2:33

So now that we know how to do fixed points, can we somehow get new insight in

the, our square root function? Previously we gave you Newton's method

essentially as fact. That's the sequence that you have to use

for your iterations. Can we somehow derive that sequence?

When in fact, yes. So here's a specification of the square

root function. What is square root of x?

Well, it would by a number y, such that y squared equals x.

That's what it means to be a square root. Now that we can, now we can do just simple

algebraic manipulation. So, we divide both sides of this equation

by y, so we get square root of x is the number y such that y equals x divided by

y. But what that means is that square of the

x is the fixed point of the function y to x over y because if you put y in and you

get x of a y and x of a y is the same as y then that equation is satisfied.

So a solution of the equation is the same as a fixed point of this function again.

5:12

So if we do that, then we see that. Our guess, oscillates between one and two

all the time. And if you do the computations for square

root, I encourage you to do that and you will find that yes indeed if you trace the

execution using the substitution model then that's precisely what will happen.

So, how can we do better? One way to control these oscillations that

we were seeing is to prevent the estimation from varying to much and one

way to do this is by averaging successive values of the original sequence.

So instead of going 1,2, 1,2 we take the average of two successive values that

would give us 1.5 and that would set us on the right path to convergence.

So let's try that. So what we want to do is, instead of

saying x over y, we want to take the average of the two values.

So that would be y plus x over y divided by two.

And we get back to the square root functions that we've seen in the first

week. In fact, if we expand the fixed point

function fixed point and plug in the definition of this average temping

function of, for, for square root, then what we will find is that we get back a

very similar square root algorithm than the one we've seen last week.

So it's no surprise that we get the same results back.

So the previous examples have shown that the expressive power of a language is

greatly increased if we can, can pass, functions as arguments.

I'm going to show you next that it's also very useful to have functions that return

other functions, and I'm going, going to show you that with fixed point as the

example. Let's go back to our observation that

square root of x is a fixed point of the function that map y to x over y.

We have taken that observation, we have seen that square root doesn't converge

that way, but we could make it converge by averaging successive values.

This technique of stabilizing by averaging is general enough to merit being

abstracted into its own function. Let's see what this function would look

like. It would have the function average down.

It takes an arbitrary function from double to double.

It takes the value x of type double and it then performs the, it computes the average

of x and f of x. Let's do an exercise.

Given fixed point and average damp, can you write a square root function that uses

both functions. So, let's see how we would solve this

example. I have given you the average damp function

that we developed on the slide. What would square root be now?

So square root would be a fixed point of a function that maps y to x over y with

initial value one. But that was the one we did that didn't

converge. So, what we do is we use average damp.

On that function here. And if we run it, we can, we get the

expected square root result. So what average stamp is, is it's a

function that takes a function, namely this function that is at, at the root of

the square root specification, and it returns another function, namely the

function that is essentially the same iteration but with averaged stamping

applied. So it's a function that takes a function

to another function. If you look at this definition of square

root, then I think you would agree that it's probably very difficult to write

something that is clearer and shorter than this very definition.

It came from the specification of what it means to be a square root.

We said we derive that the solving the equations for square roots means taking

the fixed point of this function. We determined that we needed to take

averages to dampen the oscillation, so we use this average damp function in the

middle, and we are done. So in summary, we saw last week that

functions are essential abstractions. Because they allow us to introduce general

methods to perform computations, as explicit and named arguments in our

programming language. And this see, week we've seen that these

abstractions can be combined with high auto functions to create new abstractions.

So, that's a very powerful way to compose and abstract programs.

And as a programmer it's always good to look for opportunities to extract entry

use. It's not true that the highest level of

abstraction is always the best one. Sometimes it's actually better to stay a

little bit more concrete. But it's important to know the techniques

of abstraction so that you can use them when they're appropriate.