====== Math 445 lecture 9: for and while loops ======

Today's main topic is //loop// constructions. Loops perform a given action repeatedly. There are two main flavors of loops: the ''for'' loop and the ''while'' loop.

===== for loops =====

A ''for'' loop repeats a given action a fixed number of times. The general form of a ''for'' loop is  
<code matlab>
for value = vector
  action
end
</code>
The ''action'' will be performed ''length(vector)'' times, with ''value'' set to each element of 
''vector'' successively. It's probably clearer to see this by example. 

**example 1:** Print the numbers from 1 to 10.
<code matlab>
for n = 1:10
  fprintf('The value of n is %i\n', n);
end
</code>

**example 2:** Print the numbers from 10 to 0.
<code matlab>
for n = 10:-1:0
  fprintf('%i ');
end
fprintf('blast off!\n');
</code>

**example 3:** Write a ''mysum'' function that computes the sum of the elements of a vector.
<code matlab>
function s = mysum(x)
% return the sum of the components of vector x
  
  s = 0;               % start with partial sum s = 0
  for i = 1:length(x)  % loop over all elements in x
    s = s + x(i);      % add x(i) to the previously computed partial sum
  end
  
end
</code>

**example 4:** Write a function that computes the factorial of //n//.

<code matlab>
function p = factorial(n)
% return the factorial of n (assume n is a positive integer)
  
  p = 1;               % start with partial product p = 1
  for k = 1:n          % loop k over all integers from 1 to n
    p = p*k;           % multiply previous partial product by k
  end
  
end
</code>

**The for loop is probably the single most important programming construct in numerical mathematics.**

===== while loops =====

A ''while'' loop repeats a given action as long as its condition is true. The general form is
<code matlab>
while condition
  action
end
</code>

In order for the ''while'' loop to terminate, the action must modify variables in the condition.
For example, the factorial function above could also be written this way...