Implementation of a procedure that evolves an iterative exponentiation process that uses successive squaring and uses a logarithmic number of steps.
Category: Functional Programming
Sicp Exercise 1.16
Exercise 1.16. Design a procedure that evolves an iterative exponentiation process that uses successive squaring and uses a logarithmic number of steps, as does fast-expt.
(Hint: Using the observation that (bn/2)2 = (b2)n/2, keep, along with the exponent n and the base b, an additional state variable a, and define the state transformation in such a way that the product a bn is unchanged from state to state. At the beginning of the process a is taken to be 1, and the answer is given by the value of a at the end of the process. In general, the technique of defining an invariant quantity that remains unchanged from state to state is a powerful way to think about the design of iterative algorithms.)
Sicp Exercise 1.12
Exercise 1.12. The following pattern of numbers is called Pascal’s triangle.
The numbers at the edge of the triangle are all 1, and each number inside the triangle is the sum of the two numbers above it.35 Write a procedure that computes elements of Pascal’s triangle by means of a recursive process.
Sicp Exercise 1.11
Exercise 1.11. A function f is defined by the rule that f(n) = n if n<3 and f(n) = f(n - 1) + 2f(n - 2) + 3f(n - 3) if n> 3. Write a procedure that computes f by means of a recursive process. Write a procedure that computes f by means of an iterative process.
Sicp Exercise 1.09
Exercise 1.9. Each of the following two procedures defines a method for adding two positive integers in terms of the procedures inc, which increments its argument by 1, and dec, which decrements its argument by 1.
(define (+ a b)
(if (= a 0)
b
(inc (+ (dec a) b))))
(define (+ a b)
(if (= a 0)
b
(+ (dec a) (inc b))))
Using the substitution model, illustrate the process generated by each procedure in evaluating (+ 4 5). Are these processes iterative or recursive?
Square-root using Newton’s Method
The following uses Newton’s Method to find square root.
It keeps record of how guess changes from one iteration to the next and stops when the change is a very small fraction of the guess.
Continued Fraction For Tangent Function
A continued fraction representation of the tangent using J.H. Lambert’s formula:
where x is in radians.
k specifies the number of terms to compute:
Sicp Exercise 1.08
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
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.)
Solution
Sicp Exercise 1.07
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?
Solution:
Sicp Exercise 1.39
Exercise 1.39. A continued fraction representation of the tangent function was published in 1770 by the German mathematician J.H. Lambert:
where x is in radians. Define a procedure (tan-cf x k) that computes an approximation to the tangent function based on Lambert’s formula. K specifies the number of terms to compute, as in exercise 1.37.
Solution: