Higher Order Functions in Java

In order to understand future examples, we first have to discuss how to use higher-order functions in Java, and how to write anonymous inner classes. This post will have nothing to do with multi-stage programming.

Let’s write a program that can print out data tables for different mathematical functions. For example, for a function that multiplies by two, f(x) = 2x, we want to print something like this:

x                      f(x)
       -5.0000000000       -10.0000000000
       -4.0000000000        -8.0000000000
       -3.0000000000        -6.0000000000
       -2.0000000000        -4.0000000000
       -1.0000000000        -2.0000000000
        0.0000000000         0.0000000000
        1.0000000000         2.0000000000
        2.0000000000         4.0000000000
        3.0000000000         6.0000000000
        4.0000000000         8.0000000000
        5.0000000000        10.0000000000

We can write a function like this:

public static void printTableTimesTwo(double x1, double x2, int n) { assert n>1;

double x = x1; double delta = (x2-x1)/(double)(n-1); System.out.println("x f(x)"); System.out.printf("%20.10f %20.10f\n", x, x*2); for(int i=0; i<(n-1); ++i) { x += delta; System.out.printf("%20.10f %20.10f\n", x, x*2); } }

The parameter x1 determines the lower end of the interval, x2 the upper end, and n determines how many values should be printed. n needs to be at least 2 to print out the values at x1 and x2. We can generate the table above with this call:

printTableTimesTwo(-5, 5, 11);

What if we want to print out the values of a different function, for example f(x) = x + 4? We can write a new function:

public static void printTablePlusFour(double x1, double x2, int n) { assert n>1;

double x = x1; double delta = (x2-x1)/(double)(n-1); System.out.println("x f(x)"); System.out.printf("%20.10f %20.10f\n", x, x+4); for(int i=0; i<(n-1); ++i) { x += delta; System.out.printf("%20.10f %20.10f\n", x, x+4); } }

This involves a lot of code duplication, though. The only parts that actually differ are the two occurrences of x*2 and x+4. How can we factor that difference out?

Let's write an interface that we can use for any kind of function that takes in one parameter and returns one parameter f(x) = y is an example of such a function.

public interface ILambda; { public R apply(P param); }

This interface is called ILambda and it has one method, apply. We used Java generics and didn't specify the return type and the type of the parameter; instead, we just called them R and P, respectively. A function that takes in a Double and that returns a Double, like f(x) = y, can be expressed using a ILambda. A function taking a String and returning an Integer would use ILambda.

Now we can write our f(x) = 2x and f(x) = x + 4 functions using ILambda:

public class TimesTwo implements ILambda { public Double apply(Double param) { return param*2; } } public class PlusFour implements ILambda { public Double apply(Double param) { return param+4; } }

Now we can write one printTable method that takes in an ILambda called f representing the function, in addition to the parameters x1, x2 and n, as before:

public static void printTable(ILambda f, double x1, double x2, int n) { assert n>1;


        double x = x1;

        double delta = (x2-x1)/(double)(n-1);

// f.apply(x) just means what f(x) means in math! double y = f.apply(x); System.out.println("x f(x)"); System.out.printf("%20.10f %20.10f\n", x, y); for(int i=0; i<(n-1); ++i) { x += delta; y = f.apply(x); System.out.printf("%20.10f %20.10f\n", x, y); } }

Note that when we want to print out the y-value, we just write f.apply(x), which looks very similar to f(x) in mathematics. It means exactly the same.

We can print out the tables for our two functions using:

printTable(new TimesTwo(), -5, 5, 11); printTable(new PlusFour(), -5, 5, 11);

We have to create new objects for the functions: The first time we call printTable we pass a new TimesTwo object; the second time, we pass a new PlusFour object.

We can now define as many functions as we like without having to rewrite the printTable function. For example, we can easily write a square root function and use it very easily:

public class SquareRoot implements ILambda { public Double apply(Double param) { return Math.sqrt(param); } }


// ...

printTable(new SquareRoot(), -5, 5, 11);

The really neat thing is that we can even define a new function on-the-fly, without having to give it a name. We do that using anonymous inner classes in Java. Here, we call printTable and pass it a new object that implements ILambda.

printTable(new ILambda() { public Double apply(Double param) { return param*param; } }, -5, 5, 11);

We define a new ILambda from Double to Double without giving it a name. When we use anonymous inner classes, we need to fill in all the methods that are still abstract. Here, it is just the apply method.

The method printTable is now a "higher order function", because conceptually it is a function that takes another function as input.

Questions:

What does the anonymous ILambda in the example above compute? What's the mathematical function it represents?
How would you print a table for the function f(x) = x² + 2x?

You can download the complete source code for the examples here: