Consider the following three procedures. The first computes the sum of the integers from a through b:
(define (sum-integers a b) (if (> a b) 0 (+ a (sum-integers (+ a 1) b))))The second computes the sum of the cubes of the integers in the given range:
(define (sum-cubes a b) (if (> a b) 0 (+ (cube a) (sum-cubes (+ a 1) b))))The third computes the sum of a sequence of terms in the series
(define (pi-sum a b) (if (> a b) 0 (+ (/ 1.0 (* a (+ a 2))) (pi-sum (+ a 4) b))))
These three procedures clearly share a common underlying pattern. They are for the most part identical, differing only in the name of the procedure, the function of a used to compute the term to be added, and the function that provides the next value of a. We could generate each of the procedures by filling in slots in the same template:
(define (name a b) (if (> a b) 0 (+ (term a) (name (next a) b))))
The presence of such a common pattern is strong evidence that there is a useful abstraction waiting to be brought to the surface. Indeed, mathematicians long ago identified the abstraction of summation of a series and invented ``sigma notation,'' for example
to express this concept. The power of sigma notation is that it allows mathematicians to deal with the concept of summation itself rather than only with particular sums--for example, to formulate general results about sums that are independent of the particular series being summed.
Similarly, as program designers, we would like our language to be powerful enough so that we can write a procedure that expresses the concept of summation itself rather than only procedures that compute particular sums. We can do so readily in our procedural language by taking the common template shown above and transforming the ``slots'' into formal parameters:
(define (sum term a next b) (if (> a b) 0 (+ (term a) (sum term (next a) next b))))Notice that sum takes as its arguments the lower and upper bounds a and b together with the procedures term and next. We can use sum just as we would any procedure. For example, we can use it (along with a procedure inc that increments its argument by 1) to define sum-cubes:
(define (inc n) (+ n 1)) (define (sum-cubes a b) (sum cube a inc b))Using this, we can compute the sum of the cubes of the integers from 1 to 10:
(sum-cubes 1 10) 3025With the aid of an identity procedure to compute the term, we can define sum-integers in terms of sum:
(define (identity x) x)Then we can add up the integers from 1 to 10:
(define (sum-integers a b) (sum identity a inc b))
(sum-integers 1 10) 55We can also define pi-sum in the same way:
(define (pi-sum a b) (define (pi-term x) (/ 1.0 (* x (+ x 2)))) (define (pi-next x) (+ x 4)) (sum pi-term a pi-next b))Using these procedures, we can compute an approximation to :
(* 8 (pi-sum 1 1000)) 3.139592655589783
Once we have sum, we can use it as a building block in formulating further concepts. For instance, the definite integral of a function f between the limits a and b can be approximated numerically using the formula
for small values of dx. We can express this directly as a procedure:
(define (integral f a b dx) (define (add-dx x) (+ x dx)) (* (sum f (+ a (/ dx 2.0)) add-dx b) dx)) (integral cube 0 1 0.01) .24998750000000042 (integral cube 0 1 0.001) .249999875000001(The exact value of the integral of cube between 0 and 1 is 1/4.)
Simpson's Rule is a more accurate method of numerical integration than
the method illustrated above. Using Simpson's Rule, the integral of a
function f between a and b is approximated as
where h=(b-a)/n, for some even integer n, and yk =f(a+kh). (Increasing n increases the accuracy of the approximation.) Define a procedure that takes as arguments f, a, b, and n and returns the value of the integral, computed using Simpson's Rule. Use your procedure to integrate cube between 0 and 1 (with n=100 and n=1000), and compare the results to those of the integral procedure shown above.
Exercise. The sum procedure above generates a linear recursion. The procedure can be rewritten so that the sum is performed iteratively. Show how to do this by filling in the missing expressions in the following definition:
(define (sum term a next b) (define (iter a result) (if ?? ?? (iter ?? ??))) (iter ?? ??))
aThe sum procedure is only the simplest of a vast number of
similar abstractions that can be captured as higher-order procedures.
Write an analogous procedure
called product that returns the product of the values of a
function at points over a given range.
Show how to define
factorial in terms of
product. Also use product to compute approximations to
using the formula
bIf your product procedure generates a recursive process, write one that generates an iterative process. If it generates an iterative process, write one that generates a recursive process.
Exercise. a. Show that sum and product (exercise ) are both special cases of a still more general notion called accumulate that combines a collection of terms, using some general accumulation function:
(accumulate combiner null-value term a next b)Accumulate takes as arguments the same term and range specifications as sum and product, together with a combiner procedure (of two arguments) that specifies how the current term is to be combined with the accumulation of the preceding terms and a null-value that specifies what base value to use when the terms run out. Write accumulate and show how sum and product can both be defined as simple calls to accumulate.
b. If your accumulate procedure generates a recursive process, write one that generates an iterative process. If it generates an iterative process, write one that generates a recursive process.
Exercise. You can obtain an even more general version of accumulate (exercise ) by introducing the notion of a filter on the terms to be combined. That is, combine only those terms derived from values in the range that satisfy a specified condition. The resulting filtered-accumulate abstraction takes the same arguments as accumulate, together with an additional predicate of one argument that specifies the filter. Write filtered-accumulate as a procedure. Show how to express the following using filtered-accumulate:
a. the sum of the squares of the prime numbers in the interval a to b (assuming that you have a prime? predicate already written)
b. the product of all the positive integers less than n that are relatively prime to n (i.e., all positive integers i < n such that ).