780.20: 1094 Session 9
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,
sample .bashrc file
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 leftover Session 8 tasks, plus
look at some additions.
 Learn or review some things about the bash shell.
 Try out rsync.
 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.
Followup to Session 8
Work on these tasks for the first 30 minutes of the session only,
then move on.
 Take a look at the private_vs_public.cpp code, which
is a selfcontained class and main program. Notice how the main
program can access and change the x value and use the xsq
function. But try changing from x to y in the statements getting,
printing, and changing the value of x.
What happens? Why does the get_y function work?
 Complete the Session 8 sections on "Damped,
Driven Pendulum" and "Looking for Chaos".
 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.
Bash Shell Stuff
Here are some things you should know about using the Bash shell.
(I'm assuming everyone is using this. If not, type bash
at a prompt and you will be in the Bash shell!)
 Bash aliases. The .bashrc file sits in your home directory and
is "sourced" when you start an interactive terminal.
Take a look at the sample .bashrc file printout (the file bashrc,
without the ".", is
in the session 9 zip file). Copy any aliases of interest to
your own .bashrc (or the whole thing using cp bashrc
~/.bashrc if you don't have one already)
and activate the changes with the
command: source ~/.bashrc. Try some aliases such as
ll (for "long listing") and df,
which shows info about the disk file system.
 Create your own alias in .bashrc
to automatically log you in (via ssh) to
a computer you
use a lot (e.g., fox.mps.ohiostate.edu). What line
did you add to .bashrc?
 Finding a command. Suppose you want to know both the defined
alias and the original command for du. Use "type a du" and you should
be told all of the possibilities. Try this on some other commands.
 The ulimit command. Type "ulimit a". Can you figure out how to
change the stack size to "unlimited"? [Note: changing the stack
size is not allowed on Cygwin.]
 History. To reexecute the last command that contained "string",
use !string. E.g., !make will rerun the last make command.
If you type "history", you'll get a list of recent commands, with
numbers. To reexecute number 25, type !25. Put the H
function from the sample bashrc in your own .bashrc and activate the
change. This command searches your history for matches, e.g., "H
string" will find those lines with "string" somewhere. You can then
use the numbers to reexecute the one you want. Try it!
(E.g., try "H ulimit")
 Line editing. If you use the uparrow to bring back a line that
you want to change, you can use emacsstyle commands to move around on
the line. For example,
Ctrla takes you to the start of the line while
Ctrle takes you to the end.
Ctrlb and Ctrlf move you back and
forward a character at a time,
while Altb and Altf move you back and forward a
word at a time. Ctrlk kills to the end of the line.
Ctrld deletes the current character and Altd deletes the current
word. Try these out!
Introduction to Rsync
Rsync is a backup or mirroring tool that only copies the
differences of files that have changed. It can do this transfer
in compressed form and using ssh for security (both recommended!).
So updates are fast, efficient, and secure.
Here we'll make a trial run so that you get the basic idea.
 Take a look in the bashrc file for the definitions "rsyncbackup"
and "rsyncmirror". Make sure they are in your own .bashrc file (and
run "source .bashrc" to activate them). These aliases use some of
the common rsync options. The difference is that rsyncmirror deletes
files at the destination that don't exist at the source; be careful
of this!
 Create a directory in your home directory
called "780_backup" to use for testing rsync (i.e., "mkdir
780_backup").
 We'll create a backup of the session 9 directory on fox.
(Since all your files are mounted on all of the physics public
computers, you don't need to copy to another computer, but this is
a demonstration!)
Go to the
parent directory of session_09.
Give this command:
rsyncbackup session_09 fox.mps.ohiostate.edu:780_backup
If you are asked if you want to continue connecting, answer yes.
You'll probably be asked for your password as well (it is the same one
everywhere). You should see a list of files transferred and some
statistics.
[If you don't have a Department unix account, backup to your home
directory with
rsyncbackup session_09 ~
(where ~ is your home directory; to go there use cd ~).]
 Check that the directory was transferred.
Now go back to the original session_09 directory
and change one file. (E.g.,
edit one of the .cpp files and add a comment.)
Then repeat the rsync transfer (from the appropriate directory),
using "!rsync" to avoid typing the entire alias.
You should find that only the changed file is updated.
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 9 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.
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 [Save for PS#4]
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 9.
Last modified: 07:34 am, February 11, 2009.
furnstahl.1@osu.edu