Computational Methods Junior Honours : Checkpoint 1

The purpose of this checkpoint is to refresh you on the mechanics of editing, compiling and running java code, which you learned in
Physics 2 Scientific Programming

You should refresh your memory of with the Physics 2 Scientific Programming course and associated documents.

A list of the most needed Unix commands is here

Particular attention should be payed to

· Loops

· Methods

· Objects

Some useful Java documentation (increasingly advanced).

• http://java.sun.com/docs/books/tutorial/getStarted/index.html
• http://java.sun.com/docs/books/tutorial/index.html#basics
• http://java.sun.com

The checkpoint will also teach you how to write numerical data into text files, and display them as presentation quality graphs
using the popular xmgrace tool.

Example: CosinePlotter class

• Make a directory for this course, we'd suggest
• mkdir CompMeth
• cd CompMeth
• We recommend that you use a single directory for all your java code in this course.
• You should keep copies of your plots and datafiles in a subdirectory for each checkpoint
• Download java files CosinePlotter.java
• Read it using the "emacs" editor

CosinePlotter.java defines a class that outputs the values of cos(x) for one period.

The cosine is evaluated in a method that is accessed by a method (aka function) call.

• Compile it using the following

javac CosinePlotter.java

• Run the test

java CosinePlotter

• Understand how the output is generated.

Your task:

• Create a file called HarmonicPlotter.java in a similar style to CosinePlotter.java. Write a class HarmonicPlotter to

• Open a file using

• PrintWriter out = new PrintWriter( new FileWriter("harmonic.dat"));
• Here we declare a variable out, of type PrintWriter.
• PrintWriter is a class defined in java.io, which is imported at the top of the program
• out is assigned with a new PrintWriter value.
• We could have called “out” anything we want – variable names are the programmers choice
• The PrintWriter constructor method is called with a new FileWriter class.
• The FileWriter constructor is called to make it write a file called “harmonic.dat”
• NOTE – once classes are written they can then be used as a type of variable.
• You can now write to this "PrintWriter" the same way you print to the screen
• out.println(A+" "+B);
• System.out is related to PrintWriter (actually it is a member variable of type PrintStream in class System)

• Remember to close the file using out.close() or the last data written may not appear

• View the file using the graphing tool xmgrace

• xmgrace harmonic.dat

• Modify the function computed

• The function printed now should be the sum  sin x + 1/3 sin 3x + 1/5 sin 5x + 1/7 sin 7x

• Use a for loop to add together the terms (i.e. the order).
• Remember that a for loop consists of
1. for( initial-assignment ; loop-end-check ; variable-update ) { loop-body }
2. e.g.

for ( int i=0;i<10;i++){

System.out.println(“i = “+i);

}

• Write this in a style you can change easily to compute 100 , or 1000 terms.

• Math.sin(n*x) is likely required

• Learn to use xmgrace

• Use xmgrace to label the axes (menu plot/axis properties), set  the symbols (double click on the lines) and change colors

Save the file as a postscript file (harmonic.ps) using the print menu and print it out. 

o Play with the zoom (magnifying glass) and autoscale (AS) buttons on the left hand side

o Try playing with logarithmic scales, and change the plot title.

• Produce a plots containing only the first one, two, three and four terms of the sum by inserting an outer loop over the order.
• Printing a spare blank line between each order will enable xmgrace to plot these overlayed in different colours

· Can you tell what is happening?

• This is the lowest terms in the fourier series for a square wave
• As more terms are added it gets more accurate and therefore more square-like
• You will cover Fourier series later this term in Physical Mathematics
• For the purposes of this Checkpoint it is sufficient to see this as practical proof that an arbitrary function can be very well described by the right set of sines and cosines.
• This should provide you with a concrete experience when you cover Fourier transforms in detail.

· Advanced:

• Compute by hand the fourier coefficents for a step function centred at the origin.
• Use the canonical form in this link to compute fourier coefficents a(n) and b(n)
• Show that for even n the sin  nx coefficient is zero
• Show that for odd n the sin nx coefficient is  4/(n PI).
• Produce a harmonic plot that approximates a square wave of amplitude 1.0