| 1 | Key concerns of numerical methods | |
| 2 | Performance: Arithmetic vs. memory | |
| 3 | Memory optimization and cache obliviousness | |
| 4 | Accuracy and floating-point arithmetic | |
| 5 | Floating-point and numerical stability | |
| 6 | Backwards stability of summation, norms | Problem set 1 due |
| 7 | Condition numbers and eigenvalues | |
| 8 | The singular-value decomposition | |
| 9 | Least-square problems and QR factors | |
| 10 | Gram-Schmidt stability and householder QR | |
| 11 | Householder, Gaussian, and Cholesky factorization | Problem set 2 due |
| 12 | Eigenproblems, characteristic polynomials, and Shur factors | |
| 13 | Hessenberg factorization and its applications, power methods | |
| 14 | QR iteration for eigenproblems | |
| 15 | Overview of iterative and sparse solvers | Problem set 3 due |
| 16 | Arnoldi and Lanczos iterations | |
| 17 | Restarting Lanczos iterations | |
| 18 | GMRES and MINRES | |
| 19 | Steepest-descent and conjugate-gradient methods | Problem set 4 due |
| 20 | Preconditioning and condition numbers of PDE-like matrices | |
| 21 | Biconjugate-gradient methods, sparse-direct solvers | |
| 22 | Nonlinear conjugate gradient, and conjugate-gradient eigensolvers | Final project proposal due |
| 23 | Overview of optimization problems | |
| 24 | Adjoint methods and sensitivity analysis | Midterm taken before Ses #24 |
| 25 | Adjoint methods for recurrences, CCSA algorithms | |
| 26 | Lagrange dual functions and KKT conditions | |
| 27 | Quasi-Newton methods | Problem set 5 due |
| 28 | BFGS updates | |
| 29 | Derivative-free optimization: Linear and quadratic models | |
| 30 | Global optimization and the DIRECT algorithm | |
| 31 | Numerical integration and accuracy of the trapezoidal rule | |
| 32 | Clenshaw-Curtis quadrature | |
| 33 | Chebyshev approximation | |
| 34 | Fast Fourier transforms - the Cooley-Tukey algorithm | |
| 35 | FFTW and FFT implementation in practice | Final project due at end of term |
