ability to write simple programs.
In particular, we're going to introduce the concept of iteration, or
how to repeat a method multiple times in order to reuse the computation to
execute something an arbitrary number of times.
Once we have iteration, we can start doing some interesting classes of
Starting with guess and check methods, the guess and answer, check it, and
then use the results to improve the guess.
This will lead naturally to the idea of loop constructs, or mechanisms in
our language that let us generalize the idea of repeating a computation
until a particular condition is reached.
This will let us generalize our guess and check methods to a broader group
of algorithms called successive approximation, and especially to a
very common and very powerful method called bisection search.
At the end of this lecture you will be able to write simple algorithms that
solve numerical problems by improving approximate estimates of the answer to
get successively better guesses.
ERIC GRIMSON: In the last section, we introduced a
couple concepts in terms of writing programs or scripts.
We had straight-line programs, where we had just a linear
sequence of instructions and we executed them 1
at a time in order.
And we added the idea of branching, or conditionals,
where we could do a test, and depending on whether that test
was true or false, we might skip to 1 piece of code or a
different piece of code.
But in both cases, we're doing each instruction at most once.
Useful, but not great.
To get to the ability to write programs or scripts with
arbitrary complexity, we need 1 more key concept, and that
concept is the idea of being able to reuse a piece of code
an arbitrary number of times.
We might have a set of things we want to do once or
depending on some value, twice, or depending on some
value, 27 times or 1,000 times.
And rather than having to copy the code that many times, we'd
like to be able to have the computer automatically reuse
the code as many times as desired.
That notion of iteration is going to be incredibly
valuable, and we're going to talk about some constructs to
help us make that happen.
to an iteration loop, which is this chunk right here, we
start with a test.
That will be Boolean that will return either True or False.
If it's True, we're going to go down and execute some set
of instructions, the body of the loop, and then go back
around and do the test again.
And we'll keep doing that.
As long as the test is True, we'll execute that same set of
instructions in the body over, and over, and over, until
finally, the test is False, in which case we skip around the
loop body and pick up the computation elsewhere.
This notice allows me, based on this Boolean test, to be
able to do the code some arbitrary number of times, and
that's really great.
Let's look at an example.
Here's a simple example that shows that idea, and I want to
talk about the syntax of the construct, but also about what
the example's going to do.
This is a simple piece of code that squares a number.
successively adding x to itself x times, which is of
course what x squared actually does.
I add x copies together.
Notice what we've got.
We're going to set up some value of x.
I'm going to do a binding there.
We're going to bind a variable ans for answer, which is where
we're going to add up all the instances of x to get where we
want to be.
And we're going to need to keep track of how many
versions of x do we still have to add in.
So we're going to set up another variable called
itersLeft, initially bound to x.
The looping construct, the iteration construct, we have
is called a while loop.
And there's the key word while, and it has the
It has a Boolean test-- in this case, we're checking to
see whether itersLeft is equal to 0 or not--
followed by a colon, and then it has a set of indented
instructions, which of course, are the body of the loop.
The way the while loop works is it
first tests that Boolean.
If that Boolean is True, itersLeft is not equal to 0 in
this case, it will then execute each of the
instructions in the body in sequence.
And when it gets to the end of that sequence of instructions,
it's going to go back up and retest the Boolean again.
So it will cycle through this code multiple times until that
Boolean is finally False.
When it is False, it will skip down to the end of the loop,
which is shown by where the indentation stops, and pick up
the new set of instructions and continue.
So there's the structure, let's see if this does the
Well, here's my code, and let's just walk through this.
We won't run it on idle, we'll just walk through it.
Initially, x is bound to 3, ans is bound to 0, and
itersLeft is bound to 3.
The while loop says, test to see if this is equal to 0.
Since it is not equal to 0, the test is True, and
therefore, I take the current value of ans, the current
value of x, add them together, and rebind that to ans.
I then take itersLeft, subtract one from it, and
rebind that to itersLeft.
So I've decremented itersLeft, and I've incremented ans.
I go back up, and again, I check.
Is that not equal to 0?
It is not equal to 0.
Therefore, I take ans and I take x, I add them together,
and I rebind those to be the new value of ans.
I change itersLeft by 1.
That's my new value there.
And again, I go back through the loop.
That is not equal to 0.
Well, the test is still True, so again, I take ans, I take
x, add them together, create that to be my new binding for
ans, which is 9.
I take itersLeft, subtract 1 from it, there's my new value
And again, I go back to the top of the loop.
At this point, 0 being not equal to 0 is False.
Terrible way of saying it, but that test is no longer True.
In which case, I will skip to the end of this loop, and
print out that x * x, or if you like, 3 *
3, is equal to 9.
A little slow, but it does what I want.
Notice I have reused this code an arbitrary number times.
And in fact, if I were to change x to be something else,
I will reuse that piece of code a
different number of times.
There's my iteration that I really want.
You can also see some properties of
an iterative loop.
First of all, we need to set an iteration variable outside
of the loop.
In this case, it's x and itersLeft.
Actually, the one I really care about here is itersLeft.
I also need to test that variable to
determine when I'm done.
There's the use of itersLeft inside of the test.
Now, it could be a simple test, it could be a more
compound test, but that's basically what I need to test.
And finally, I need to be changing that variable inside
of the loop, right there.
If I didn't, then that test value would never change,
which means I would never stop the loop.
But there's a property I need.
When I set up an iterative loop, I need to say what's the
variable I'm setting outside, how am I testing it, and am I
making sure to change it somehow inside of the loop in
order to ensure that I get the pieces that I want.
So what do we have now?
We already saw that branching structures, things like
conditionals, will let us jump to different pieces of code
based on a test.
Once we add in the idea of looping over the code, things
like a while loop, that allows us to repeat pieces of code
until the condition is satisfied.
So it's a generalization of a conditional.
On simple branching structures, we've already said
that programs like that are constant time, that is, we
execute each instruction at most, once.
With looping structures, notice
that things are different.
Now, the program is going to take an amount of time that
depends on values of variables as well as the amount of code
or the number of instructions inside the loop, because it's
going to depend how many times we walk through the loop, and
that depends on the variable.
Nonetheless, these loops are going to be really valuable,
and we're going to turn to that next.