Last modified on 30 May 2011, at 09:27

The Way of the Java/Iteration

IterationEdit

Multiple assignmentEdit

I haven't said much about it, but it is legal in Java to make more than one assignment to the same variable. The effect of the second assignment is to replace the old value of the variable with a new value.

int fred = 5;
System.out.print (fred);
fred = 7;
System.out.println (fred);

The output of this program is 57, because the first time we print fred his value is 5, and the second time his value is 7.

This kind of multiple assignment is the reason I described variables as a container for values. When you assign a value to a variable, you change the contents of the container.

When there are multiple assignments to a variable, it is especially important to distinguish between an assignment statement and a statement of equality. Because Java uses the = symbol for assignment, it is tempting to interpret a statement like a = b as a statement of equality. (a==b) is a statement of equality!

First of all, equality is commutative, and assignment is not. For example, in mathematics if a = b then b = a. But in Java, a = 7 is a legal assignment statement, and 7 = a; is not.

Furthermore, in mathematics, a statement of equality is true for all time. If a==b now, then a will always equal b. In a computer program, however, variables are intrinsically inconsistent. In Java, an assignment statement can make two variables equal, but they don't have to stay that way!

int a = 5;
int b = a;     // a and b are now equal
a = 3;         // a and b are no longer equal

The third line changes the value of a but it does not change the value of b, and so they are no longer equal. In many programming languages an alternate symbol is used for assignment, such as <- or :=, in order to avoid this confusion.

Although multiple assignment is frequently useful, you should use it with caution. If the values of variables are changing constantly in different parts of the program, it can make the code difficult to read and debug.

LoopsEdit

One of the things computers are often used for is the automation of repetitive tasks. Repeating identical or similar tasks without making errors is something that computers do well and people do poorly.

We have already seen programs that use recursion to perform repetition, such as nLines and countdown. This type of repetition is called iteration, and Java provides several language features that make it easier to write iterative programs.

The two features we are going to look at are the while statement and the for statement.

The while statementEdit

Using a while statement, we can rewrite countdown:

public static void countdown (int n) 
{
  while (n > 0) 
  {
    System.out.println (n);
    n = n - 1;
  }
  System.out.println ("Blastoff!");
}

You can almost read a while statement as if it were English. What this means is, ``While n is greater than zero, continue printing the value of n and then reducing the value of n by 1. When you get to zero, print the word `Blastoff!

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

  1. Evaluate the condition in parentheses, yielding true or false.
  2. If the condition is false, exit the while statement and continue execution at the next statement.
  3. If the condition is true, execute each of the statements between the squiggly-brackets, 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. Notice that if the condition is false the first time through the loop, the statements inside the loop are never executed. The statements inside the loop are sometimes called the body of the loop.

The body of the loop should change the value of one or more variables so that, eventually, the condition becomes false 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 will terminate because we know that the value of n is finite, and we can see that the value of n gets smaller each time through the loop (each iteration), so eventually we have to get to zero. In other cases it is not so easy to tell:

public static void sequence (int n) 
{
  while (n != 1) 
  {
    System.out.println (n);
    if (n % 2 == 0)
    {
      n = n / 2;
    }
    else                   // n is odd
    {
      n = n * 3 + 1;
    }
  }
}

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

At each iteration, the program prints the value of n and then checks whether it is even or odd. If it is even, the value of n is divided by two. If it is odd, the value is replaced by n * 3 + 1. For example, if the starting value (the argument passed to sequence) is 3, the resulting sequence is 3, 10, 5, 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 will terminate. For some particular values of n, we can prove termination. For example, if the starting value is a power of two, then the value of n will be even every time through the loop, until we get to 1. The previous example ends with such a sequence, starting with 16.

Particular values aside, the interesting question is whether we can prove that this program terminates for all values of n. So far, no one has been able to prove it or disprove it!

TablesEdit

One of the things loops are good for is generating and printing tabular data. For example, before computers were readily available, people had to calculate logarithms, sines and cosines, and other common mathematical functions by hand.

To make that easier, there were books containing long tables where you could find the values of various functions. Creating these tables was slow and boring, and the result tended to be full of errors.

When computers appeared on the scene, one of the initial reactions was, This is great! We can use the computers to generate the tables, so there will be no errors. That turned out to be true (mostly), but shortsighted. Soon thereafter computers (and calculators) were so pervasive that the tables became obsolete.

Well, almost. It turns out that for some operations, computers use tables of values to get an approximate answer, and then perform computations to improve the approximation. In some cases, there have been errors in the underlying tables, most famously in the table the original Intel Pentium used to perform floating-point division.

Although a log table is not as useful as it once was, it still makes a good example of iteration. The following program prints a sequence of values in the left column and their logarithms in the right column:

double x = 1.0;
while (x < 10.0) 
{
  System.out.println (x + "   " + Math.log(x));
  x = x + 1.0;
}

The output of this program is:

1.0   0.0
2.0   0.6931471805599453
3.0   1.0986122886681098
4.0   1.3862943611198906
5.0   1.6094379124341003
6.0   1.791759469228055
7.0   1.9459101490553132
8.0   2.0794415416798357
9.0   2.1972245773362196

Looking at these values, can you tell what base the log function uses by default?

Since powers of two are so important in computer science, we often want to find logarithms with respect to base 2. To find that, we have to use the following formula:

_2 x = log_e x / log_e 2

Changing the print statement to

System.out.println (x + "   " + Math.log(x) / Math.log(2.0));

yields:

1.0   0.0
2.0   1.0
3.0   1.5849625007211563
4.0   2.0
5.0   2.321928094887362
6.0   2.584962500721156
7.0   2.807354922057604
8.0   3.0
9.0   3.1699250014423126

We can see that 1, 2, 4 and 8 are powers of two, because their logarithms base 2 are round numbers. If we wanted to find the logarithms of other powers of two, we could modify the program like this:

double x = 1.0;
while (x < 100.0) 
{
  System.out.println (x + "   " + Math.log(x) / Math.log(2.0));
  x = x * 2.0;
}

Now instead of adding something to x each time through the loop, which yields an arithmetic sequence, we multiply x by something, yielding a geometric sequence. The result is:

1.0   0.0
2.0   1.0
4.0   2.0
8.0   3.0
16.0   4.0
32.0   5.0
64.0   6.0

Log tables may not be useful any more, but for computer scientists, knowing the powers of two is! Some time when you have an idle moment, you should memorize the powers of two up to 65536 (that's 2^16).

Two-dimensional tablesEdit

A two-dimensional table is a table where you choose a row and a column and read the value at the intersection. A multiplication table is a good example. Let's say you wanted to print a multiplication table for the values from 1 to 6.

A good way to start is to write a simple loop that prints the multiples of 2, all on one line.

int i = 1;
while (i <= 6) 
{
  System.out.print (2 * i + "   ");
  i = i + 1;
}
System.out.println ("");

The first line initializes a variable named i, which is going to act as a loop counter. As the loop executes, the value of i increases from 1 to 6, and then when i is 7, the loop terminates. Each time through the loop, we print the value 2*i followed by three spaces. Since we are using the print command rather than println, all the output appears on a single line.

As I mentioned in Section printing, in some environments the output from print gets stored without being displayed until println is invoked. If the program terminates, and you forget to invoke println, you may never see the stored output.

The output of this program is:

2   4   6   8   10   12

So far, so good. The next step is to encapsulate and generalize.

Encapsulation and generalizationEdit

Encapsulation usually means taking a piece of code and wrapping it up in a method, allowing you to take advantage of all the things methods are good for. We have seen two examples of encapsulation, when we wrote printParity in Section alternative and isSingleDigit in Section boolean.

Generalization means taking something specific, like printing multiples of 2, and making it more general, like printing the multiples of any integer.

Here's a method that encapsulates the loop from the previous section and generalizes it to print multiples of n.

public static void printMultiples (int n) 
{
  int i = 1;
  while (i <= 6) 
  {
    System.out.print (n * i + "   ");
    i = i + 1;
  }
  System.out.println ();
}
  

To encapsulate, all I had to do was add the first line, which declares the name, parameter, and return type. To generalize, all I had to do was replace the value 2 with the parameter n.

If I invoke this method with the argument 2, I get the same output as before. With argument 3, the output is:

3   6   9   12   15   18

and with argument 4, the output is:

4   8   12   16   20   24 

By now you can probably guess how we are going to print a multiplication table: we'll invoke printMultiples repeatedly with different arguments. In fact, we are going to use another loop to iterate through the rows.

int i = 1;
while (i <= 6) 
{
  printMultiples (i);
  i = i + 1;
}

First of all, notice how similar this loop is to the one inside printMultiples. All I did was replace the print statement with a method invocation.

The output of this program is:

1   2   3   4   5   6   
2   4   6   8   10   12   
3   6   9   12   15   18   
4   8   12   16   20   24   
5   10   15   20   25   30   
6   12   18   24   30   36   

which is a (slightly sloppy) multiplication table. If the sloppiness bothers you, Java provides methods that give you more control over the format of the output, but I'm not going to get into that here.

MethodsEdit

In the last section I mentioned all the things methods are good for. About this time, you might be wondering what exactly those things are. Here are some of the reasons methods are useful:

  • By giving a name to a sequence of statements, you make your program easier to read and debug.
  • Dividing a long program into methods allows you to separate parts of the program, debug them in isolation, and then compose them into a whole.
  • Methods facilitate both recursion and iteration.
  • Well-designed methods are often useful for many programs. Once you write and debug one, you can reuse it.

More encapsulationEdit

To demonstrate encapsulation again, I'll take the code from the previous section and wrap it up in a method:

public static void printMultTable () 
{
  int i = 1;
  while (i <= 6) 
  {
    printMultiples (i);
    i = i + 1;
  }
}

The process I am demonstrating is a common development plan. You develop code gradually by adding lines to main or someplace else, and then when you get it working, you extract it and wrap it up in a method.

The reason this is useful is that you sometimes don't know when you start writing exactly how to divide the program into methods. This approach lets you design as you go along.

Local variablesEdit

About this time, you might be wondering how we can use the same variable i in both printMultiples and printMultTable. Didn't I say that you can only declare a variable once? And doesn't it cause problems when one of the methods changes the value of the variable?

The answer to both questions is no, because the i in printMultiples and the i in printMultTable are not the same variable. They have the same name, but they do not refer to the same storage location, and changing the value of one of them has no effect on the other.

Variables that are declared inside a method definition are called local variables because they are local to their own methods. You cannot access a local variable from outside its home method, and you are free to have multiple variables with the same name, as long as they are not in the same method.

It is often a good idea to use different variable names in different methods, to avoid confusion, but there are good reasons to reuse names. For example, it is common to use the names i, j and k as loop variables. If you avoid using them in one method just because you used them somewhere else, you will probably make the program harder to read.

More generalizationEdit

As another example of generalization, imagine you wanted a program that would print a multiplication table of any size, not just the 6x6 table. You could add a parameter to printMultTable:

public static void printMultTable (int high) 
{
  int i = 1;
  while (i <= high) 
  {
    printMultiples (i);
    i = i + 1;
  }
}

I replaced the value 6 with the parameter high. If I invoke printMultTable with the argument 7, I get:

1   2   3   4   5   6   
2   4   6   8   10   12   
3   6   9   12   15   18   
4   8   12   16   20   24   
5   10   15   20   25   30   
6   12   18   24   30   36   
7   14   21   28   35   42   

which is fine, except that I probably want the table to be square (same number of rows and columns), which means I have to add another parameter to printMultiples, to specify how many columns the table should have.

Just to be annoying, I will also call this parameter high, demonstrating that different methods can have parameters with the same name (just like local variables):

public static void printMultiples (int n, int high) 
{
  int i = 1;
  while (i <= high) 
  {
    System.out.print (n*i + "   ");
    i = i + 1;
  }
  newLine ();
}
public static void printMultTable (int high) 
{
  int i = 1;
  while (i <= high) 
  {
    printMultiples (i, high);
    i = i + 1;
  }
}

Notice that when I added a new parameter, I had to change the first line of the method (the interface or prototype), and I also had to change the place where the method is invoked in printMultTable. As expected, this program generates a square 7x7 table:

1   2   3   4   5   6   7   
2   4   6   8   10   12   14   
3   6   9   12   15   18   21   
4   8   12   16   20   24   28   
5   10   15   20   25   30   35   
6   12   18   24   30   36   42   
7   14   21   28   35   42   49

When you generalize a method appropriately, you often find that the resulting program has capabilities you did not intend. For example, you might notice that the multiplication table is symmetric, because n*i==i*n, so all the entries in the table appear twice. You could save ink by printing only half the table. To do that, you only have to change one line of printMultTable. Change

printMultiples (i, high);

to

printMultiples (i, i);

and you get:

1   
2   4   
3   6   9   
4   8   12   16   
5   10   15   20   25   
6   12   18   24   30   36   
7   14   21   28   35   42   49  

I'll leave it up to you to figure out how it works.

GlossaryEdit

  • loop A statement that executes repeatedly while or until some condition is satisfied.
  • infinite loop A loop whose condition is always true.
  • body The statements inside the loop.
  • iteration One pass through (execution of) the body of the loop, including the evaluation of the condition.
  • encapsulate To divide a large complex program into components (like methods) and isolate the components from each other (for example, by using local variables).
  • local variable A variable that is declared inside a method and that exists only within that method. Local variables cannot be accessed from outside their home method, and do not interfere with any other methods.
  • generalize To replace something unnecessarily specific (like a constant value) with something appropriately general (like a variable or parameter). Generalization makes code more versatile, more likely to be reused, and sometimes even easier to write.
  • development plan A process for developing a program. In this chapter, I demonstrated a style of development based on developing code to do simple, specific things, and then encapsulating and generalizing. In Section distance I demonstrated a technique I called incremental development. In later chapters I will suggest other styles of development.