# 1.45

## Exercise 1.45: We saw in 1.3.3 that attempting to compute square roots by naively finding a fixed point of y↦x/y 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 $y↦x/y^2$. Unfortunately, the process does not work for fourth roots—a single average damp is not enough to make a fixed-point search for $y↦x/y^3$ converge. On the other hand, if we average damp twice (i.e., use the average damp of the average damp of $y↦x/y^3$) the fixed-point search does converge. Do some experiments to determine how many average damps are required to compute nth roots as a fixed-point search based upon repeated average damping of $y↦x/y^{n−1}$. Use this to implement a simple procedure for computing nth roots using fixed-point, average-damp, and the repeated procedure of Exercise 1.43. Assume that any arithmetic operations you need are available as primitives.

### Reusable Procedures

#### fixed-point

```
(define tolerance 0.00001)
(define (fixed-point f first-guess)
(define (close-enough? v1 v2)
(< (abs (- v1 v2))
tolerance))
(define (try guess)
(let ((next (f guess)))
(if (close-enough? guess next)
next
(try next))))
(try first-guess))
```

#### average-damp

```
(define (average x y) (/ (+ x y) 2))
(define (average-damp f)
(lambda (x)
(average x (f x))))
```

#### repeated

```
(define (compose f g)
(lambda (x) (f (g x))))
(define (square x) (* x x))
(define (repeated op n) (if (= n 0) (lambda (x) x)
(compose op (repeated op (- n 1)))))
((repeated square 2) 5)
```

#### n-th-root

```
(define (pow x n) (if (= n 0) 1 (* x (pow x (- n 1)))))
(define (n-th-root x n) (fixed-point (average-damp (lambda (y) (/ x (pow y (- n 1))))) 1))
(n-th-root 9 2)
```

Given The above implementation,

```
(n-th-root 16 4) will make the page NOT responding!
```

##### repeated average-damp twice will make (n-th-root 16 4) work

```
(define (pow x n) (if (= n 0) 1 (* x (pow x (- n 1)))))
(define (n-th-root x n) (fixed-point ((repeated average-damp 2) (lambda (y) (/ x (pow y (- n 1))))) 1))
(n-th-root 16 4)
```

The above implementation will make (n-th-root 666 8) freeze the page!

##### Guess that repeated average-damp n/2 times will make (n-th-root x n) work

```
(define (pow x n) (if (= n 0) 1 (* x (pow x (- n 1)))))
(define (n-th-root x n) (fixed-point ((repeated average-damp (floor (/ n 2))) (lambda (y) (/ x (pow y (- n 1))))) 1))
(n-th-root 666 8)
```