October 5, 2004 --- Class 11 --- Running Adapt, Oscillatory Motion, Conservation
			of Energy in the Euler Algorithms

Activities:

	Comparing secant and false position methods

		We has some more comments about the comparison of these methods
		but these comments and the output have been moved into notes
		for previous class.

        Solving for the height
 
                To find the height at which the ball must be dropped, a
                root finding algorithm is added to the code.  It is based
                on the secant method.  The secant method can be unstable,
                so the loop is terminated upon a maximum number of
                iterations or when the desired accuracy is obtained.  This
                is done in the code adapt.c.

                The secant method is based on a linear fit to the function.
                The linear fit is used to estimate where the function
                vanishes.  This estimate is used in subsequent estimates.
                It is easy to implement since you only need the value of
                the function, not its derivative.

		We used the code with different starting values and also
		adjusted the accuracy to which we want 
		      v(height)/v_naive(height) = 0.99

		The code keeps adjusting the height from which it drops the
		object until the ratio of the velocity at which the object
		hits the ground, to the velocity it would have had if the
		acceleration were the constant g is 0.99.

        Chapter 5---Oscillatory Motion

                We used the code ~sg/chap5/sho2.c and executable sho2 to 
		calculate the motion of a simple harmonic oscillator.  
		This code is based on the Euler method.  We found that energy 
		is not conserved.  We will see in the class after next that 
		with the Euler-Cromer algorithm, the energy
                varies during the motion of the body, but it is not
                monotonically increasing as it is for the Euler method.

		The error is order dt for these methods, but the error
		for the Euler method is unstable, i.e., the errors
		grow with time.

		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.

                For homework #3, the Euler-Richardson and Runge-Kutta methods
		will be used.

	Conservation of Energy in Euler algorithms

		By calculating the energy after a single step for the
		harmonic oscillator, we were able to show that for Euler
		algorithm the change in energy is proportional to the
		current energy, thus it is increasing exponentally.  For
		the Euler-Cromer algorithm, the change in energy is
		proportional to the difference between potential and
		kinetic energy.  (This should be derived by the student.)
		Since for a harmonic oscillator the time
		average of kinetic and potential energies is the same, the
		Euler-Cromer algorithm conserves energy on average, i.e.,
		the energy ocsillates around the correct mean value.