Computational Modeling
Fall 2005

For today: HomeworkSixteen (Zipf meets Pareto)

Today's outline:

1) Discrete FT

2) Fast (Discrete) FT

3) Complex arithmetic in Python

For next time: HomeworkSeventeen  (FFT and start Gould)


Complex arithmetic in Python
----------------------------

>>> x = complex(3, 4)
>>> x
(3+4j)

>>> x.real
3.0
>>> x.imag
4.0

>>> x.conjugate
<built-in method conjugate of complex object at 0xb7f8f398>
>>> x.conjugate()
(3-4j)

>>> abs(x)
5.0

>>> from math import exp
>>> exp(x)
Traceback (most recent call last):
  File "<stdin>", line 1, in ?
TypeError: can't convert complex to float; use abs(z)

>>> from cmath import exp
>>> exp(x)
(-13.128783081462158-15.200784463067954j)


Implement FFT
-------------

1) Download my FFT framework from http://wb/cm/code/Fourier1.py

2) Fill in the body of fft

3) Test your program with various input frequencies, including
   some that exceed the Nyquist frequency.


Analysis of FFT
---------------

Why is FFT faster than the naive implementation?

Option 1: solve the recurrence relation

Option 2: analyze work per level of recursion and # levels



Epistomology
------------

From http://en.wikipedia.org/wiki/Epistemology

"Epistemology, from the Greek words episteme (knowledge) and logos
(word/speech) is the branch of philosophy that deals with the nature,
origin and scope of knowledge. It has historically been one of the
most investigated and most debated of all philosophical
discourses. Much of this debate has focused on analysing in detail the
nature and variety of knowledge, and how it relates to similar notions
such as truth and belief. Much of this discussion concerns the
justification of knowledge claims.

Not surprisingly, the way that knowledge claims are justified both
leads to and depends on the general approach to philosophy one adopts,
and so philosophers have developed a range of epistemological theories
to accompany their general philosophical positions. More recent
studies have re-written centuries-old assumptions, and the field of
epistemology continues to be vibrant and dynamic."

The rest of this page is a good read, and will prepare us for our
next topic, Bayesian Epistemology.