where x is in radians. Define a procedure
(tan-cf x k)
that computes an approximation to the
tangent function based on Lambert’s formula.
K
specifies the number of terms to compute, as in
exercise 1.37.
1.3.4 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 1.3.3. We formulated a new version of the square-root procedure as a fixed-point search,
starting with the observation that x is a fixed-point of the function y x/y. 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
is a procedure that takes as its argument a procedure
f
and returns as its value a
procedure (produced by the
lambda
) that, when applied to a number
x
, produces the average of
x
and
(f x)
. For example, applying
average-damp
to the
square
procedure produces a procedure
whose value at some number x is the average of x and x
2
. Applying this resulting procedure to 10
returns the average of 10 and 100, or 55:
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((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 y x/y. It is instructive to compare this formulation of the square-root
method with the original version given in section 1.1.7. 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 x is a fixed point of the
function y x/y
2
, so we can immediately generalize our square-root procedure to one that extracts
cube roots:
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(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 1.1.7, we mentioned that this was a
special case of Newton’s method. If x g(x) is a differentiable function, then a solution of the
equation g(x) = 0 is a fixed point of the function x f(x) where
and Dg(x) is the derivative of g evaluated at x. Newton’s method is the use of the fixed-point method
we saw above to approximate a solution of the equation by finding a fixed point of the function f.
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For many functions g and for sufficiently good initial guesses for x, Newton’s method converges very
rapidly to a solution of g( x) = 0.
62
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 x x
3
is the function x 3x
2
. 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 dx to be, say, 0.00001) as the procedure
(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 x x
3
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
newton-transform
procedure expresses the formula at the beginning of this section, and
newtons-method
is readily defined in terms of this. It takes as arguments a procedure that
computes the function for which we want to find a zero, together with an initial guess. For instance, to
find the square root of x, we can use Newton’s method to find a zero of the function y y
2
- x starting
with an initial guess of 1.
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This provides yet another form of the square-root procedure:
(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
g
that computes some function, a
procedure that transforms
g
, and an initial guess. The returned result is a fixed point of the
transformed function.
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 y x/y) 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 y y
2
- x) as
(define (sqrt x)
(fixed-point-of-transform (lambda (y) (- (square y) x))
newton-transform
1.0))
We began section 1.3 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.
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