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| Week | Contents |
|---|---|
| Week 1 | Course overview, discretization of ODE(motivation for linear system of equations), Reiview of basic properties about matrices |
| Week 2 | Reivew of diagonalizability of matrices, spectral theorem symmetric matrices, positive square root of positive semidefinte matrices, Generalized eigenvalue problem, Sensitivity and condtioning, absolute and relative error, floating point arithemetic |
| Week 3 | Norms, matrix norms, induced norms, Matrix p-norms, condtion number |
| Week 4 | condition number, perturbing the coefficient matrix A and/or the vector b in the linear system Ax=b, Scaling and condition number |
| Week 5 | Back substitution, Gaussinan elimination, LU decomposition, pivoting |
| Week 6 | Cholesky decomposition, Plane rotator(Givens rotator), Reflector(Householder transformation), QR decomposition(Proofs using rotator and reflectors) |
| Week 7 | Least squares problems, revision for mid-semester |
| Mid semester | 18.02.2019 to 26.02.2019 |
| Week 8 | Singular value decomposition(SVD) |
| Week 9 | SVD and least squares problem, Low rank approximation |
| Week 10 | Pseudo inverse, sensitivity analysis for SVD |
| Week 11 | Eigenvalue problems, Gershgorin's theorem, Improved Gershgorin's theorem |
| Week 12 | Improved Gershgorin's theorem(cont.), Rayleigh principle, Courant–Fischer min-max principle |
| Week 13 | Sylvester's law of inertia, Bauer-Fike theorem, Sensitivity analysis for eigenvalues |
| Week 14 & 15 | The power method, inverse iteration by von Wielandt, Jacobi method, Householder reduction to Hessenberg form, QR algorithm. |