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

# 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.