Revision of 1.1.7 Example: Square Roots by Newton's Method from 30 June 2009 - 7:51pm

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Exercise 1.6

Alyssa P. Hacker doesn’t see why if needs to be provided as a special form. “Why can’t I just define it as an ordinary procedure in terms of cond?” she asks. Alyssa’s friend Eva Lu Ator claims this can indeed be done, and she defines a new version of if:

(define (new-if predicate then-clause else-clause)
  (cond (predicate then-clause)
        (else else-clause)))

Eva demonstrates the program for Alyssa:

(new-if (= 2 3) 0 5)

(new-if (= 1 1) 0 5)

Delighted, Alyssa uses new-if to rewrite the square-root program:

(define (sqrt-iter guess x)
  (new-if (good-enough? guess x)
          (sqrt-iter (improve guess x)

What happens when Alyssa attempts to use this to compute square roots? Explain.

Exercise 1.7

The good-enough? test used in computing square roots will not be very effective for finding the square roots of very small numbers. Also, in real computers, arithmetic operations are almost always performed with limited precision. This makes our test inadequate for very large numbers. Explain these statements, with examples showing how the test fails for small and large numbers. An alternative strategy for implementing good-enough? is to watch how guess changes from one iteration to the next and to stop when the change is a very small fraction of the guess. Design a square-root procedure that uses this kind of end test. Does this work better for small and large numbers?

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:

(x/(y^2) + 2y)/3.

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



REAL :: Input, X, NewX, Tolerance
INTEGER :: Count

READ(*,*) Input, Tolerance

Count = 0 ! count starts with 0
X = Input ! X starts with the input value
DO ! for each iteration
Count = Count + 1 ! increase the iteration count
NewX = 0.5*(ans + Input/X) ! compute a new approximation
IF (ABS(X - NewX) < Tolerance) EXIT ! if they are very close, exit
X = NewX ! otherwise, keep the new one

WRITE(*,*) ‘After ‘, Count, ’ iterations:’
WRITE(*,*) ’ The estimated square root is ‘, NewX
WRITE(*,*) ’ The square root from SQRT() is ‘, SQRT(Input)
WRITE(*,*) ’ Absolute error = ‘, ABS(SQRT(Input) - NewX)



Do you have more great aritlces like this one?

Clojure Functions of this article

(defn square [x] (* x x))

(defn abs [x] (if (< x 0) (- x) x ))

(defn average [x y] (/ (+ x y) 2))

(defn improve [guess x] (average guess (/ x guess)))

(defn good-enough? [guess x] (< (abs (- (square guess) x)) 0.001))

(defn sqrt-iter [guess x] (if (good-enough? guess x) guess (sqrt-iter (improve guess x) x) ))

(defn sqrt [x] (sqrt-iter 1.0 x))

Now you can run the examples.

- bmentges (Bruno Carvalho)

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