Exercises for Computational Physics I.

 

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Exercises

Course Description

This course addresses the students participating in the Lecture of Computational Physics I held by Professor Allen Caldwell. The topics presented in the lecture will be discussed here in greater detail, supplemented by additional material and examples. Problems will be described and discussed during the recitation. The homework problems assigned during the lecture will be discussed after the hand-in date.

Contact

Allen Caldwell | Manuela Jelen | Jing Liu

This course is given by Manuela Jelen together with me under the supervision of Professor Allen Caldwell. If you have any question, Please do not hesitate to contact us.

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Reference


Exercise I

Lecture 1 | Exercise (pdf) | Homework | back to top

Questions raised from the recitation

Q: How can I run ROOT in a CIP computer?

A: you have to do some special configurations before you can run ROOT in CIP. Please type after your shell prompt (e.g. ">"):
> source /etc/profile.local
> ini ROOT
> root
Don't type the shell prompt ">" in your console.

You can also intall ROOT by yourselves. Please follow the installation guide in ROOT homepage.

Q: Is ROOT macro the same as normal C++ file?

A: No. ROOT embeds a C/C++ interpreter named CINT to be able to execute scripts (macro) and command line input which have syntax similar to C/C++. The codes in the macros and the command lines have looser syntax requirement than normal C++ source, which makes it easier for part-time programmers to develop their small size codes to deal with special problems. But meanwhile, it means that CINT scripts are non-trivial to convert to compilable C++ source. Though there are some criticisms for CINT, it does make life easier for the physicists who don't care too much about the syntax.

Q: Are the codes in the ROOT macro executed interpretively?

A: Yes, if you execute the macro in the ROOT command line, CINT will interpret the codes line by line. Of course the execution speed is much slower than that of the compiled C++ codes. But the advantage is that when you change something in your macro you can rapidly get the result by using CINT. To sum up, execution time is not critical for physicists, but rapid problem-solving is.

Q: Can I write a standalone macro?

A: Yes, you can. It means that you don't use CINT to interpret it, but use common compiler to compile it and then execute the executable file. To do so, you have to write it exactly like a C++ program, including header files that you need, linking ROOT libraries during the compiling. There are some examples that you can follow in $ROOTSYS/test directory.

Q: I don't see any relation between the objects of TGraph and TCanvas in freefalling.C. How can the object of TGraph know where to draw the result?

A: Please check the source codes of the Draw() function in TGraph. Pay attention to the last two lines. You may find the reason.

Exercise II

Lecture 2 | Pi.C | a4b4.C | back to top

Linux Basic

Exercise III

Lecture 3 | Exercise | interpolation.C | back to top

Exercise IV

Lecture 4 | Bicycle.C | back to top

Reading Material of "C++ Class"

Reading Material of "C++ namespace"

Ideas from You

Fabian designed a set of classes to deal with the first order ODE of the form dx/dt = function(x, t) in general. This is a nice try. We will keep picking up some nice parts from your homework. Long for your contributions :-)

Exercise V

Lecture 5 | back to top

Adaptive Step Size

AdaptiveStepSize.C | Graphic Result

Something need to notice:

  • You don't need a very complex method to search for the most suitable step size. For this special purpose, You can just use a simple one, like Euler Method. And once you get the right step size, you can use any method that you like to calculate the results.
  • In this case, we try to solve dx2/d2t = -k/m*x, you have two variables dx and dv that you can use to compare to the specified accuracy ("eps" in the codes). Try both of them and see what would happen.

2nd and 4th Order Runge-Kutta

Oscillation.C | Graphic Result

Something need to notice:

the 2nd order Runge-Kutta method works better than the 4th one in this special case.

Planetary Motion

planetary.C | Graphic Result

Something need to notice:

I don't have time to comment this macro well. Check it if you are interested in it enough. In most of the case we will have a circle like orbit, which means sometimes the orbit changes rapidly on "x", sometimes on "y". If you compare dx with specified accuracy to search suitable step size, you will get good step size for "x" not for "y", you will lose accuracy on some parts of the whole orbit. If you compare dy with specified accuracy, you will lose accuracy on other parts of the whole orbit. Seems not need to implement adaptive step method in this special case.

Exercise VI

Lecture 6 | back to top

The Sun, Earth and Moon

sem.C | Graphic Result

The main tip here is that you can use a class named TVector3 provided by ROOT to simplify the codes. You can calculate the problem by using vectors, and don't need to loop over all 3 axises, X, Y, Z.

sem.C is written as a standard C++ program rather than a ROOT macro. It might show you how you use ROOT as another C++ library. To compile it, please put Makefile.arch and Makefile into the same directory where you put sem.C, and type make. Good luck!

Reading Material of GNUmake:

The Cylinder Capacitor

This is a really good example that shows you a clear physics picture in mind is very important for you to solve the problem correctly. I put the discussion of this problem on three pieces of paper ex621.JPG, ex622.JPG, ex623.JPG

Sorry, I am afraid that I made a mistake here. Some of you used the to solve this problem. I thought it couldn't be correct, but I was wrong. I didn't prove it mathematically, but I got a easy-to-think picture in mind, which showed that if we didn't simplify the problem into 1 dimension, we might get the correct result by using relaxation method. You can find my explaination in another piece of paper here ex624.JPG --- Jing 14/12/06

According to the suggestion from the professor, if one want to use the Cylinder Polar Coordinate to solve the problem in one dimension, he should approximate the form of Laplace Equation Cylinder Polar Coordinate, not in the Cartesian Coordinate. I wrote a ROOT macro to realize that. You can see the graphic result here.

Exercise VII

Guest Lecture - Kalus Dolag | back to top

Fabian gave a very nice talk about C++ in our recitation. Those who couldn't manage to join us in that afternoon can have a look of the slides here.

As we said before, you are encouraged to share your special knowledge with your classmates. We believe you can also benefit a lot from your sharing experience. Since we are not personally familiar with you, we don't know exactly who we can ask for contribution. Would you please stand out by yourselves if you are ready?

Exercise VIII

Lecture 7 | chargesInBox.C | back to top

The new ROOT class that we used in our solution is TTree. We created a object of TTree, saved the numeric results into it, then save the whole object to a root file. After that, we can use ROOT to open it, and use the plot function provided by TTree to show the results as we like.

Of course, this is not the only solution. Frederik Beaujean used Gnuplot to get a very nice plot of his numeric results (as showed below). Those who are interested in how to realize this may contact him :-)

Exercise IX

Lecture 8 | ex1 | ex2 | ex3 & swap.h | back to top

Exercise X

Lecture 9 | back to top

There is no universal method to calculate the eigenpairs (eigenvalues ai and their corresponding eigenvectors xi) of a matrix A. According to the Lecture,

  • if A can be diagonalized, one can use Jacobi Method to get its eigenpairs.
  • one can try to solve equation |A-aE|=0 to get the eigenvalues ai, and then solve (A-aiE)xi=0 to get the corresponding eigenvectors. If (A-aiE) is singular, you cannot solve (A-aiE)xi=0 by using the methods mentioned in the previous lecture. But if you happen to know the singular value decomposition of (A-aiE), let's say (A-aiE)=UΣVT, the eigenvector is just one of the columns of V. We'll figure it out how this works in the exercise 1.
  • If |a1|>|a2|≥|a3|≥...≥|an|, one can use Power Method to get the eigenvector corresponding to the largest eigenvalue a1, and use some modified Power Method to get the others. We'll go into detail in exercise 2
  • The last method mentioned in the lecture is variational technique, which allows the determination of the smallest eigenvalue and its eigenvector.

ex1. Singular Value Decomposition

Theorem. every m x n matrix has a singular value decomposition. [Ref. tutorial from University of Wisconsin-La Crosse]

From its SVD, one can know many properties of the original matrix A. One of the properties which is useful here is that the columns of V whose Σii are zero are the solution of Ax=0 [Ref. tutorial from Brown Univ.]

There are a lot of routines used to do the SVD, such as the web app. mentioned by the professor. ROOT also provides functions to calculate this [Ref. Section Linear Algebra of ROOT in ROOT Users' Guide]

If you are curious about how to do the singular value decomposition, please refer to the Algorithm of Singular Value Decomposition.

Remark. There are several conflicting notational conventions in use in the literature. The one in the lecture is just one of them.

ex2. Power Method

In the lecture, we've got the eigenvectors corresponding to the eigenvalues 8 and 1. In order to get the eigenvector corresponding to eigenvalue 7, we need some modified Power Methods to find the eigenvectors corresponding to the nondominant eigenvalues:

Markus Hanl wrote some nice comments in his codes. You can see them here.

Exercise XI

Lecture 10 | integration.C | back to top

Exercise XII

Lecture 11 | secant.C | Χ2 | back to top

Frederik Beaujean gave us a very nice introduction to Gnuplot. You can find here the talk and some simple examples to play with.

Exercise XIII

Lecture 12 | back to top

Exercise XIV