Functional programming is becoming increasingly widespread in industry. This trend is driven by the adoption of Scala as the main programming language for many applications. Scala fuses functional and object-oriented programming in a practical package. It interoperates seamlessly with both Java and Javascript. Scala is the implementation language of many important frameworks, including Apache Spark, Kafka, and Akka. It provides the core infrastructure for sites such as Twitter, Tumblr and also Coursera. In this course you will discover the elements of the functional programming style and learn how to apply them usefully in your daily programming tasks. You will also develop a solid foundation for reasoning about functional programs, by touching upon proofs of invariants and the tracing of execution symbolically. The course is hands on; most units introduce short programs that serve as illustrations of important concepts and invite you to play with them, modifying and improving them. The course is complemented by a series programming projects as homework assignments. Recommended background: You should have at least one year programming experience. Proficiency with Java or C# is ideal, but experience with other languages such as C/C++, Python, Javascript or Ruby is also sufficient. You should have some familiarity using the command line.
Lecture 1.5 - Example: square roots with Newton's method
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来自 École Polytechnique Fédérale de Lausanne 的课程
Functional Programming Principles in Scala
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École Polytechnique Fédérale de Lausanne
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课程 1（共 5 门，Specialization Functional Programming in Scala）
从本节课中
Getting Started + Functions & Evaluation
Get up and running with Scala on your computer. Complete an example assignment to familiarize yourself with our unique way of submitting assignments. In this week, we'll learn the difference between functional imperative programming. We step through the basics of Scala; covering expressions, evaluation, conditionals, functions, and recursion
与讲师见面
Martin OderskyProfessor
Computer Science
In this session, we are going to apply what we learned so far in a larger
example. The task is to develop a square root
function using Newton's method. That's the first time that we need more
than a couple of lines to actually express a program and we will learn new techniques
and new tools for doing so. So, what that means is that we want to
define a function with the signature. It takes a double and gives me back a
double, and return the value is the square root of its parameter x.
The classical way to compute that is by successive approximations using the method
of Isaac Newton. So, let's see what this method is.
To compute square root of x, we start with an initial estimate value.
It doesn't really matter what that estimate is as long as it's positive.
So, let's pick y = one. And then, we repeatedly improve the
estimate by taking the mean of the old estimate y and the value of x divided by
y, where x is the value we want to draw the root from.
So, it's, let's see that in action. For the value of x, in our example, would
be two. Our initial estimation is one, as usual,
so the initial quotient is two divided by one, that's two, the mean is 1.5.
Now, we plug in 1.5 to our estimation, we get a new quotient of 1.33, a new mean
that you'd see here and you'll see that it converges very quickly to the actual value
of two. So now, we have an abstract algorithm.
Let's see how we would put that into a function that computes that algorithm.
A typical way to code algorithms in functional languages is to go step by
step. So, we take a small task and formulate it
as a function and then probably, that task will need further tasks that then would be
defined in their own function. The first function we want to define is
the actual iteration called the square root Iter, which says, well, if you have a
guess and the value would be one to draw the root from, what do we do?
Well, either we stop the iteration and return the result or we go and do another
iteration step. The predicate that controls that we give
it a name, let's call it, isGoodEnough. So, if my guess is good enough, then I
just can return the guess. If my guess is not yet good enough, then I
have to improve my guess. Let's do that with another function in
prove and call, square root Iter again with the improved guess.
Note that square root Iter is recursive so its right-hand side calls itself.
One peculiarity in Scala is that the return type of a recursive function needs
to be given always whereas, for other functions, it's optional.
The reason for that is that to compute that return type, the Scala compiler would
have to look at the right-hand side and because the square root Iter function is
recursive it's stacked in a cycle. It would need the return type for that to
compute the type for expression. To break the cycle, we require that
recursive functions always have explicit return types.
It actually makes sense even for other publicly used functions to give an
explicitly return type because that's good documentation but it's not required by the
language. Okay.
So, let's implement this in practice. What we're going to see is a new way to
write programs. So far, we have done everything in the
repo. Now, for larger programs, we have
something that is better adapted. We go into Eclipse and we start a
worksheet. So, I take you through the steps one by
one. Here, we have an empty Eclipse thing.
We first start a new project the Scala project.
Call it, demands a name, call it [unknown] for the course.
And then, in the project what we can do is define a package, which is essentially a
directory in which we are going to put all the stuff from this week.
Call this Week one. We don't need to define packages but it's
good practice to keep stuff tidy and ordered.
And then, in that package, we define a new sheet.
It needs a name, let's call it Session. In a worksheet, what we can do is we can
type expressions and have them evaluated, just like in a repo.
So, we can do one plus two and expressions get reevaluated when we hit Save, that
will become [unknown] or Control+S. But the expressions show up on the right
in a different column that you see here. We can also define functions, let's say,
the absolute, absolute function that we've seen before.
So, we could say, well, if x is less than zero - x else x And the repo would answer
the, with the value of the absolute function, well, not the repo, the
worksheet but the two are actually related.
It's just a different mode of presentation.
One thing we can do which is nice, nicer in the worksheet is we can actually go
back and edit expressions and the reevaluation will then show the edited
value. So, it's not just a single line editing
window but a whole window that we can use. So, let's use the worksheet to complete
our square root functions. The first thing I'm going to do is take it
from the slide. So, I select square root Iter here and
paste it inside the worksheet. I can reformat with Control+Shift+F.
So, that would reformat things nicely. But the worksheet still doesn't compile
because, of course, I have two functions here that I haven't yet defined that would
be as good enough and improve. So, for isGoodEnough, let's do that first.
So, is, isGoodEnough takes these same parameters as square root Iter, and it
returns a bullion that says whether we can stop the iteration or whether we need more
steps. So, what could that predicate be?
Well, one way to do it would be to say, well, if we take the current guess here
and we square it. If we are close enough to the original
value x, then guess by itself would be close enough to the desired square root
value. So, let's code that up.
We would say, guess, guess - x, we would have to take the absolute value of that.
And it would have to be smaller than some epsilon value of, say, 001 for the moment.
Okay. So, that was number one.
The second function we need to define is Improve.
And that's now the core of the Newton method, we say, would say, well, it could
take a guess and an X. And to improve the guess, what do we do?
Well, we say, we need to take the mean of the value the current value of guess and
the quotient of x divided by guess. So, to take the average, we would first
sum the two values and then, divide by two.
Good. Now, we have a worksheet that compiles.
You see on the right-hand side, you see all of the types.
The last function we need to define is square root itself.
So, let's do that, what would that be? Well, it would be the square root
iteration. With our first initial value, we said,
that was 1.0 and x. Well, now, we have all the elements in
place and we can start to test our function.
Let's start with the square root of two and we get 1.4142.
So, that looks right. Let's do another one.
Square root of four, and we, we get something close to two, as we would
expect. So, as an exercise, I would like you to
think about the implementation we have so far and try to improve it.
One thing to improve is that the, is good enough test is not very precise for small
numbers. And it can lead to non-termination for
very, very large numbers. So here, let me demonstrate that.
And let's take a very small number. Let's say, one^-6.
So, one / 1,000,000.
So our square root, the square root we would expect to see is one over minus
three. And actually what we see is 0.03.
So, that's pretty far away from where we want to be.
And the other problem we have is if we take a large one, let's say, one to the
60th power. What would we see then?
Well, actually what we'd see is non-termination.
You see this here, with this little bar? That's as well, it doesn't find the square
root. Not so good.
Let me take that out with an escape. So, can you explain why that is?
Why is the, is good enough test not precise for small numbers and why can it
lead to non-termination for very large numbers?
And secondly, can you define a different version of isGoodEnough that doesn't have
these problems? Here are some numbers to test your, your
version, some very small and some very large numbers.
So, to answer that exercise, let's go back to our worksheet.
The problem, we had for both small numbers and, and large numbers was that our
isGoodEnough test is, takes an absolute difference.
It asks whether the absolute difference between the square of guess and X is less
than some threshold value. And that's not very good for very, very
small numbers because the number we want to find in the square root might, might
even be smaller than that. So, that, relatively speaking, that
epsilon value might be huge compared to the number we want to find.
And for very large numbers, we have the opposite problem, which is that very large
floating point numbers have, can actually be further apart than this epsilon value.
That means in the 52-bits, they have available for the [unknown].
It could be that the distance between one number and the next is actually larger
than 0.001. And in that case, of course the iteration
can never stop because, simply because there is no value that's ever good enough.
So, we can solve both problems by making the test proportional to x.
And that's very simply done by saying, okay, let's divide the absolute difference
by the current value of x and let's make that be smaller than the threshold value.
So, once I've done that, I can simply reevaluate my spreadsheet.
And now, I will see that everything computes nicely.
And for the small number, I get something very close to one^e-3 as expected, and for
the large numbe, I get something like e1 to e30, and that's also as expected.
