Continued Fraction This assignment involves computing continued fractions, and c

Question

Continued Fraction This assignment involves computing continued fractions, and contains three parts. The general form of a continued fraction is: A finite continued fraction s an expression of the form where a0 is an integer, all other a, are positive integers, and n is a non-negative integer. We can specify a continued fraction by an array of integers containing the values a0 .. a... 1. Write a function that takes an array of integers as specified above (and ending with a -1) and returns the value of the fraction as a double. 2. Write a function that takes an array of integers as specified above (and ending with a -1). Your function will represent the value of the continued fraction as a regular'' fraction, p/q, in lowest terms. We are looking for the result of doing the fractional arithmetic and keeping everything in integers as we work our way through. The function returns a 2-element integer array, y, with v[0] = p and v[1] = q. 3. Write recursive functions that, given the above representation of a continued fraction, returns p and q such that p/q represent the value of the continued fraction as a ''regular'' fraction in lowest terms, as in question 2 above.

Explanation / Answer

/* calculate approximations for continued fractions */
#include <stdio.h>

/* kind of function that returns a series of coefficients */
typedef double (*coeff_func)(unsigned n);

/* calculates the specified number of expansions of the continued fraction
* described by the coefficient series f_a and f_b */
double calc(coeff_func f_a, coeff_func f_b, unsigned expansions)
{
   double a, b, r;
   a = b = r = 0.0;

   unsigned i;
   for (i = expansions; i > 0; i--) {
       a = f_a(i);
       b = f_b(i);
       r = b / (a + r);
   }
   a = f_a(0);

   return a + r;
}

/* series for sqrt(2) */
double sqrt2_a(unsigned n)
{
   return n ? 2.0 : 1.0;
}

double sqrt2_b(unsigned n)
{
   return 1.0;
}

/* series for the napier constant */
double napier_a(unsigned n)
{
   return n ? n : 2.0;
}

double napier_b(unsigned n)
{
   return n > 1.0 ? n - 1.0 : 1.0;
}

/* series for pi */
double pi_a(unsigned n)
{
   return n ? 6.0 : 3.0;
}

double pi_b(unsigned n)
{
   double c = 2.0 * n - 1.0;

   return c * c;
}

int main(void)
{
   double sqrt2, napier, pi;

   sqrt2 = calc(sqrt2_a, sqrt2_b, 1000);
   napier = calc(napier_a, napier_b, 1000);
   pi = calc(pi_a, pi_b, 1000);

   printf("%12.10g %12.10g %12.10g ", sqrt2, napier, pi);

   return 0;
}

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