December 1, 2006 --- Class 26 --- Feedback Shift Register Random Number
			Generator, Fitting Data

Activities:
	
	Running box_rand

	We ran a few tests with the fsr random number generator and saw that
	the number of unoccupied cells was decreasing.  Then using a 
	foreach shell command, we ran the code 10 times, putting the reults
	in a file.  Using awk and aver, it was easy to get the average of
	the ten trials and plot the result with a semilog scale using axis.
	The result clearly shows exponential decay of the number of unoccupied
	boxes.

	Understanding the Feedback Shift Register Random Number Generator

	We examined how the feedback shift register (FSR) code makes a new 
	random 28 bit integer every time it is called.  The reason this random
	number generator has a period longer than 2^28 is because the period
	is determined by the entire state of the feedback shift register
	which has about 140 bits.

	Feedback shift number random generators are based on bitwise exclusive
	or operations.  The recursion relation is

		b_n = b_{n-p} ^ b_{n-q}

	So the new bit depends on bits earlier in the sequence by p and q.
	There are certain values of p and q that result in good randomness.
	You can find good pairs of p and q in the book "Seminumerical
	Algorithms" by Donald Knuth in Table 1.  Our pair (p,q) = (97,127).

	Within the code, 28 bits of each 32 bit word are used to store the
	state of the FSR.  rshift[1] to rshift[5] hold 140 bits of state.
	The next 28 bits, numbered 141-168 are created in rshift[0].

	Using b_n = b_{n-97} ^ b_{n-127}, we see that with n=141 we need
	28 bit words starting with bit 44 (=141-97) and bit 14 (= 141-127).

	Here are how the bits are stored in the rightmost 28 bits of rshift[i].

	rshift[5] = 1-28
	rshift[4] = 29-56
	rshift[3] = 57-84
	rshift[2] = 85-112
	rshift[1] = 113-140

	From the code, t2= (rshift[5]<<13))|(rshift[4]>>15)
	contains the bits 14-41 that we need.  In the C language x<<n
	shift the bits of the variable x to the left by n places.  Similarly,
	>> denotes a shift to the right.

	We can get bits 44-56 from t1 = (rshift[4]<<15)|(rshift[3]>>13).
	We then take the exclusive or of t1 and t2 and mask of the rightmost
	28 bits.  Finally, the shift register is moved to higher values,
	dropping off the old value of rshift[5], and the new 28 bit integer
	we created is divided by 2^28 to get a float between 0 and 1.  All
	the details can be found in ~sg/src/newfsr.c.

	Fitting Data

	We looked at what one should do to fit data to a model function.
	If the data has a Gaussian distribution about its true value, then
	we maximize the probability for the set of observations by
	adjusting the parameters of the model to minimize \chi^2.

	\chi^2= \Sum_{i=0}^N}  (y_i- y_model(x_i)^2/ \sigma_i^2

	The case of a linear model was considered in great detail.  We
	obtained specific formula for the slope and intercept parameters of
	the fit.  We also found the error on the slope and intercept, and
	the covariance of those quantities.

	The next thing we need to do is consider whether or not our fit is a
	good one.  We minimized chi-squared as best we could, but how small
	should it be?