8

Exercise 1.8: Newton’s method for cube roots is based on the fact that if y is an approximation to the cube root of x, then a better approximation is given by the value

$$\frac{x/y^2+2y}{3}$$

Use this formula to implement a cube-root procedure analogous to the square- root procedure. (In section Section 1.3.4 [1-3-4], page 69 we will see how to implement Newton’s method in general as an abstraction of these square-root and cube-root procedures.)

(define (cube-root lastGuess guess x)
    (if (good-enough? lastGuess guess)
        guess
        (cube-root guess (improve-cube-root guess x) x)
)
)

(define (improve-cube-root guess x)
    (/
        (+
            (/ x (* guess guess))
            (* 2 guess)
        )
    )
    3
)

(define (good-enough? lastGuess guess)
    (< (abs (- lastGuess guess)) 0.001)
)

(cube-root 100 1 27)