Math 56 project ideas.

Project 1-page descriptions with at least one reference due Tues May 7.
Presentations Tues May 28, write-ups due Friday 31st May at noon.

---- Alex Barnett  5/2/13

Take any topic from class and investigate further, with a
computational component. (See references on the Schedule page).

Non-uniform FFT, using Gaussian convolution as in Greengard-Lee.
Use for spectral or autocorrelation analysis of spike data
eg prime distribution.

Study other matrix norms and singular value decomposition (SVD)
including implementing applications to data analysis. (Trefethen-Bau book)

Go more in depth into image deconvolution algorithms that
handle noisy images.

Study and implement methods for dense matrix eigenvalues (Trefethen-Bau).

Implement BBP algorithm for arbitrarily high hexadecimal digits of pi.
Discuss the PSLQ algorithm used to find the BBP formula.

Implement the search for asymptotics of the number of unique entries
in the n*n multiplication table, using random factored integers and
Monte-Carlo, as in Brent-Pomerance 2012.
See:  http://maths-people.anu.edu.au/~brent/pd/multiplication.pdf

Riemann zeta function: how is it computed, and how can its Fourier
transform tell you about prime numbers?

Plot fractals from Indria's pearls book; teach us some about the group
theory underlying them.

Ulam spiral - how dense are the "arms"? What number theory
conjectures underlie them, and what is known?

Explore and solve for coeffs of high-order finite-difference stencils,
following LeVeque book. Do the same in 2D.
Use to solve ODEs on regular grids, understand how to do boundary
conditions in 1D and maybe 2D.

Study conjectures on pattern-avoiding permutations.

Study of 3n+1 conjecture (eg number of iterations, largest n).  Do
large-scale computations, statistics of distributions (eg Benford's
Law), analyse the graph following the iteration backwards.  See
Lagarias reviews, and Miller-Takloo-Bighash book.

Dig into attempts to prove in literature.

Study FFT for other audio applications.

Install Chebfun, teach us about what it does,
and write a new demo, upload to the chebfun demos site.

Read on how non-power-of-two FFTs work. How does it work for N a prime?
Also discuss bit-reversal for in-place FFT.
Code and test some of the algorithms.
Or maybe focus on history of Gauss and the FFT (Heideman et al article).

Study and implement/compare algorithms for large-integer multiplication
in Ch. 9 of Crandall-Pomerance. What does mpmath use? For what N is
each algorithm best?

History of computation of digits of pi. How does polygon method work?
What is its complexity?

Compute highest digits of pi you can. Compare different methods,
eg Brent-Salamin vs Ramanujan, other recent cubic etc methods.
Statistics of digits of pi - uniform distribution, auto-correlation?
Do some large-scale tests of whether they are an iid random string.

Quadrature for integrals of functions with singularities.

Eigenvalue spacing distributions of random matrices, or for singular
values. Extremal distributions and Tracy-Widom.

Benford's Law (not so exciting)

Project ideas on links in http://experimentalmath.info

Implement tanh-sinh quadratures and evaluate integrals to 1e4 digits.
Compare various other schemes.
Look for relations via PSLQ

Parallelize codes via openMP or MPI.

Build a sage module.

Upload a working utility to matlabcentral.

Spectral methods in 2D, or time-stepping of 1D problems (Trefethen book).

Create and upload a new chebfun demo to the chebfun site.

Make adaptive BVP spectral-element solver, or ODE spectral solver via
time grid. (?)

Explore pyOpenCL, go through tutorials, then code something useful on a GPU.

Review some advanced techniques & history. (although don't just paraphrase
Bailey, Gourdon, etc). Large computing architectures.