sicp 2.34

 

Exercise 2.34.  Evaluating a polynomial in x at a given value of x can be formulated as an accumulation. We evaluate the polynomial

 

using a well-known algorithm called Horner's rule, which structures the computation as

 

In other words, we start with a_n, multiply by x, add a_n-1, multiply by x, and so on, until we reach a₀.¹⁶ Fill in the following template to produce a procedure that evaluates a polynomial using Horner's rule. Assume that the coefficients of the polynomial are arranged in a sequence, from a₀ through a_n.

 

(define (horner-eval x coefficient-sequence)
  (accumulate (lambda (this-coeff higher-terms) <??>)
              0
              coefficient-sequence))

 

For example, to compute 1 + 3x + 5x³ + x⁵ at x = 2 you would evaluate

 

(horner-eval 2 (list 1 3 0 5 0 1))

 

 

 

(define (accumulate op initial sequence)
  (if (null? sequence)
      initial
      (op (car sequence)
          (accumulate op initial (cdr sequence)))))

(define (horner-eval x coefficient-sequence)
  (accumulate (lambda (this-coeff higher-terms) (+ (* higher-terms x) this-coeff))
              0
              coefficient-sequence))

(horner-eval 2 (list 1 3 0 5 0 1))

 

79