Math Software Programming

This is not a course in programming. On the other hand, the whole point of using mathematical software is to save labor. Computers love to perform repetitive and tedious tasks, and are good at doing simple operations without mistakes. Those are the kinds of things that people are not good at doing. For that reason, we should know how to do some simple programming. Here are some naive principles of programming in Matlab and Maple.

Functions

It is best to make programs modular. Tasks that will be performed many times can be put in "functions"; code objects that are almost self contained. They take some input arguments, operate on those for a while, and return some outputs. A simple sketch of a function follows.

	function(arg1,arg2)
	Now use arg1 and arg2 to compute a variable called output
	return output
	end

Arguments

Most programs take arguments - variables and values that you type in for them to use. Typically arguments appear in parentheses after the function name. For example, in Matlab you pass the argument x to the function norm as

		norm(x)

Return values

Most programs return variable values. In other words, certain things it computes are available after the program finishes. We can assign variables to the things returned so we can use them later. For example, the following line takes the result of the norm function and saves it in a variable called mynorm.

		mynorm = norm(x)

Programming is just like using the interactive interface

You are comfortable working with a Matlab command line or a Maple worksheet. Programming these packages is almost the same. The chief difference is that instead of writing things line by line, you write many lines into a file. Running that command file executes all your commands sequentially. Still, the Matlab or Maple commands you know are the ones you use.

To do one thing a certain number of times, use a "for" loop

The most powerful thing about computers is their high tolerance for boredom. Make the computer do the boring things, and save your brain for the tricky things by using "for" loops. They are called loops because the execution goes around and around. These always have the form

	  for i from start_value to end_value,
	      tasks to be repeated go here
		  they frequently depend on the variable i
		  when the "end" is reached, the program loops back to the "for" statement until end_value is passed
	  end

To do something as long as a condition is true, use a "while" loop

Sometimes you don't know beforehand how many times you want to do a task. Instead, you want to do it as long as some condition is true. In this case, you use a "while" loop, instead of a "for". This is key: the "for" loop does some task a fixed number of times; the "while" loop does some task for just as long as some condition is true.

	  while (some condition is true),
	      tasks to be repeated go here
		  they may or may not depend on a counter i
		  when the "end" is reached, the program loops back to the "while"
	  end

To make decisions, use an "if" statement

Your program can choose to do different things based on e.g. whether some expression yields a positive or negative result. To do this use an "if" statement. If constructions always look something like the following.

	  if expression
	     do the tasks in here if expression was true
		 only one of these branches is executed
	  else
	     do these tasks if the expression was false
		 when the end is reached, we are done - no loop
	  end

The expression can take many forms - examples in Matlab notation follow.

	  if x+5<10
	  if sin(i*x)==0
	  if n/5 <= 100

The last test will take place at the final exam time on Tuesday, 12 December, from 1:30-3:30. It will be written as a one-hour (not 50 minute) exam, but you may have the full two hours for it. In other respects it will be very like the other tests, but comprehensive - it will emphasize Python, but cover all the topics we have seen. There is a Sample Exam, but be aware that things will have changed somewhat with the advent of ChatGPT.

Assignment A is posted.

Department of Mathematics

Math 300: Mathematical Computing