September 4, 2007 --- Class 3  --- AWK, Intro to Euler Method

Activities:

	AWK
		AWK is a useful scripting language that can help you to manipulate
		information in a file.  "The AWK Programming Language" is by Alfred Aho,
		Brian Kernighan and Peter Weinberger.  You should have already read the 
		tutorial I asked you to print and read in the first class.  You can
		also find useful information at www.oreilly.com for a variety of
		topic regarding computers and computing.

		PERL is another useful scripting language which is probably more
		popular than AWK now.  However, AWK is so simple that it is very
		convenient for class.  You may eventually decide to learn PERL.

		If you have \delta t values in a file and you want to have
		\delta t and 0.5*(\delta t)^2 on each line, it is easy to
		do this with an awk command.  Assume the \delta t are in
		a file with one value per line and the file is called col.

		awk '{print $1,0.5*$1*$1}' col
		
		will result in two columns with the desired numbers.  The output
		may be redirected to a file if desired.

		awk may be used to create a list of numbers, as well
		awk 'END{for(i=1;i<$1; ++i) print 0.01*i}'
		can be used and you may interactively type a number and
		then on the next line ^D (which stands for pressing d, while
		holding down the control key).

		AWK is a very versatile pattern matching utility that can be
		used to manipulate files.  Each AWK command consists of a
		pattern followed by an action.  If the pattern is matched, the
		action is performed.  The action is placed inside curly
		braces.  A very simple awk command is:
		  awk '{print}' file_name
		The single quotes protect the curly braces from being 
		interpreted by the shell as special characters.  In this
		AWK command, there is no pattern to match, so the print
		command is always invoked.
		  awk '/1/ {print}' file_name
		has a pattern that matches any line containing the character 1.
		Only such lines will be printed.
		  awk '/^1/ {print}' file_name
		In this example, "^" is a special character to indicate the 
		beginning of a line.  Thus, only lines that begin with "1"
		will be printed.
		  awk '{print $2}' file_name
		will result in the second word of each line being printed.
		(Words are separated by white space, i.e., spaces or tabs.)
		  awk '{print $2,$1}' file_name
		prints the second word followed by the first word.

		You might like to try the above examples with the file
		cols that you are welcome to copy from my home directory.

     Introduction to the Euler Method

	Tea or coffee cooling problem (CSM, Sec. 2.1)
		This is the problem discussed in chapter 2.  Should you add
		the milk to the tea as soon as it is poured, or just before
		you want to drink it in order to cool it as quickly as possible?
		To solve this problem, we are going to learn to integrate a
		differential equation numerically via the Euler method.

	Euler method
		The Euler method is based on a very simple approximation.
		You know that  dy/dx = lim_dx->0 (y[x+dx]-y[x])/dx.  In the
		Euler method, we just don't take the limit as dx->0, we use
		a small value of dx.  In this case, if dy/dx = f[x],
			(y[x+dx]-y[x])/dx = f[x] (approximately) and so,

			y[x+dx] = y[x] + dx * f[x]

		In this way we can update y to successive values of x by
		updating dx at each step.