week | date | reading | daily topics, demos & codes | worksheets |
---|---|---|---|---|

1 | Mar 25 Tu | [R] 1.4, [BG] 2.1 | Syllabus, introduction. Algebraic and exponential convergence (converge.m). Big-O and little-o. | asymp. |

26 W X-hr | Resources page | Dan: Matlab (intro56.m, and tom.m); LaTeX (test.tex which needs squiggle.eps, and gives test.pdf). | ||

27 Th | [R] 1.5, [BG] 2.8.3 | Taylor series convergence rate and the complex plane. Effective 2d plots. Super-exponential convergence. | ||

2 | Apr 1 Tu | [R] 5.3, 2.1-2, 3.1, [H] 1.4-1.8, [TB] Ch.13, [M126] p.12, [BG] 2.2.2 | Newton and sqrt iteration, quadratic convergence. Floating point (1-page summary), summing series. | newton |

2 W X-hr | ||||

3 Th | [M126] p.10, [BG] 2.2.3, [R] Ch. 8 | HW1 due.
Rules of floating point operations. Catastrophic cancellation
(catastrophic.m).
Condition number. Derivatives by finite differences.
| condnum | |

3 | 8 Tu | [GC] Ch.6, [TB] Ch.14-15, [R] Ch.4, [BG] 2.3, 3.2, 3.5 | Balancing finite difference errors. Stability of algorithms, backwards stability. | backstab |

9 W X-hr
| Coding finite differencing and testing its error performance. | |||

10 Th | [TB] Ch.15,12, [GC] 7.4.2, [BG] 4.3-4.3.2 | HW2 due; Quiz 1
(study topics).
Accuracy of stable algorithms.
Condition number of matrix-vector multiplication,
condition number of a matrix
(linsys).
| ||

4 | 15 Tu | [BG] 7.2, [R] 7.7 | Stability of linear systems. Fourier series with complex exponentials. | fourier |

16 W X-hr | Dan's practise problems for midterm 1. (Also see topics wk 1-3 and practise problems). | |||

17 Th | [BG] 7.2.2 |
HW3 due.
Deriving Discrete Fourier Transform (DFT) via quadrature approximation.
Trigonometric interpolation. Band-limited functions.
Getting to know DFT matrix. Roots of unity.
| dftsum | |

Midterm 1: Thurs April 16, 6-8pm,
Kemeny 004.
| ||||

5 | 21 Tu | [BG] 7.2.2 | Sum lemma. Inversion formula, unitarity. Aliasing formula, Nyquist sampling theorem. | alias |

22 W X-hr | (none) | |||

23 Th | [BG] 7.2.3, history | HW4 due.
Audio signal analysis application, physical frequency units
(audiofft.m)
The Fast Fourier Transform (Danielson-Lanczos lemma,
Cooley-Tukey algorithm).
| ||

6 | 29 Tu | Gourdan 1, 2, [CP] 9.5, this | Review smoothness and Fourier decay, super-algebraic convergence. Applications of FFT: Convolution and deconvolution, Acyclic convolution. Large integer addition (bigintadd.m, testbigintadd.m). Strassen's fast multiplication. | arbprec |

May 1 W X-hr | Dan: filtering and convolution | |||

2 Th | Gourdan, Sandifer, BBB, App. 12-15, Salamin |
HW5 due.
Fast division via Newton iteration for reciprocal.
[Ingredients of arbitrary precision arithmetic library.]
Error bounds in trigonometric polynomial interpolation.
Computing digits of pi: Taylor with Machin formulae
(atanCplane).
High accuracy floating-point computation in Python/SAGE/mpmath.
| machin | |

7 | 7 Tu | quest, BBP | Project 1-page description due this week.
Brent-Salamin quadratic convergent iteration.
Regularization for deconvolution in presence of noise.
Borwein-Bailey-Plouffe algorithm for binary digits of log 2, and pi.
| matlabvspython |

8 W X-hr | Quiz 2, and bit of review for Midterm 2.
| |||

9 Th | [S] Ch. 1, [H] 2.3-2.4, Brent |
HW6 due.
Computational number theory: basics and applications of factoring,
modular arithmetic, trial division, complexity thereof.
| factorbasic | |

8 | 13 Tu | Brent, 2 sieves | GCD via Euclid. Finding large factors: Fermat's method. | kraitchik |

Midterm 2: Tues May 13, 6-8pm, Kemeny 120
Topics
| ||||

15 Th | [EMA] Ch. 6; [CP] Ch. 6.1 | Kraitchik's method, linear algebra mod 2. Quadratic sieve, frequency of smooth numbers (smoothhist.py, its plot). | ||

9 | 20 Tu | HW7 due.
Numerical integration: periodic trapezoid rule, error analysis.
Clenshaw-Curtis quadrature theory.
| clencurt | |

21 W X-hr | Coding and testing quadrature schemes | |||

22 Th | Clenshaw-Curtis in practice (democlencurt.m which needs clencurt.m). Adaptive quadrature (adaptivequad.m), oscillatory quadrature via complex contour integration (oscquad.m and gauss.m), higher dimensions. | |||

10 | 27 Tu | Student project presentations (may use poster rather than slides) in lecture
slot..
| ||

28 W | 10-11am in Kemeny 209:
Remaining student presentations.
| |||

June 1 Su | Project write-ups due by end of day | |||

30 F - June 3 Tu: Exam period (no final exam :) ) |

- [S] B. Southworth's reasonably complete course notes LaTeXed by him from M56, 2013. Please edit and extend them; to help you here's the source folder.
- [R] J. Restrepo, Introduction to Scientific Computing (draft, 2001), part I, part II
- [GC] A. Greenbaum and T. Chartier.
Numerical Methods:
Design, Analysis, and Computer Implementation of Algorithms
(Princeton, 2012).
*I have ordered this for the library.* - [H] N. J. Higham, Accuracy and Stability of Numerical Algorithms, 1st ed (SIAM, 1996). local copy
- [TB] L. N. Trefethen and D. Bau, Numerical Linear Algebra (SIAM, 1997)
- [SM] L. N. Trefethen, Spectral Methods in MATLAB (SIAM, 2000)
- [ATAP] L.N. Trefethen, Approximation Theory and Approximation Practice (SIAM, 2013). 2011 draft
- [BG] D. Bindel and J. Goodman, Principles of Scientific Computing, (draft, 2009)
- [B] G. Bal, Lecture notes on numerical analysis (2008)
- [CP] R. Crandall and C. Pomerance, Prime numbers: a computational perspective, 2nd ed. (2005)
- [S] W. Stein. Elementary Number Theory: A Computational Perspective. course notes (2007).
- [H] J. Hutchinson. An Introduction to Contemporary Mathematics. course notes (2010).
- [EMA] D. J. Bailey, J. M. Borwein, N. J. Calkin, R. Girgensohn. D. R. Luke and V H. Moll. Experimental Mathematics in Action (2006).
- [M126] A. Barnett, Handwritten lecture notes for Math 126, Winter 2012.