| 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 |