November 6, 2007 --- Class 20 --- Simpson's Rule Rederived, AHO, NDSolve, 
			Hit or Miss Method

	Simpson's Rule Rederived

	Starting from the trapezoid rule, we used the formulae below
	for correcting the leading 1/N^2 errors.

	S_N  = A + E/N^2
	S_2N = A + E/(2N)^2

	so   A = 4/3 S_2N -1/3 S_N

	Applying this to a one-dimensional integral, we will easily find that 
	the weighting of the sum of two expressions on the right hand side
	exactly coincides with the Simpson's rule formula.  We previously
	found that Simpson's Rule has a leading error of order 1/N^4.  Since
	the approach above removes the leading 1/N^2 error and since the
	error of the trapezoid rule only has even powers of 1/N (asserted, but
	not proved), this is not surprising.

	Anharmonic Oscillator: Analytic Considerations

	For homework you were asked to study the period of the 
	harmonic oscillator and how it depends on the force constant.
	It turns out that a lot can be learned about this problem by
	analytic means.  We discussed how to use time translation to
	get simple initial conditions.  We then considered how we
	could rescale x and t to simplify the equation of motion.
	When we had done so, we found that we could determine the
	dependence of the period on k, m and the initial amplitude,
	up to a single unknown constant.  This constant was the period
	with unit mass, k and amplitude.  We just found that period using
	Mathematica.

	Solving Differential Equations with Mathematica: Anharmonic Oscillator

	If you have a simple differential equation to solve, you might want
	to use Mathematica rather than a numerical code you develop yourself.
	To solve a differential equation numerically, we use NDSolve.  To solve
	one analytically, use DSolve.  For example:

	NDSolve[ {x''[t] == - x[t]^3, x[0] == 1, x'[0] == 0}, x, {t, 0, 10}]

	is the command required to solve the anharmonic oscillator with k/m=1.
	x''[t] stands for the second derivative at time t.  x'[0] stands for
	the first derivative, or velocity, at t=0.  Within the list that is
	the first argument to NDSolve, we have the differential equation
	itself and the initial conditions on position and velocity that are
	required to specify the solution.  The solution will be an 
	"interpolating function" for x[t] over the range
	0 <= t <= 10.

	If your next command is

	Plot[ Evaluate[x[t] /. %], {t, 0, 10}]

	Mathematica will display a plot of the solution.  The command
	Evaluate[x[t] /. %] tells Mathematica to use the output of the 
	previous command as a substitution rule and then to evaluate the
	function x.  Assuming that In[1] was the command to solve the
	differntial equation, Out[1] is the rule for the interpolating function
	that defines the solution.  You can then find the solution at a 
	particular time with a command like:

	x[7.5] /. Out[1]

	I do not find the following commands intuitive, but they can be used
	to find the period:

	x /. First[Out[1]]

	FindRoot[ %[t] == 1, {t, 7.5}]

	The output of the first command is the interpolating function itself.
	In the second command, % stands for the interpolating function, we
	make it a function of t and then ask Mathematica to find the value of
	t that makes the function 1.  We have the search start near 7.5 because
	looking at the plot of the solution to the anharmonic oscillator, we
	see the the period is close to 7.5.  The result from Mathematica is
	7.4163.

	Higher Dimensional Integrals

	Monte Carlo methods like hit or miss and the sample mean methods
	have errors that fall like n^{-1/2}.  This is not very good.  To
	cut the error in half you need to do 4 times a much work.  For
	Simpson's rule the error goes like n^{-4}, so if you do twice a
	much work, the error falls by a factor of 16.  However, for 
	multidimensional integrals, a method with an error like h^a,
	where h is the spacing between grid points, will have an error like
	n^{-a/d} because the number of points on the grid goes like h^{-d}.
	As an example, if a=4 and d=10, a/d=0.4 so the error is falling
	more slowly than n^{-1/2}.  So, for high dimension integrals,
	Monte Carlo methods may be bad, but other methods are worse.

	Hit-or-Miss Method

	The Hit-or-Miss method and Sample Mean methods are described in CSM
	Sec. 11.2.  I introduced the hit-or-miss method.  In the next class,
	we will test this method.