]

Least Squares Approximations III. Discrete expansions
 Examples of discrete least squares.
 From discrete leastsquares to interpolation
Lecture notes: Spectral Expansions
[ pdf file ]
 Interpolation I. Lagrange and Newton interpolants
 Divided differences
 Equidistant points
 Synthetic calculus
 Forward backward and centered formulae

Interpolation II. Error estimates.
 Runge effect
 region of analyticity
Polynomial vs. spline interpolation
[ demonstration ]

Interpolation III. Interpolation with derivatives
 Hermite interpolation
 piecewise interpolation
 Splines
Lecture notes: Interpolation Error
[ pdf file ]
Assignment #2
[ pdf file ]
 Interplolation IV. Trigonometric interpolation
 FFT
 truncation + aliasing
 error estimates
 elliptic solvers
 fast summations (  discrete convolution)

MiniMax Approximation
 Modern aspects of approximation
 Error Estimates  Jackson, Bernstein and Chebyshev
 smoothness and regularity spaces
Numerical Differentiation and Numerical Integration

Numerical differentiation
 Polynomial and spline interpolants  Error estimates
 Equidistant points: synthetic calculus; Richardson extrapolation
 Spline and trigonometric interpolation
Assignment #3
[ pdf file ] and a solution to
Jackson's theorem [ pdf file]

Gauss Quadratures
Lecture notes: A Very Short Guide to Jacobi Polynomials
[ pdf file]

Numerical integration with equidistant points
 NewtonCotes formulae
 Composite Simpson's rule
 Romberg & adaptive integration
Assignment #4 [ pdf file ]
MIDTERM #1 (Q & A's  an example)
[ pdf file ]
MIDTERM #2 (Q & A's  the real one)
[ pdf file ]
Solution of Linear System of Equations  Iterative Methods
 Introduction
 Fixed point iterations
Lecture notes: Matrices  norms, eigenvalues and powers
[ pdf file]
 The basic algorithms:
 Jacobi, GaussSeidel and SOR methods
 Steepest descent; Conjugate gradient method
 ADI and dimensional splitting methods
 Multigrid
 Preconditioners
Assignment #5 [ pdf file ]
Eigensolvers
 Introduction
 Similarity transformations, Rayleigh quotations
 Power and inverse power method
 Householder transformations
 Hessenberg form and the QR method
Assignment #6 [ pdf file ]
 Singular Value decomposition
 Preconditioners
 The divide and conquer method
FINAL (+ answers)
[ pdf file ]
References
GENERAL TEXTBOOKS
K. Atkinson, An INTRODUCTION to NUMERICAL ANALYSIS, Wiley, 1987
S. Conte & C. deBoor, ELEMENTARY NUMERICAL ANALYSIS, McGrawHill
User friendly; Shows how 'it' works; Proofs, exercises and notes
G. Dahlquist & A. Bjorck, NUMERICAL METHODS, PrenticeHall,
User friendly; Shows how 'it' works; Exercises
E. Isaacson & H. Keller, ANALYSIS of NUMERICAL METHODS, Wiley
The 'First'; Proofs; outdated in certain aspects; Encrypted
message in Preface
A. Ralston & P. Rabinowitz, FIRST COURSE in
NUMERICAL ANALYSIS, 2nd ed., McGrawhill,
Detailed; Scholarly written; Comprehensive; Proofs exercises and notes
J. Stoer & R. Bulrisch, INTRODUCTION TO NUMERICAL ANALYSIS, SpringerVerlag
detailed account on approximation, linear solvers & eigensolvers,
ODE solvers,..
B. Wendroff, THEORETICAL NUMERICAL ANALYSIS, Academic Press, 1966
Only the 'Proofs'; elegant presentation
APPROXIMATION THEORY
E. W. Cheney, INTRODUCTION TO APPROXIMATION THEORY
Classical
P. Davis, INTERPOLATION & APPROXIMATION, Dover
Very readable
T. Rivlin, AN INTRODUCTION to the APPROXIMATION of FUNCTIONS
Classical
R. DeVore & G. Lorentz, CONSTRUCTIVE APPROXIMATION, Springer
A detailed account from classical theory to the modern theory; everything; Proofs exercises and notes
NUMERICAL INTEGRATION
F. Davis & P. Rabinowitz, NUMERICAL INTEGRATION,
Everything...
(mainly) ITERATIVE SOLUTION OF LINEAR SYSTEMS
A. Householder, THE THEORY OF MATRICES IN NUMERICAL ANALYSIS
The theoretical part by one of
the grand masters; Outdated in some aspects
G. H. Golub & Van Loan, MATRIX COMPUTATIONS,
The basic modern reference
Y. Saad, ITERATIVE METHODS for SPARSE LINEAR SYSTEMS,
PWS Publishing, 1996. (Available on line at
http://wwwusers.cs.umn.edu/~saad/books.html)
R. Varga, MATRIX ITERATIVE ANALYSIS,
Classical reference for the theory of iterations
J. H. Wilkinson HANDBOOK for AUTOMATIC COMPUTATIONS, 1971
Modern theory started there with the grand master...
D. Young, ITERATIVE SOLUTION OF LARGE linear SYSTEMS, Academic Press, 1971
Excellent detailed account
(mainly) EIGENSOLVERS
B. Parllett, THE SYMMETRIC EIGENVALUE PROBLEM
Recommended
J. H. Wilkinson The ALGEBRAIC EIGEVALUE PROBLEM, 1965
The classical reference
Eitan Tadmor

