MAPL/CMSC 666-667: A Tentative Syllabus for MAPL/CMSC 666-667

MAPL/CMSC 666: Numerical Analysis (I)
Fall semester, 2000
Instructor: Bo Li

A Tentative Syllabus

Note: an * indicates an optional topic.

Class Topics Remarks
1 Weierstrass Approximation Theorem, Bernstein's proof  
2 space C[a,b], best uniform approximation: existence, uniqueness  
3 Chebyshev Alternation Theorem, Chebyshev polynomials of first kind  
4 modulus of continuity, Lipschitz functions, Jackson Theorems  
5 solution of least squares poly. appr., weighted L2 space, inner product space HW 1 due
6 Gram-Schmidt process, Topeler Theorem, Bessel inequality, Parseval identity  
7 orthogonal polynomials: minimization, recurrence, zeros  
8 examples of orthogonal polynomials (trigonometric, Chebyshev, Legendre)  
9 Lagrange interpolation: existence, uniqueness, Lagrange formula, remainder HW 2 due
10 Newton formula and divided differences, *iterated linear interpolation  
11 Peono kernel and remainder theorem, optimal interpolation points  
12 convergence of Lagrange and piecewise Lagrange interpolation polynomials  
13 Hermite interpolation, generalization, divided differences with repeated points HW 3 due
14 trigonometric interpolation with various sets of points, fast Fourier transform  
15 cubic splines: reprensentation, Holladay identity, minimization property Project 1 due
16 cubic spline interpolation: existence, uniqueness, computation  
17 *B-splines: definition and computation  
18 deg. of precision, methods of undetermined coeff. basic & comp. quadrature, HW 4 due
19 Peano kernel and error formula, interpolatory quadrature, error, optimal points  
20 Newton-Cotes formulas: coefficients, error and composite error, examples  
21 Euler-Maclaurin formula, Richardson extrapolation, Romberg algorithm  
22 weighted Gaussian quadrature: degree of precision, error, coefficients HW 5 due
23 Gauss-Legendre, Gauss-Chebyshev quadrature: coefficients, errors  
24 convergence of sequences of integral approximations  
25 approximation of singular integrals, adaptive numerical integration  
26 Gaussian elimination, backward substitute, SDD matrix, partial pivoting HW 6 due
27 direct LU factorization, SPD matrix, Cholesky factorization, tridiagonal matrix  
28 vector and matrix norms, spectral radius, error bounds, condition number Project 2 due
29 least squares problem, normal equation, Gram-Schmidt orthogonalization  
30 Householder & Givens transformation, QR methods for least squares problems  
31 Jacobi, Gauss-Seidel, and relaxation methods, general iterative methods HW 7 due
32 convergent matrix, convergence of the basic iterative methods  
33 conjugate gradient methods (CGM): algorithms, properties  
34 convergence analysis for CGM, PCG, incomplete Cholesky factorization  
35 Gershgorin Circle Theorem, power and inverse power methods HW 8 due
36 *Hessenberg reduction by Householder transformation, Hyman's method  
37 QR algorithm and its convergence, Schur decomposition, shifting Project 3 due
38 singular value decomposition, computation of singular values  
39 reduction of a Hermitian matrix: Householder method, Givens method HW 9 due
40 eigenvalues of a tridiagonal Hermitian matrix, Sturm sequence, QR method  
41 Rayleigh quotient iteration and its convergence  
42 *eigenvalue perturbation theory HW 10 due

Last updated by Bo Li on April 18, 2001. © Bo Li, 2000, 2001.