No. | Date | Topics |
---|---|---|
Approximation and interpolation | ||
1 | 8/23 | introduction to class; norms, seminorms, and function spaces |
2 | 8/25 | approximation problems; existence and uniqueness of best approximation |
3 | 8/28 | Weierstrass Approximation Theorem; Bernstein polynomials |
4 | 8/30 | Bernstein polynomials, Jackson Theorem in C^{1}_{2} |
5 | 9/1 | Jackson Theorem in C^{k}_{2}, Jackson Theorem inC^{1} |
6 | 9/6 | Jackson Theorem inC^{k}, polynomial approximation to analytic functions |
7 | 9/8 | Characterization and uniqueness of the minimax approximation |
8 | 9/11 | Lagrange interpolation |
9 | 9/13 | theory of Lagrange interpolation, Lesbesgue constants |
10 | 9/15 | Chebyshev polynomials, interpolation at the Chebyshev points, |
11 | 9/18 | least squares approximation, normal equations, Gram matrices, Gram-Schmidt |
12 | 9/20 | Legendre polynomials, error estimates in L^{2} and max norm for least squares approximation |
13 | 9/22 | weighted least squares |
14 | 9/25 | piecewise polynomial spaces in 1D |
15 | 9/27 | error analysis for piecewise Lagrange interpolation and piecewise Hermite cubic interpolation, cubic spline interpolation |
16 | 9/29 | Bramble-Hilbert lemma |
17 | 10/2 | analysis of cubic spline interpolation, concluded |
18 | 10/4 | piecewise polynomial approximation in 2D |
19 | 10/6 | analysis piecewise polynomial interpolation in 2D |
20 | 10/11 | analysis piecewise polynomial interpolation in 2D |
21 | 10/13 | fast Fourier transform |
Numerical quadrature | ||
22 | 10/16 | quadrature rules |
23 | 10/18 | Peano kernel theorem |
24 | 10/20 | Euler-Maclaurin expansion |
25 | 10/23 | Romberg integration |
26 | 10/23 | Gaussian quadrature |
27 | 10/25 | convergence of Gaussian quadrature, weighted Gaussian quadrature |
28 | 10/27 | adaptive quadrature |
29 | 10/30 | adaptive quadrature |
Direct methods of numerical linear algebra | ||
30 | 11/1 | basic algorithms of numerical linear algebra |
31 | 11/3 | Midterm exam |
32 | 11/6 | elimination and factorization |
33 | 11/8 | Cholesky decomposition, pivoting |
34 | 11/10 | backward error analysis |
35 | 11/13 | condition of linear systems |
Numerical solution of nonlinear systems and optimization | ||
36 | 11/15 | introduction to nonlinear systems |
37 | 11/17 | one-point iterations |
38 | 11/20 | Newton's method |
39 | 11/22 | quasi-Newton methods |
40 | 11/27 | Broyden's method |
41 | 11/29 | convergence of Broyden's method |
42 | 12/1 | unconstrained minimization, steepest descents |
43 | 12/4 | line-search methods |
44 | 12/6 | global convergence of line-search methods |
45 | 12/8 | various complements |