780.20: 1094 Session 11

Handouts: gaussian_random.cpp and random_walk.cpp printouts, Monte Carlo excerpt from chpt. 7.

Today we'll play some games with the GSL random number generators.

• Generate some random walks and verify their properties.
• Try out primitive Monte Carlo integration.
• Take a look at an alternative version of the random walk code using classes.

Please work in pairs (more or less). The instructors will bounce around 1094 and answer questions.

Random Number Generation

The program gaussian_random.cpp calls GSL routines to generate both uniformly distributed and gaussian distributed numbers.

1. Look at the gaussian_random.cpp code (there is a printout) and identify where the random number generators are allocated, seeded, and called. Compile and link the code (use make_gaussian_random) and generate pairs of uniformly and gaussian distributed numbers in random_numbers.dat.
2. Think of a way to use gnuplot to roughly check that the random numbers are uniformly distributed. [Hint: Your eye is a good judge of nonuniformity.]

3. You can check the distributions more quantitatively by making histograms of the random numbers. Think about how you would do that. Then take a look at gaussian_random_new.cpp, which has added histogramming (as well as automatic seeding). [Feel free to criticize the implementation!)] Use gnuplot to check the uniform and gaussian distributions. Do they look random?

4. Run gaussian_random_new.plt to plot and fit the gaussian distributions with gnuplot. Do you reproduce the parameters of the gaussian distribution?

5. Take a look at (and run) rolling_dice.nb to see how to do random numbers in Mathematica. [For your information only.]

Random Walking

We'll generate random walks in two dimensions using method 2 from the list in section 6.10 of the Landau/Paez handout. In particular we'll start at the origin: (x,y) = (0,0) and for each step select Delta_x at random in the range [-sqrt(2), sqrt(2)] and Delta_y in the same range. So positive and negative steps in each direction are equally likely. The code random_walk.cpp implements this plan.

1. Why do the ranges for x and y go up to sqrt(2) instead of 1?

2. Look at the random_walk.cpp code and identify where the random number generator is allocated, seeded, and called. Compile and link the code (use make_random_walk) and generate a random walk of 6400 steps.
3. Plot the random walk (stored in "random_walk.dat") using gnuplot (use "with lines" to connect the points). Repeat a couple of times to get some intuition for what the walks look like.
4. Check for the endpoints of a few walks (you can use the command "tail random_walk.dat" to do this easily) and explain to your partner why the distances R from the origin are what you might expect. (Can you reproduce the derivation of how R scales with N?)

5. Now we'll study how the final distance from the origin R = sqrt(x_final^2 + y_final^2) scales with the number of steps N. Note that now we don't need to save anything from a run except the value of R. The value of R will fluctuate from run to run, so for each N we want to average over a number of trials. How many? The book claims about sqrt(N); does this make sense?

6. Edit the code to make sqrt(N) runs for each value of N and takes the average of R. Make an appropriate plot that reveals the dependence of R on N. [The code random_walk_length.cpp and plot file random_walk_length.plt implement this task. Try it yourself before looking at those.] Does it agree with expectations?

Monte Carlo Integration: Uniform Sampling

Your goal is to calculate the 10-dimensional integral of

```  (x1 + x2 + x3 + ... + x10)^2
```
where each of the variables ranges from 0 to 1. The exact answer is 155/6. The basic Monte Carlo method is described in Section 7.12 (see handout). In particular, equations (7.14) and (7.16) show that the integral is given approximately by the range(s) times the average of the function evaluated at N random vectors. (So for a 5-dimensional integral, each vector is a set of 5 random numbers {x1,x2,x3,x4,x5}.)

[If you are running short of time, use mc_integration.cpp and mc_integration.plt below instead of writing your own.]

1. Modify gauss_random.cpp or random_walk.cpp (copy one of them to mc_int.cpp) to make a Monte Carlo estimate of the 10-dimensional integral using N vectors (where N is input as in random_walk.cpp). Run it a few times with different seeds.
2. Now for each value of N, make 16 trials and take the average as your official estimate of the integral.
3. Try sample sizes of N = 10, 20, 40, ..., (i.e., from 10 to some suitably large number by multiples of 2).
4. Plot the absolute value of the error versus N on a log-log plot and try to identify the approximate dependence.

Monte Carlo Integration: GSL Routines

Run a test program to do the same integral as in the last section but with vegas and miser.

1. Take a look at the program gsl_monte_carlo_test.cpp while also looking at Monte Carlo integration in the online GSL library.
2. The integral above is not a great test. After running the program, change the integrand to something more interesting (your choice!). Don't worry about knowing the exact answer; compare the results from the different routines.

C++ Class for a Random Walk

The random walk code random_walk.cpp is basically written as a C code with C++ input and output. Here we reimplement the code as a C++ class (originally in anticipation of using it in a Qt application).

1. In the RandomWalk directory, compile and link RandomWalk_test (using make_RandomWalk_class_test). Run it to generate "RandomWalk_test.dat", which you should plot with gnuplot to verify that the output looks the same as from random_walk.cpp.
2. Compare the old and new code (you have printouts of each). Discuss with your partner the advantages (and any disadvantages) of the definition of RandomWalk as a class. List some.

3. An advantage of programming with classes is the ease of extending or generalizing a code. List two ways to extend the class definition.

4. For the experts (or anyone willing to try!), modify the code to do the following:
• Extend the code to make available the x- and y-components of the last step (what are called delta_x and delta_y internally).
• Allow for the upper and lower limits of the step size to be initialized by the user. (And you still want to be able to use the current version that doesn't require these.) [Hint: Can you have more than one constructor?]

• How would you dynamically create and destroy random walks? [Hint: new and delete]

• How would you allow the random number generator to be changed?