Numerical Analysis II Math 447/847 Spring 2011 Class Log

1. Jan 10, Mon. class cancelled due to snow
2. Jan 12, Wed. course overview and logistics; initial student survey; Lec 1: matrix-vector multiplication; Vandermonde matrix; associated subspaces
3. Jan 14, Fri. invertibility; characterizations; view as change of basis; Lec 2: adjoint; inner product; orthogonality; orthogonal sets are linearly independent
Jan 17, Mon. Martin Luther King Day, no class.
4. Jan 19, Wed. writing a vector as a linear combination of orthogonal vectors; unitary matrices; Lec 3: norms; p-norms; associated unit balls; matrix norms induced by vector norms
5. Jan 21, Fri. No class.
6. Jan 24, Mon. examples of matrix norms; examples of computing p-norm of a matrix; Hölder and Cauchy-Schwarz inequalities; examples
7. Jan 26, Wed. bounding matrix norm of products; Frobenius norm; Lec 4: singular value decomposition; geometric interpretation
8. Jan 28, Fri. existence and uniqueness of SVD
9. Jan 31, Mon. Lec 5: properties of the SVD
10. Feb 2, Wed. low-rank approximations using the SVD; introduction to Sage
11. Feb 4, Fri. Lec 6: projectors; decomposition of space into complementary subspaces; projectors are orthogonal iff hermitian
12. Feb 7, Mon. constructing projectors for subspaces with arbitrary bases; Lec 7: QR factorization; Gram-Schmidt orthogonalization
13. Feb 9, Wed. existence and uniqueness of QR factorization; application to solving Ax=b; Lec 8: thinking of Gram-Schmidt as using projectors
14. Feb 11, Fri. modified Gram-Schmidt algorithm; performance vs classical Gram-Schmidt
15. Feb 14, Mon. discussion of HW3 #4 and recursive calls; Gram-Schmidt as "triangular orthogonalization"; Lec 10: Householder orthogonalization; "orthogonal triangularization"; construction of Householder reflectors
16. Feb 16, Wed. Householder algorithm; algorithm analysis and floating point operation counts; operation count for classical Gram-Schmidt, modified Gram-Schmidt, and Householder
17. Feb 18, Fri. more about operation count for Householder; Lec 11: formulation of least squares problem; polynomial interpolation
18. Feb 21, Mon. discussion of HW3; polynomial fitting; solving least squares problems; pseudoinverse
19. Feb 23, Wed. solving normal equations in practice; discussion of midterm; Lec 12: absolute conditioning
20. Feb 25, Fri. relative conditioning; examples; matrix-vector multiplication; condition number of matrix
21. Feb 28, Mon. condition number of solving systems of equations; Lec 13: floating point numbers and scientific notation; types of problems with restricted representation
22. Mar 2, Wed. floating point arithmetic; machine epsilon; 2 fundamental axioms
23. Mar 4, Fri. discussion of HW4 and midterm take-home; Lec 14: accuracy; stability
24. Mar 7, Mon. backward stability; Big Oh notation; independence of stability notions from the norm used; Lec 15: backwards stability of floating point arithmetic
25. Mar 9, Wed. more examples that are not backwards stable: outer product, adding 1; accuracy of backward stable algorithms
26. Mar 11, Fri. Lec 16: backwards stability of Householder algorithm
27. Mar 14, Mon. discussion of midterm; importance of orthogonality; orthogonality and function spaces
28. Mar 16, Wed. Lec 17: backwards stability of back substitution; informal feedback survey
29. Mar 18, Fri. No class.
Mar 21 to Mar 25. Spring break - no classes.
30. Mar 28, Mon. summary of informal feedback survey; Lec 18: conditioning of least squares problems
31. Mar 30, Wed. perturbing b and its effect on y and x
32. Apr 1, Fri. tilting the range of A and the effect on y
33. Apr 4, Mon. perturbations of A and the effect on x
34. Apr 6, Wed. Lec 19: stability of least squares algorithms; inaccuracy of using the normal equations; Lec 20: Gaussian elimination and the LU decomposition
35. Apr 8, Fri. operation count; solving linear systems; example of instability
36. Apr 11, Mon. Lec 21: pivoting; PA=LU factorization; Lec 22: statement of stability dependent on L and U
37. Apr 13, Wed. growth factor of a matrix; bounds on growth factor; def of backward stability for Gaussian elimination with partial pivoting; empirical observation that Gaussian elimination with partial pivoting performs well in practice; discussion
38. Apr 15, Fri. Lec 24: properties of eigenvalues
39. Apr 18, Mon. Schur factorization; existence; Lec 25: overview of eigenvalue algorithms; power iteration; solving char poly; equivalence with finding roots of polynomials; all eigenvalue solvers are iterative; two phases of using Schur factorization
40. Apr 20, Wed. discussion of inverting unit lower triangular matrices; Lec 26: reduction to upper Hessenberg form using Householder reflectors; operation count; applied to Hermitian matrices; stability
41. Apr 22, Fri. Lec 27: Rayleigh quotient; error estimate; power iteration; convergence
42. Apr 25, Mon. inverse iteration; convergence; Rayleigh quotient iteration; cubic convergence; example
43. Apr 27, Wed. Lec 28: QR algorithm without shifts; unnormalized simultaneous iteration; convergence
44. Apr 29, Fri. simultaneous iteration; equivalence with QR algorithm
Thurs May 5, 1:00pm-3:00pm. Final Exam

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