Revision of 1.3.3 Procedures as General Methods from 30 June 2009 - 8:11pm

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Exercises

Exercise 1.35

Show that the golden ratio φ (section 1.2.2) is a fixed point of the transformation x → 1 + 1/x, and use this fact to compute φ by means of the fixed-point procedure.

Modify fixed-point so that it prints the sequence of approximations it generates, using the newline and display primitives shown in exercise 1.22. Then find a solution to x^x = 1000 by finding a fixed point of x → log(1000)/log(x). (Use Scheme’s primitive log procedure, which computes natural logarithms.) Compare the number of steps this takes with and without average damping. (Note that you cannot start fixed-point with a guess of 1, as this would cause division by log(1) = 0.)

Exercise 1.37

An infinite continued fraction is an expression of the form

As an example, one can show that the infinite continued fraction expansion with the N_i and the D_i all equal to 1 produces 1/φ, where φ is the golden ratio (described in section 1.2.2). One way to approximate an infinite continued fraction is to truncate the expansion after a given number of terms. Such a truncation — a so-called k-term finite continued fraction — has the form

Suppose that n and d are procedures of one argument (the term index i) that return the N_i and D_i of the terms of the continued fraction. Define a procedure cont-frac such that evaluating (cont-frac n d k) computes the value of the k-term finite continued fraction. Check your procedure by approximating 1/φ using
```
(cont-frac (lambda (i) 1.0)
           (lambda (i) 1.0)
           k)
```
for successive values of k. How large must you make k in order to get an approximation that is accurate to 4 decimal places?
If your cont-frac procedure generates a recursive process, write one that generates an iterative process. If it generates an iterative process, write one that generates a recursive process.

Exercise 1.38

In 1737, the Swiss mathematician Leonhard Euler published a memoir De Fractionibus Continuis, which included a continued fraction expansion for e - 2, where e is the base of the natural logarithms. In this fraction, the N_i are all 1, and the D_i are successively 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, …. Write a program that uses your cont-frac procedure from exercise 1.37 to approximate e, based on Euler’s expansion.

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.

Revision of 1.3.3 Procedures as General Methods from 30 June 2009 - 8:11pm

Exercise 1.35

Exercise 1.36

Exercise 1.37

Exercise 1.38

Exercise 1.39

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