# 780.20: 2082 Session 3

Handouts: Background notes for Session 3; Landau/Paez chapter 4 excerpts; Gnuplot fitting example; Integrals with singularities; C++ formatting

Your goals for today and next time (in order of priority):

• Finish leftover tasks from Session 2
• Look at a quick demo about comparing floating-point numbers
• Use a makefile to compile a project with multiple .cpp files and a header (.h) file
• Duplicate the results in Fig. 4.3 with a log-log plot, then fit the slopes with Gnuplot
• Practice coding an algorithm: Add a new subroutine with the 3/8 rule and analyze
• Add an appropriate integration routine from the GSL and analyze the error
• Modify the code to use a different function, then evaluate an integral with a singularity
• Use Mathematica to check accuracy of your code

Please work in pairs (more or less). Dick and Daniel will bounce around 2082 and answer questions (don't be shy to ask about anything!).

You should make or change to a 780 sub-directory, download session03.tarz from the 780 homepage and unpack it ("tar xfvz session03.tarz").

## Leftover Tasks from Session 2

Here are the priority tasks to finish from Session 2. Don't worry about other items; when you complete these, move on to Session 3.

1. Summing in Different Orders. Finish the entire section. Be careful in part 5: There is one bug that cannot be caught by the compiler (check against the answer in the comment to track it down).
2. Bessel 1-3 Skip Bessel 4.
3. Mathematica Scavenger Hunt. Do the first one only.

## Makefiles for multiple project files (including header file)

Last year, we took one of the "all-in-one" C programs from the Landau/Paez text, integ.c, and converted it to C++: integ.cpp, and split it up:

• integ_test.cpp, which has the main program and the function to be integrated
• integ_routines.cpp, which has the integration functions themselves
• integ_routines.h, which has function prototypes
There is also a function in gauss.cpp and there is make_integ_test to compile it all. This year, we've also modified the codes so that the function is passed as an argument to the integration functions.

The idea is that the integration routines should be isolated in a file by themselves. The main program just invokes these routines. The header file conveys the prototype information about the integration routines to the main program (and to any other functions that might call the routines).

1. Look at the original file (integ.c) and see how it was converted and split up into integ_test.cpp, integ_routines.cpp, and integ_routines.h (the cpp files are printed out for you). Also look in ORIGINALS to see the change to passing the function for the integrand. Why do these changes improve the code?

2. Change integ_test.cpp to output relative (rather than absolute) errors.
3. Create the executable integ_test using the makefile make_integ_test and run it to generate the integ.dat output file. Note that only the files that have changed are recompiled by the makefile!
4. Use Gnuplot to reproduce Figure 4.3 on page 59 of the Landau-Paez handout. Explain what you can learn from the plot to your partner (and then an instructor). Consider all regions of the graphs.

5. Can you change the loop in integ_test.cpp so that the points on the log-log plot are evenly spaced?

## Finding the Approximation Error From a Log-Log Plot

From the plot in the previous section, we can estimate the approximation errors by eye. Now we want to actually fit lines to find the slope using Gnuplot. Use the handout as a guide.

1. Modify the code so that it outputs the logarithm base 10 (log10) of N and the relative errors.
2. Find the slopes of the trapezoid and Simpson's rule plots in the regions where they are linear.
3. Are the results consistent with the analysis in the text? Can you fit the round-off error region?

## Coding an Algorithm

This is primarily practice converting an algorithm to working code. Your goal is to add a new function to integ_routines.cpp, called three_eighth, which implements the 3/8 rule from Table 4.1.

1. From the rule and the discussion in the text, write some pseudocode to explain how you would integrate a function with this rule.
2. Implement your pseudocode in C++ by adding a new routine (called three_eighth) to integ_routines.cpp (be sure to add a prototype to integ_routines.h).
3. Modify integ_test.cpp to output to a new file the results from the new routine, and add them to the previous plot. Can you explain the approximation error? [Warning: If you don't change the way that integ_test.cpp loops through the number of intervals, you will likely run into a subtle error!]

## GSL Scientific Library Yet Again

1. Go to the web page with the GSL manual (it is linked from the 780.20 web page). Find an appropriate integration routine for the test integral we've been working on.
2. Add another calculation to integ_test.cpp (with output to a file) using this GSL routine.
3. Compare the accuracy of the GSL routine to the others on an error plot.

## More Complicated Integrals

1. Choose one of the integrals with singularities from the handout [Eq.(9) is a good choice!].
2. Determine an "exact" answer analytically or using Mathematica (see instructors for assistance).
3. Modify your code so that you can analyze the integral with one of the integration rules several ways, using an error plot to compare (these may not all apply to a given integral):
1. Brute force (unmodified)
2. Subtracting the singularity
3. Changing variables