October 6, 2005 --- Class 10 --- Error Analysis, Higher Order Algorithms

Activities:
		Error Analysis of the Euler Algorithm

		We can use awk to compare our numerical integration of the
		equations of motion with the exact answer.  Here is an
		awk script check.a that returns the error for the case when
		omega=3

BEGIN{omega=3.0}
{t=$1; y=$2; yexact=cos(omega*t); print t, (y-yexact) }

		The script may be found in ~sg/chap5/check.a.  I like to use
		suffix .a to indicate when a file contains awk commands.
		Note that when you put awk commands in a file, you don't need
		all those confusing quotes to keep the shell from interpreting
		various characters like "{" and "}".

		An alternative is to omit the BEGIN line and define omega
		in the awk command.  To define omega call awk this way:

		awk -f check2.a omega=3. filename

		In the same directory, you will find a shell script plot2.csh
		for putting two axis graphs on the same page.  The -s in the
		second command keeps axis from trying to clear the page at
		the beginning of the second plot.  The parentheses concatenate
		the output of the two axis commands so they are joined together
		in the pipe to the plot filter.

	Higher order algorithms

		We talked about several algorithms including leap frog
		Adams-Bashforth and Runge-Kutta.  Euler-Richardson is
		basically the 2nd order Runge-Kutta algorithm.  We also
		talked about the 4th order Runge-Kutta algorithm.  Many of
		these algorithms are presented in Appendix 5A of Chapter 5.
		On page 128.  There are two typos in the book regarding
		the 4th order R-K algorithm.  In Eq. (5.59a), the right-hand-
		side should be multiplied by Delta t.  In Eq. (5.59b), k_3x
		on the RHS should be k_3v
		
		You may also find more about these algorithms in "Numerical
		Analysis" by Burden and Faires, or "Numerical Recipes" by
		Press et al.
		
		If you missed class and did not get the handout, please come to
		my office to get one.