780.20: 1094 Session 10
Handouts: Printout of nonlinear.nb notebook,
excerpt from Landau/Paez Chapter 5, printout of
GslSpline and test files, "Using GSL Interpolation Functions",
ode_test.cpp printout
In this session,
we'll look at using a GSL
adaptive differential equation solver and then look at interpolation.
Your goals for today and ...
 Finish up some leftover Session 8/9 tasks
 Try out a GSL differential equation solver.
 Try out various GSL interpolation functions on a simple example.
Please work in pairs (more or less).
The instructors will bounce around 1094 and answer questions.
Followups to Sessions 8 and 9
Work on these tasks for the first half of the session only,
then move on to Session 10 tasks.
 The Mathematica notebook nonlinear.nb looks at the same
type of analysis as diffeq_pendulum.cpp only it uses the
Duffing equation. Look through it and follow along with the printout.
This exercise is mostly just to expose you to how to do lots of useful
things
with Mathematica, for future reference.
What questions do you have
about Poincare sections or the power spectrum?
 If you're familiar with Mathematica and have time, you could
convert nonlinear.nb to study the pendulum instead.
But for now, just run the "answer" notebook
called pendulum.nb. Try different parameter choices;
what do you see with p2 vs. p5?
(For the Poincare plot with p5, try removing the PlotRange.)
 Complete the Session 8 sections on "Damped,
Driven Pendulum" if not already completed.
 Continue with unfinished Session 9 tasks.
GSL Differential Equation Solver
The program ode_test.cpp demonstrates the GSL adaptive
differential equation solver by solving the Van der Pol oscillator,
another nonlinear differential equation
(see the Session 10 background notes for the equation).
 Take a look at the code and figure out where the values of mu
and the initial conditions are set. Change mu to 2 and the
initial conditions to x0=1.0 and v0=0.0 (y[0] and y[1]). Note the
different choices for "stepping algorithms", how the function is set up
and that a Jacobian is defined, and how the equation is stepped along
in time. Next time we'll see how to rewrite this code with classes.
 Use the makefile to compile and link the code. Run it.
 Create three output files using
the initial conditions [x0=1.0, v0=0.0], [x0=0.1, v0=0.0], and
[x0=1.5, v0=2.0] (just change values and recompile each time).
Notice how we've used a stringstream to uniquely name each file.
 Use gnuplot to make phasespace plots of all three cases on a
single plot. Print it out and attach it.
What do you observe? This is called an
isolated attractor.
 Think about how you would restructure this code using classes.
Next time we'll explore a possible implementation.
GSL Interpolation Routines
We'll use the Landau/Paez text's example of a
theoretical scattering cross section as
a function of energy to try out the GSL interpolation routines.
The (x,y) data, with x>E and y>sigma_{th}, is given in the
bottom row of Table
5.1 (see the Chapter 5 excerpt; note we are NOT fitting
sigma_{exp}).
 Start with the gsl_spline_test_class.cpp code (and corresponding
makefile). Take a look at the printout and try running the code.
Note that we've used a Spline class as a "wrapper" for the GSL functions,
just as we did earlier with the Hamiltonian class. Compare the
implementation to the example on the "Using GSL Interpolation Functions"
handout.

Instead of the sample function in the code,
you will change the program to interpolate
the Table 5.1 data from the handout. Set npts and the (x,y) arrays equal to
the appropriate values when you declare them. Declare them on
separate lines. An array x[4] can be initialized with the values
1., 2., 3., and 4. with the declaration:
double x[4] = {1., 2., 3., 4. };
 Use the code to generate a cubic spline
interpolation for the cross section from 0 to 200 MeV in steps of
5 MeV. Output this data to a file for plotting with gnuplot
and try it out.
 Now modify the Spline class to allow for a polynomial
interpolation and change the gsl_spline_test_class.cpp main
program to
generate linear and polynomial
interpolations as well and add the results to the output
file.
 Generate a graph with all three interpolations plotted.
For each curve, how does
the resonance energy E_{r} (the peak position) and
gamma (full width at half maximum) as seen on
the graph compare to the theoretical
predictions of (78,55)?
Cubic Splining [if time permits]
Here we'll look at how to use cubic splines to define a function
from arrays of x and y values.
A question that always arises is: How many points do we need?
Or, what may be more relevant, how accurate will our function (or
its derivatives) be for a given spacing of x points?
 We'll reuse the Spline class from the last section and
the original gsl_spline_test_class.cpp function, which splined
an array.
 The goal is
to modify the code so that it splines the groundstate hydrogen
wave function:
u(r) = 2*r*exp(r)
 Your task is to determine how many (equally spaced) points to use
to represent the wave function. Suppose you need the derivative of the wave
function to be accurate to one part in 10^{6}
for 1 < r < 4 (absolute, not relative
error) Devise (and carry out!) a plan that will tell you
the spacing and the number of points needed to reach this goals.
What did you do?
 Now suppose you need integrals
over the wave function to be accurate to 0.01%.
Devise (and carry out!) a plan that will tell you
the spacing and the number of points needed to reach this goals.
(To try out integrals,
use one of the GSL integration routines on an integral involving u(r)
that you know the answer to; there's an obvious one!)
780.20: 1094 Session 10.
Last modified: 07:25 am, February 15, 2010.
furnstahl.1@osu.edu