September 12, 2005 --- Class 5---Error Analysis for the Euler Method (II),
		Introduction to Chapter 3

Information:

	XLiveCD
		If you are using an MS Windows machine at home, you might
		want to have an X Windows server on your machine so you can
		display graphs from nations at home.  A free server is
		available here:
			http://xlivecd.indiana.edu/
		You should be able to get a CD image and burn a CD.  You
		don't need to install the software to use it.  You can run
		the software from the CD.
	Knoppix
		A free version of Linux that you can boot from a CD is
		available from
			http://www.knoppix.org/
		You can burn a knoppix CD and try out linux without
		installing it on the hard drive.

Activities:

	Error estimates
		Most of of this class was devoted to understanding the
		error in the Euler algorithm for integrating a differential
		equation.  We did a careful estimate of the error for the
		simple case 
			dy/dx = f(x).

		This is simple because the right hand side depends on x,
		not y(x).  So this is just doing an integral.

		If we use a step size dx, the error in a bin with left hand
		edge at x is:

	(1/2!) (dx)^2 f'(x) + (1/3!) (dx)^3 f''(x) + higher order terms

		In our first example of the Euler program, f(x)=2x.  Thus,
		f'(x)=2 and f''(x)=0, so the second term above is 0.  Why
		is the error order dx, not (dx)^2?  That is because the
		above expression is for each bin.  In our example, we
		integrate x from 1 to 2, so the number of bins n=(1/dx).
		Thus, the total error is dx for this case.

		In our second example, f(x)=3x^2. f'(x)=6x and f''(x)=6,
		so the second error term does not vanish.  If you very
		carefully evalute the two error terms you will find that
		the first term is 
			3 (dx)^2 * n (1 + (n-1)*dx/2 )
		where n=1/dx.  Using the fact that n*dx=1, this first
		term can be written as
			3 dx (3/2 -dx/2) or 9/2 *dx - 3/2 * (dx)^2.
		Actually, only this term requires care.  The f'' term is
		easily seen to be 
			(dx)^2.
		Adding the two terms, our final result is
			9/2 *dx - 1/2 * (dx)^2.

		Before we did the careful estimate, we plotted the error
		divided by dt and found that the coefficient of the second
		term is negative, not positive and we would get from just
		looking at the second term in the Taylor expansion.

		You might like to write an awk script to show that the error 
		for the function f(x)=3*x^2 agrees perfectly with the 
		expression above.
		We would like to combine the two steps of running 
		ans_vs_step.csh and running awk on the output into a single
		script.  Note that a simple solution is to pipe the output of
		ans_vs_step.csh to the desired awk command.

		However, if one wants a single shell script there is the
		challenge of dealing with the fact that ans_vs_step.csh
		used two separate commands, echo -n and euler to print
		a single line.  Here are two ways to solve the problem.

		Append output to a temporary file and then awk the file.
		This is found in ~sg/chap2/err_vs_step.csh

		#!/bin/csh
		set noclobber
		rm -f temp.ans
		foreach step ( 0.5 0.2 .1 .05 .02 .01 .005 )
        		echo -n $step "  "  >>! temp.ans
        		euler $step |tail -1|awk '{print $2}'  >>temp.ans
		end
		awk '{print $1,(8-$2),4.5*$1,4.5*$1-0.5*$1*$1}' temp.ans

		Notes: >> appends the standard output to the filename that
		follows.  >>! does it even when the file does not already
		exist.  This is necessary if you set the noclobber variable.
		In this example we first remove the temporary file temp.ans
		with the rm -f command.  The -f option forces the file to
		be removed without displaying permissions, asking questions or
		reporting errors.  So, if the file does not already exist,
		you will not have an error message.  After the end of the
		foreach loop, we run the single awk command to create the
		display we want.  You must run this command in a directory
		where you can create the file temp.ans.

		An alternative solution uses the csh () to join two commands.
		It can be found in ~sg/chap2/err2_vs_step.csh

		#!/bin/csh
		set noclobber
		foreach step ( 0.5 0.2 .1 .05 .02 .01 .005 )
       		(echo -n $step "  "; euler $step |tail -1|awk '{print $2}') | \
          		awk '{print $1,(8-$2),4.5*$1,4.5*$1-0.5*$1*$1}' 
		end

		Notes: In this case, we use two unix commands that both
		print to the standard output and we put the two commands
		within parentheses.  The two commands are separated by a
		semicolon.  We then pipe that single output 
		stream to the awk command.  It is on a separate line, but the
		line that starts with "(echo" ends with a backslash.  That tells
		the shell that the next line is a continuation.  Important:
		don't put any spaces after that backslash!

	Introduction to Chapter 3

	Solving Newton's Law of Motion

		We started Chapter 3 and discussed how to turn a 2nd order
		differential equation into a series of coupled first order
		differential equations.  We also discussed why position,
		velocity and acceleration are so important, and the next
		derivative, called the jerk, is not well known.