   This HTML version of Think Perl 6 is provided for convenience, but it is not the best format of the book. You might prefer to read the PDF version.

# Chapter 6  Iteration

This chapter is about iteration, which is the ability to run a block of statements repeatedly. We saw a kind of iteration, using recursion, in Section ??. We saw another kind, using a for loop, in Section ??. In this chapter we’ll see yet another kind, using a while statement. But first we want to say a little more about variable assignment.

## 6.1  Assignment Versus Equality

Before going further, I want to address a common source of confusion. Because Perl uses the equals sign (=) for assignment, it is tempting to interpret a statement like \$a = \$b as a mathematical proposition of equality, that is, the claim that \$a and \$b are equal. But this interpretation is wrong.

First, equality is a symmetric relationship and assignment is not. For example, in mathematics, if a=7 then 7=a. But in Perl, the statement \$a = 7 is legal and 7 = \$a is not.

Also, in mathematics, a proposition of equality is either true or false for all time. If a=b now, then a will always equal b. In Perl, an assignment statement can make two variables equal, but they don’t have to stay that way:

```> my \$a = 5;
5
> my \$b = \$a;   # \$a and \$b are now equal
5
> \$a = 3;       # \$a and \$b are no longer equal
3
> say \$b;
5
```

The third line changes the value of \$a but does not change the value of \$b, so they are no longer equal.

In brief, remember that = is an assignment operator and not an equality operator; the operators for testing equality between two terms are == for numbers and eq for strings.

## 6.2  Reassignment

As you may have discovered, it is legal to make more than one assignment to the same variable. A new assignment makes an existing variable refer to a new value (and stop referring to the old value):

```> my \$x = 5;
5
> say \$x;
5
> \$x = 7;
7
> say \$x
7
```

The first time we display \$x, its value is 5; the second time, its value is 7.

Figure ?? shows what reassignment looks like in a state diagram.

Reassigning variables is often useful, but you should use this feature with some caution. If the values of variables change frequently, it can make the code difficult to read and debug. Figure 6.1: State diagram.

## 6.3  Updating Variables

A common kind of reassignment is an update, where the new value of the variable depends on the old:

```> \$x = \$x + 1;
```

This means “get the current value of \$x, add one, and then update \$x with the new value.”

If you try to update a variable that has not been given a value, you get a warning, because Perl evaluates the right side of the assignment statement before it assigns a value to \$x:

```> my \$x;
> \$x = \$x + 1;
Use of uninitialized value of type Any in numeric context
in block <unit> at <unknown file> line 1
```

Before you can update a variable, you have to declare it and initialize it, usually with an assignment statement:

```> my \$x = 0;
> \$x = \$x + 1;
```

Updating a variable by adding 1 is called an increment; subtracting 1 is called a decrement.

As mentioned earlier in Section ??, Perl has some shortcuts for increment and decrement :

```\$x += 1; # equivalent to \$x = \$x + 1
\$x++;    # also equivalent

\$x -= 1; # equivalent to \$x = \$x - 1
\$x--;    # also equivalent
```

## 6.4  The while Statement

Computers are often used to automate repetitive tasks. Repeating identical or similar tasks without making errors is something that computers do well and people do poorly. In a computer program, repetition is also called iteration.

We have already seen two functions, countdown and `print-n-times`, that iterate using recursion (see Section ??). Because iteration is so common, most programming languages including Perl provide language features to make it easier. One is the for statement we saw in Section ??. We’ll get back to that later.

Another is the while statement. Here is a version of countdown that uses a while statement:

```sub countdown(Int \$n is copy) {
while \$n > 0 {
say \$n;
\$n--;
}
say 'Blastoff!';
}
```

You can almost read the while statement as if it were English. It means, “While \$n is greater than 0, display the value of n and then decrement \$n. When you get to 0, display the word Blastoff!

More formally, here is the flow of execution for a while statement:

1. Determine whether the condition is true or false.
2. If false, exit the while statement and continue execution at the next statement.
3. If the condition is true, run the body and then go back to step 1.

This type of flow is called a loop because the third step loops back around to the top.

The body of the loop should change the value of one or more variables so that the condition becomes false eventually and the loop terminates. Otherwise, the loop will repeat forever, which is called an infinite loop. An endless source of amusement for computer scientists is the observation that the directions on shampoo, “Lather, rinse, repeat,” are an infinite loop.

In the case of countdown, we can prove that the loop terminates: if \$n is zero or negative, the loop never runs. Otherwise, \$n gets smaller each time through the loop, so eventually we have to get to 0.

For some other loops, it is not so easy to tell whether the loop terminates. For example:

```sub sequence(\$n is copy) {
while \$n != 1 {
say \$n;
if \$n %% 2 {        # \$n is even
\$n = \$n / 2;
} else {            # \$n is odd
\$n = \$n*3 + 1
}
}
return \$n;
}
```

The condition for this loop is \$n != 1, so the loop will continue until \$n is 1, which makes the condition false.

Each time through the loop, the program outputs the value of \$n and then checks whether it is even or odd. If it is even, \$n is divided by 2. If it is odd, the value of \$n is replaced with \$n*3 + 1. For example, if the argument passed to sequence is 42, the resulting values of n are 42, 21, 64, 32, 16, 8, 4, 2, 1.

Since \$n sometimes increases and sometimes decreases, there is no obvious proof that \$n will ever reach 1, or that the program terminates. For some particular values of n, we can prove termination. For example, if the starting value is a power of two, n will be even every time through the loop until it reaches 1. The previous example ends with such a sequence of powers of two, starting with 64.

The hard question is whether we can prove that this program terminates for all positive values of n. So far, no one has been able to prove it or disprove it! (See http://en.wikipedia.org/wiki/Collatz_conjecture.)

As an exercise, you might want to rewrite the function `print-n-times` from Section ?? using iteration instead of recursion.

The while statement can also be used as a statement modifier (or postfix syntax):

```my \$val = 5;
print "\$val " while \$val-- > 0;   # prints 4 3 2 1 0
print "\n";
```

The while loop statement executes the block as long as its condition is true. There is also an until loop statement, which executes the block as long as its condition is false:

```my \$val = 1;
until \$val > 5 {
print \$val++;                 # prints 12345
}
print "\n";
```

## 6.5  Local Variables and Variable Scoping

We have seen in Section ?? that variables created within a subroutine (with the my keyword) are local to that subroutine. The my keyword if often called a declarator, because it is used for declaring a new variable (or other identifier). It is by far the most common declarator. Other declarators include our or state, briefly described later in this chapter.

Similarly, subroutine parameters are also usually local to the subroutine in the signature of which they are declared.

We briefly mentioned that the term lexically scoped is probably more accurate than local, but it was too early at that point to really explain what this means.

Declaring a variable with my gives it lexical scope. This means it only exists within the current block. Loosely speaking, a block is a piece of Perl code within curly brackets or braces. For example, the body of a subroutine and the code of a while or for loop or of an if conditional statement are code blocks. Any variable created with the my declarator exists and is available for use only between the place where it is declared and the end of the enclosing code block.

For example, this code:

```if \$condition eq True {
my \$foo = "bar";
say \$foo;  # prints "bar"
}
say \$foo;      # ERROR: "Variable '\$foo' is not declared ..."
```

will fail on the second print statement, because the `say` function call is not in the lexical scope of the \$foo variable, which ends with the closing brace of the condition block. If we want this variable to be accessible after the end of the condition, then we would need to declare it before the if statement. For example:

```my \$foo;
if \$condition eq True {
\$foo = "bar";
say \$foo;  # prints "bar"
} else {
\$foo = "baz";
}
say \$foo;      # prints "bar" or "baz" depending on \$condition
```

If a lexical variable is not declared within a block, its scope will extend until the end of the file (this is sometimes called a static or a global variable, although these terms are somewhat inaccurate). For example, in the last code snippet above, the scope of the \$foo variable will extend until the end of the file, which may or may not be a good thing, depending on how you intend to use it. It is often better to reduce the scope of variables as much as possible, because this helps reduce dependencies between various parts of the code and limits the risk of subtle bugs. In the code above, if we want to limit the scope of \$foo, we could add braces to create an enclosing block for the sole purpose of limiting the scope:

```{
my \$foo;
if \$condition eq True {
\$foo = "bar";
say \$foo;  # prints "bar"
} else {
\$foo = "baz";
}
say \$foo;      # prints "bar" or "baz" depending on \$condition
}
```

Now, the outer braces create an enclosing block limiting the scope of \$foo to where we need it. This may seem to be a somewhat contrived example, but it is not uncommon to add braces only for the purpose of precisely defining the scope of something.

Lexical scoping also means that variables with the same names can be temporarily redefined in a new scope:

```my \$location = "outside";
sub outer {
say \$location;
}
sub inner {
my \$location = "inside";
say \$location;
}
say \$location;   # -> outside
outer();         # -> outside
inner();         # -> inside
say \$location;   # -> outside
```

We have in effect two variables with the same name, \$location, but different scopes. One is valid only within the inner subroutine where it has been redefined, and the other anywhere else.

If we add a new subroutine:

```sub nowhere {
my \$location = "nowhere";
outer();
}
nowhere();       # -> outside
```

this will still print “outside,” because the outer subroutine knows about the “outside” version of the \$location variable, which existed when outer was defined. In other word, the outer code that referenced to the outer variable (“outside”) knows about the variable that existed when it was created, but not about the variable existing where it was called. This is how lexical variables work. This behavior is the basis for building closures, a form of subroutine with some special properties that we will study later in this book, but is in fact implicitly present everywhere in Perl 6.

While having different variables with the same name can give you a lot of expressive power, we would advise you to avoid creating different variables with the same name and different scopes, at least until you really understand these concepts well enough to know what you are doing, as this can be quite tricky.

By far, most variables used in Perl are lexical variables, declared with the my declarator. Although they are not declared with my, parameters declared in the signature of subroutines and parameters of pointy blocks also have a lexical scope limited to the body of the subroutine or the code block.

There are other declarators, such as our, which creates a package-scoped variable, and state, which creates a lexically scoped variable but with a persistent value. They are relatively rarely used.

One last point: although they are usually not declared with a my declarator, subroutines themselves also have by default a lexical scope. If they are defined within a block, they will be seen only within that block. An example of this has been given at the end of the solution to the GCD exercise of the previous chapter (see Subsection ??). That being said, you can declare a subroutine with with a my declarator if you wish:

```my sub frobnicate {
# ...
}
```

This technique might add some consistency or some form of self-documenting feature, but you won’t buy very much added functionality with that.

## 6.6  Control Flow Statements (last, next, etc.)

Sometimes you don’t know it’s time to end a loop until you get half way through the body. In that case, you can use a control flow statement such as last to jump out of the loop.

For example, suppose you want to take input from the user until they type done. You could write:

```while True {
my \$line = prompt "Enter something ('done' for exiting)\n";
last if \$line eq "done";
say \$line;
}
say 'Done!';
```

The loop condition is True, which is always true, so the loop runs until it hits the last statement.

Each time through, it prompts the user to type something. If the user types done, the last statement exits the loop. Otherwise, the program echoes whatever the user types and goes back to the top of the loop. Here’s a sample run:

```\$ perl6 while_done.pl6
Enter something ('done' for exiting)
Not done
Not done
Enter something ('done' for exiting)
done
Done!
```

This way of writing while loops is common because you can check the condition anywhere in the loop (not just at the top) and you can express the stop condition affirmatively (“stop when this happens”) rather than negatively (“keep going until that happens”).

Using a while loop with a condition that is always true is a quite natural way of writing an infinite loop, i.e., a loop that will run forever until something else in the code (such as the last statement used above) forces the program to break out of the loop. This is commonly used in many programming languages, and this works well in Perl. There is, however, another common and more idiomatic way of constructing infinite loops in Perl 6: using the loop statement, which we will study in Section ?? (p. ??). For now, we’ll use the while True statement, which is fairly legitimate.

Sometimes, rather than simply breaking out of the while loop as with the last control statement, you need to start the body of the loop at the beginning. For example, you may want to check whether the user input is correct with some (unspecified) is-valid subroutine before processing the data, and ask the user to try again if the input was not correct. In this case, the next control statement lets you start at the top the loop body again:

```while True {
my \$line = prompt "Enter something ('done' for exiting)\n";
last if \$line eq "done";
next unless is-valid(\$line);
# further processing of \$line;
}
print('Done!')
```

Here, the loop terminates if the user types “done.” If not, the user input is checked by the is-valid subroutine; if the subroutine returns a true value, the processing continues forward; if it returns a false value, then the control flow starts again at the beginning of the body of the loop, so the user is prompted again to submit a valid input.

The last and next control statements also work in for loops. For example, the following for loop iterates in theory on a range of integer numbers between 1 and 20, but discards odd numbers by virtue of a next statement and breaks out of the loop with a last statement as soon as the loop variable is greater than \$max (i.e., 10 in this example):

```my \$max = 10;
for 1..20 -> \$i {
next unless \$i %% 2; # keeps only even values
last if \$i > \$max;   # stops loop if \$i is greater than \$max
say \$i;              # prints 2 4 6 8 10
}
```

You may have as many last and next statements as you like, just as you may have as many return statements as you like in a subroutine. Using such control flow statements is not considered poor practice. During the early days of structured programming, some people insisted that loops and subroutines have only one entry and one exit. The one-entry notion is still a good idea, but the one-exit notion has led people to bend over backwards and write a lot of unnatural code. Much of programming consists of traversing decision trees. A decision tree naturally starts with a single trunk but ends with many leaves. Write your code with the number of loop controls (and subroutine exits) that is natural to the problem you’re trying to solve. If you’ve declared your variables with reasonable scopes, everything gets automatically cleaned up at the appropriate moment, no matter how you leave the block.

## 6.7  Square Roots

Loops are often used in programs that compute numerical results by starting with an approximate answer and iteratively improving it.

For example, one way of computing square roots is Newton’s method (also known as the Newton-Raphson’s method). Suppose that you want to know the square root of a. If you start with almost any estimate, x, you can compute a better estimate y with the following formula:

y =
 x + a/x 2

For example, if a is 4 and x is 3:

```> my \$a = 4;
4
> my \$x = 3;
3
> my \$y = (\$x + \$a/\$x)/2;
2.166667
```

The result is closer than 3 to the correct answer (√4 = 2) . If we repeat the process with the new estimate, it gets even closer:

```> \$x = \$y;
2.166667
> \$y = (\$x + \$a/\$x)/2;
2.006410
```

After a few more updates, the estimate is almost exact:

```> \$x = \$y;
2.006410
> \$y = (\$x + \$a/\$x)/2;
2.000010
> \$x = \$y;
2.000010
> \$y = (\$x + \$a/\$x)/2;
2.000000000026
```

In general we don’t know ahead of time how many steps it takes to get to the right answer, but we know when we get there because the estimate stops changing:

```> \$x = \$y;
2.000000000026
> \$y = (\$x + \$a/\$x)/2;
2
> \$x = \$y;
2
> \$y = (\$x + \$a/\$x)/2;
2
```

When \$y == \$x, we can stop. Here is a loop that starts with an initial estimate, x, and improves it until it stops changing:

```my (\$a, \$x) = (4, 3);
while True {
say "-- Intermediate value: \$x";
my \$y = (\$x + \$a/\$x) / 2;
last if \$y == \$x;
\$x = \$y;
}
say "Final result is \$x";
```

This will print:

```-- Intermediate value: 3
-- Intermediate value: 2.166667
-- Intermediate value: 2.006410
-- Intermediate value: 2.000010
-- Intermediate value: 2.000000000026
-- Intermediate value: 2
Final result is 2
```

For most values of \$a this works fine, but there are a couple of caveats with this approach. First, in most programming languages, it is dangerous to test float equality, because floating-point values are only approximately right. We do not have this problem with Perl 6, because, as we have already mentioned, it is using a better representation of rational numbers than most generalist programming languages. (You may want to keep this in mind if you are using some other languages.) Even if we don’t have this problem with Perl, there may also be some problems with algorithms that do not behave as well as Newton’s algorithm. For example, some algorithms might not converge as fast and as neatly as Newton’s algorithm but might instead produce alternate values above and below the accurate result.

Rather than checking whether \$x and \$y are exactly equal, it is safer to use the built-in function abs to compute the absolute value, or magnitude, of the difference between them:

```    last if abs(\$y - \$x) < \$epsilon:
```

where `epsilon` has a very small value like 0.0000001 that determines how close is close enough.

## 6.8  Algorithms

Newton’s method is an example of an algorithm: it is a mechanical process for solving a category of problems (in this case, computing square roots).

To understand what an algorithm is, it might help to start with something that is not an algorithm. When you learned to multiply single-digit numbers, you probably memorized the multiplication table. In effect, you memorized 100 specific solutions. That kind of knowledge is not algorithmic.

But if you were “lazy,” you might have learned a few tricks. For example, to find the product of n and 9, you can write n−1 as the first digit and 10−n as the second digit. (For example, to figure out 9*7, n−1 is 6 and 10−n is 3, so that the product 9*7 is 63.) This trick is a general solution for multiplying any single-digit number by 9. That’s an algorithm!

Similarly, the techniques you learned in school for addition (with carrying), subtraction (with borrowing), and long division are all algorithms. One of the characteristics of algorithms is that they do not require any intelligence to carry out. They are mechanical processes where each step follows from the last according to a simple set of rules.

Executing algorithms is boring, but designing them is interesting, intellectually challenging, and a central part of computer science.

Some of the things that people do naturally, without difficulty or conscious thought, are the hardest to express algorithmically. Understanding natural language is a good example. We all do it, but so far no one has been able to explain how we do it, at least not in the form of an algorithm.

## 6.9  Debugging

As you start writing bigger programs, you might find yourself spending more time debugging. More code means more chances to make an error and more places for bugs to hide.

One way to cut your debugging time is “debugging by bisection.” For example, if there are 100 lines in your program and you check them one at a time, it would take 100 steps.

Instead, try to break the problem in half. Look at the middle of the program, or near it, for an intermediate value you can check. Add a say statement (or something else that has a verifiable effect) and run the program.

If the midpoint check is incorrect, there must be a problem in the first half of the program. If it is correct, the problem is in the second half.

Every time you perform a check like this, you halve the number of lines you have to search. After six steps (which is fewer than 100), you would be down to one or two lines of code, at least in theory.

In practice it is not always clear what the “middle of the program” is and not always possible to check it. It doesn’t make sense to count lines and find the exact midpoint. Instead, think about places in the program where there might be errors and places where it is easy to put a check. Then choose a spot where you think the chances are about the same that the bug is before or after the check.

## 6.10  Glossary

Reassignment
Assigning a new value to a variable that already exists.
Update
An assignment where the new value of the variable depends on the old.
Initialization
An assignment that gives an initial value to a variable that may later be updated.
Increment
An update that increases the value of a variable (often by one).
Decrement
An update that decreases the value of a variable.
Iteration
Repeated execution of a set of statements using either a recursive function call or a loop.
Infinite loop
A loop in which the terminating condition is never satisfied.
Algorithm
A general process for solving a category of problems.

## 6.11  Exercises

Exercise 1

Copy the loop from Section ?? and encapsulate it in a subroutine called `my-sqrt` that takes \$a as a parameter, chooses a reasonable value of \$x, and returns an estimate of the square root of \$a.

To test it, write a function named `test-square-root` that prints a table like this:

```[fontshape=up]
a  mysqrt(a)        sqrt(a)          diff
1  1.0000000000000  1.0000000000000  1.110223e-15
2  1.4142135623747  1.4142135623731  1.594724e-12
3  1.7320508075689  1.7320508075689  0.000000e+00
4  2.0000000000000  2.0000000000000  0.000000e+00
5  2.2360679774998  2.2360679774998  0.000000e+00
6  2.4494897427832  2.4494897427832  8.881784e-16
7  2.6457513110647  2.6457513110646  1.025846e-13
8  2.8284271247494  2.8284271247462  3.189449e-12
9  3.0000000000000  3.0000000000000  0.000000e+00

```

The first column is a number, a, the second column is the square root of a computed with `my-sqrt`, the third column is the square root computed by the sqrt built-in function of Perl, and the fourth column is the absolute value of the difference between the two estimates. Don’t worry too much about obtaining a clean tabular formatting, we haven’t seen the built-in functions to do that.

Solution: ??

Exercise 2

The mathematician Srinivasa Ramanujan found an infinite series that can be used to generate a numerical approximation of 1 / π:

 1 π
=
2
 2
9801

 ∞ ∑ k=0

 (4k)!(1103+26390k) (k!)4 3964k

Write a function called `estimate-pi` that uses this formula to compute and return an estimate of π. It should use a while loop to compute terms of the summation until the last term is smaller than 1e-15 (which is Perl notation for 10−15). You can check the result by comparing it to the built-in constant pi. Solution: ??.

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