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

1 | Mar 26 Tu | [R] 1.4, [BG] 2.1 | Introduction. Algebraic and exponential convergence (converge.m). Big-O and little-o. Taylor series. | |

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

28 Th | [R] 1.5, [BG] 2.8.3 | Convergence rate and the complex plane. Effective 2d plots. Super-exponential convergence. | converge | |

2 | Apr 2 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. Good coding (zeta.m and testzeta.m). Floating point, summing series. | newton |

4 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 | 9 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 |

10 W X-hr | HTML, [BG] 3.7 | Basic HTML. Breaking then fixing our bisection codes! (bring laptops) | ||

11 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.
Stability of linear systems, condition number of a matrix
(linsys).
| ||

4 | 14 Tu | [BG] 7.2, [R] 7.7 | Fourier series with complex exponentials. Deriving Discrete Fourier Transform via quadrature approximation. Trigonometric interpolation. | fourier |

15 W X-hr | practise problems for midterm 1. | |||

16 Th | [BG] 7.2.2 |
HW3 due. DFT: Roots of unity & sum lemma,
inversion formula, unitarity.
| ||

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

5 | 21 Tu | [BG] 7.2.2 | Aliasing formula, Nyquist sampling theorem. Getting to know DFT matrix. Audio signal analysis application, physical frequency units (audiofft.m) | dft |

22 W X-hr | (no X-hr) | |||

23 Th | [BG] 7.2.3, history | HW4 due.
The Fast Fourier Transform (Cooley-Tukey algorithm).
Applications of FFT: convolution and deconvolution.
| ||

6 | 30 Tu | Gourdan 1, 2, [CP] 9.5, this | Super-algebraic convergence, review other convergence types. Large integer addition (bigintadd.m, testbigintadd.m), Acyclic convolution. Strassen's fast multiplication. Fast division via Newton iteration for reciprocal. | arbprec |

May 1 W X-hr | (no X-hr) | |||

2 Th | Gourdan, Sandifer, BBB, App. 12-15, Salamin |
HW5 due.
Error bounds in trigonometric polynomial interpolation.
Computing digits of pi: Taylor with Machin formulae
(atanCplane), Brent-Salamin
quadratic convergent iteration.
High accuracy floating-point computation in Python/SAGE/mpmath.
| machin | |

7 | 7 Tu | quest, BBP | Project 1-page description due.
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
| |||

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

8 | 14 Tu | Brent, 2 sieves | Finding large factors: Fermat's method, Kraitchik's improvement, linear algebra mod 2. | kraitchik |

Midterm 2: Tues May 14, 6-8pm, Kemeny 004
Topics
| ||||

16 Th | [EMA] Ch. 6; [CP] Ch. 6.1 | Quadratic sieve, frequency of smooth numbers (smoothhist.py, its plot). Numerical integration: periodic trapezoid rule, error analysis. | ||

9 | 21 Tu | HW7 due.
Product quadrature, Clenshaw-Curtis quadrature
(democlencurt.m which needs clencurt.m)
| clencurt | |

22 W X-hr | ||||

23 Th | Adaptive quadrature (adaptivequad.m). Statistical tests in experimental mathematics: testing bias and correlation in digit strings (sampling). | hyptest | ||

10 | 28 Tu | Student project presentations (may use poster rather than slides) in lecture
slot..
Also strongly encouraged that day: 4-6pm undergraduate poster session.
| ||

31 Fr | Project write-ups due (noon) | |||

31 F - June 4 Tu: Exam period (no final exam :) ) |

- [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.