Procedures as Returned Values

The above examples demonstrate how the ability to pass procedures as arguments significantly enhances the expressive power of our programming language. We can achieve even more expressive power by creating procedures whose returned values are themselves procedures.

We can illustrate this idea by looking again at the fixed-point
example described at the end of
section . We formulated a new version
of the square-root procedure as a fixed-point search, starting with
the observation that
is a fixed-point of the function
.
Then we used average damping to make the
approximations converge. Average damping is a useful general
technique in itself. Namely, given a function *f*, we consider the
function whose value at *x* is equal to the average of *x* and *f*(*x*).

We can express the idea of average damping by means of the following procedure:

(define (average-damp f) (lambda (x) (average x (f x))))

((average-damp square) 10) 55

Using `average-damp`, we can reformulate the square-root procedure
as follows:

(define (sqrt x) (fixed-point (average-damp (lambda (y) (/ x y))) 1.0))Notice how this formulation makes explicit the three ideas in the method: fixed-point search, average damping, and the function . It is instructive to compare this formulation of the square-root method with the original version given in section . Bear in mind that these procedures express the same process, and notice how much clearer the idea becomes when we express the process in terms of these abstractions. In general, there are many ways to formulate a process as a procedure. Experienced programmers know how to choose procedural formulations that are particularly perspicuous, and where useful elements of the process are exposed as separate entities that can be reused in other applications. As a simple example of reuse, notice that the cube root of

(define (cube-root x) (fixed-point (average-damp (lambda (y) (/ x (square y)))) 1.0))

Newton's method

When we first introduced the square-root procedure, in
section , we mentioned that this was a special case of
*Newton's method*.
If
is a differentiable function, then a solution of
the equation *g*(*x*)=0 is a fixed point of the function
where

and

In order to implement Newton's method as a procedure, we must first
express the idea of derivative. Note that ``derivative,'' like
average damping, is something that transforms a function into another
function. For instance, the derivative of the function
is the function
.
In general, if *g* is a
function and *dx* is a small number, then the derivative *Dg* of *g* is
the function whose value at any number *x* is given (in the limit of
small *dx*) by

Thus, we can express the idea of derivative (taking

(define (deriv g) (lambda (x) (/ (- (g (+ x dx)) (g x)) dx)))along with the definition

(define dx 0.00001)

Like `average-damp`, `deriv` is a procedure that takes a
procedure as argument and returns a procedure as value. For example,
to approximate the derivative of
at 5 (whose exact
value is 75) we can evaluate

(define (cube x) (* x x x)) ((deriv cube) 5) 75.00014999664018

With the aid of `deriv`, we can express Newton's method as a
fixed-point process:

(define (newton-transform g) (lambda (x) (- x (/ (g x) ((deriv g) x))))) (define (newtons-method g guess) (fixed-point (newton-transform g) guess))The

(define (sqrt x) (newtons-method (lambda (y) (- (square y) x)) 1.0))

Abstractions and first-class procedures

We've seen two ways to express the square-root computation as an instance of a more general method, once as a fixed-point search and once using Newton's method. Since Newton's method was itself expressed as a fixed-point process, we actually saw two ways to compute square roots as fixed points. Each method begins with a function and finds a fixed point of some transformation of the function. We can express this general idea itself as a procedure:

(define (fixed-point-of-transform g transform guess) (fixed-point (transform g) guess))This very general procedure takes as its arguments a procedure

Using this abstraction, we can recast the first square-root computation from this section (where we look for a fixed point of the average-damped version of ) as an instance of this general method:

(define (sqrt x) (fixed-point-of-transform (lambda (y) (/ x y)) average-damp 1.0))Similarly, we can express the second square-root computation from this section (an instance of Newton's method that finds a fixed point of the Newton transform of ) as

(define (sqrt x) (fixed-point-of-transform (lambda (y) (- (square y) x)) newton-transform 1.0))

We began section with the observation that compound procedures are a crucial abstraction mechanism, because they permit us to express general methods of computing as explicit elements in our programming language. Now we've seen how higher-order procedures permit us to manipulate these general methods to create further abstractions.

As programmers, we should be alert to opportunities to identify the underlying abstractions in our programs and to build upon them and generalize them to create more powerful abstractions. This is not to say that one should always write programs in the most abstract way possible; expert programmers know how to choose the level of abstraction appropriate to their task. But it is important to be able to think in terms of these abstractions, so that we can be ready to apply them in new contexts. The significance of higher-order procedures is that they enable us to represent these abstractions explicitly as elements in our programming language, so that they can be handled just like other computational elements.

In general, programming languages impose restrictions on the ways in
which computational elements can be manipulated. Elements with the
fewest restrictions are said to have
*first-class* status. Some
of the ``rights and privileges'' of first-class elements are:
^{}

- They may be named by variables.
- They may be passed as arguments to procedures.
- They may be returned as the results of procedures.
- They may be included in data structures.
^{}

**Exercise.**
Define a procedure `cubic` that can be used together with the `
newtons-method` procedure in expressions of the form

(newtons-method (cubic a b c) 1)to approximate zeros of the cubic

**Exercise.**
Define a procedure `double` that takes a procedure of one
argument as argument and
returns a procedure that applies the original procedure twice. For
example, if `inc` is a procedure that adds 1 to its argument,
then `(double inc)` should be a procedure that adds 2. What
value is returned by

(((double (double double)) inc) 5)

**Exercise.** Let *f* and *g* be two one-argument functions. The *composition*
*f* after *g* is defined to be the function
.
Define a procedure `compose` that implements composition. For
example, if `inc` is a procedure that adds 1 to its argument,

((compose square inc) 6) 49

**Exercise.**
If *f* is a numerical function and *n* is a positive integer, then we
can form the *n*th repeated application of *f*, which is defined to be
the function whose value at *x* is
.
For
example, if *f* is the function
,
then the *n*th repeated application of *f* is
the function
.
If *f* is the operation of
squaring a number, then the *n*th repeated application of *f* is the
function that raises its argument to the 2^{n}th power. Write a
procedure that takes as inputs a procedure that computes *f* and a
positive integer *n* and returns the procedure that computes the *n*th
repeated application of *f*. Your procedure should be able to be used
as follows:

((repeated square 2) 5) 625Hint: You may find it convenient to use

**Exercise.**
The idea of *smoothing* a function is an important concept in
signal processing. If *f* is a function and *dx* is some small number,
then the smoothed version of *f* is the function whose value at a
point *x* is the average of *f*(*x*-*dx*), *f*(*x*), and *f*(*x*+*dx*). Write a
procedure `smooth` that takes as input a procedure that computes
*f* and returns a procedure that computes the smoothed *f*. It is
sometimes valuable to repeatedly smooth a function (that is, smooth
the smoothed function, and so on) to obtained the *n**-fold
smoothed function*. Show how to generate the *n*-fold smoothed
function of any given function using `smooth` and `repeated`
from exercise .

**Exercise.**
We saw in section
that attempting to compute square roots by naively finding a
fixed point of
does not converge, and that this can be
fixed by average damping. The same method works for finding cube
roots as fixed points of the average-damped
.
Unfortunately, the process does not work for
fourth roots--a single
average damp is not enough to make a fixed-point search for
converge. On the other hand, if we average damp twice (i.e.,
use the average damp of the average damp of
)
the
fixed-point search does converge. Do some experiments to determine
how many average damps are required to compute
*n*th roots as a
fixed-point search based upon repeated average damping of
.
Use this to implement a simple procedure for computing
*n*th roots using `fixed-point`, `average-damp`, and the `
repeated` procedure of exercise .
Assume that any arithmetic operations you need are available as primitives.

**Exercise.**
Several of the numerical methods described in this chapter are instances
of an extremely general computational strategy known as *iterative
improvement*. Iterative improvement says that, to compute something,
we start with an initial guess for the answer, test if the guess is
good enough, and otherwise improve the guess and continue the process
using the improved guess as the new guess. Write a procedure `
iterative-improve` that takes two procedures as arguments: a method
for telling whether a guess is good enough and a method for improving
a guess. `Iterative-improve` should return as its value a
procedure that takes a guess as argument and keeps improving the guess
until it is good enough. Rewrite the `sqrt` procedure of
section and the `fixed-point` procedure of
section in terms of `iterative-improve`.