v_root = func_1(root);

 
/* Program finds the roots of a function using the bisection method */

#include <stdio.h>
#include <stdlib.h>
#include <math.h>

/* Function prototypes */
double func_1(double x);
double func_2(double x);
double func_3(double x);
double bisect(double func(double), double a, double b, double epsilon);

/** Main function */
int main(void) {

  int f_choice, // Choice for user to make on deciding which function to use
      i_choice; // Choice for user to make on deciding which interval to use
  double epsilon, // the error tolerance
      a, // left endpoint
      b, // right endpoint
      root, // the root of the fuction
      v_root; // value of the function with specified root

  printf("Which function you want to choose (1, 2, 3): ");
  scanf("%d", & f_choice);

 /*Nested switch statment to figure which function and then which interval to use. For each
  * outter-case there are two inner-cases for the interval. After the error tolerance has been
  * recieved from the user the bisection method is invoked then the method to find the value of
  *  of the function at the computed root *************************************************** */
  switch (f_choice) {
    case (1):
      printf("Which interval you want to choose (1, 2): ");
      scanf("%d", & i_choice);
      switch (i_choice) {
        case (1):
          printf("What error is accepted (epsilon): ");
          scanf("%lf", & epsilon);
          a = -1.0;
          b = 0.0;
          root = bisect(func_1, a, b, epsilon);
          printf("The root of the function 1 in the interval 1 is %g\n", root);
          v_root = func_1(root);
          printf("The function value at the root is %g\n", v_root);
          break;
        case (2):
          printf("What error is accepted (epsilon): ");
          scanf("%lf", & epsilon);
          a = 0.0;
          b = 0.5;
          root = bisect(func_1, a, b, epsilon);
          printf("The root of the function 1 in the interval 2 is %g\n", root);
          v_root = func_1(root);
          printf("The function value at the root is %g\n", v_root);
          break;
        default:
          ;
          break;
      }
      break;
    case (2):
      printf("Which interval you want to choose (1, 2): ");
      scanf("%d", & i_choice);
      switch (i_choice) {
        case (1):
          printf("What error is accepted (epsilon): ");
          scanf("%lf", & epsilon);
          a = -4.0;
          b = -2.0;
          root = bisect(func_2, a, b, epsilon);
          printf("The root of the function 2 in the interval 1 is %g\n", root);
          v_root = func_2(root);
          printf("The function value at the root is %g\n", v_root);
          break;
        case (2):
          printf("What error is accepted (epsilon): ");
          scanf("%lf", & epsilon);
          a = 2.0;
          b = 4.0;
          root = bisect(func_2, a, b, epsilon);
          printf("The root of the function 2 in the interval 2 is %g\n", root);
          v_root = func_2(root);
          printf("The function value at the root is %g\n", v_root);
          break;
        default:
          ;
          break;
      }
      break;
    case (3):
      printf("Which interval you want to choose (1, 2): ");
      scanf("%d", & i_choice);
      switch (i_choice) {
        case (1):
          printf("What error is accepted (epsilon): ");
          scanf("%lf", & epsilon);
          a = 0.0;
          b = 1.0;
          root = bisect(func_3, a, b, epsilon);
          printf("The root of the function 3 in the interval 1 is %g\n", root);
          v_root = func_3(root);
          printf("The function value at the root is %g\n", v_root);
          break;
        case (2):
          printf("What error is accepted (epsilon): ");
          scanf("%lf", & epsilon);
          a = 1.0;
          b = 2.0;
          root = bisect(func_3, a, b, epsilon);
          printf("The root of the function 3 in the interval 2 is %g\n", root);
          v_root = func_3(root);
          printf("The function value at the root is %g\n", v_root);
          break;
        default:
          ;
          break;
      }
    default:
      ;
      break;
  }
  return 0;
}

/** Returns the polynomial 3x^3 - 3x^2 + 2 */
double func_1(double x) {
  return (3.0 * pow(x, 3.0) - 3.0 * pow(x, 2.0) + 0.2);
}

/** Returns the polynomial x^4 - 3x^2 - 8 */
double func_2(double x) {
  return (pow(x, 4.0) - 3.0 * pow(x, 2.0) - 8.0);
}

/** Returns the trig function 2sin(2x)-3cos(3x) */
double func_3(double x) {
  return (2.0 * sin(2.0 * x) - 3.0 * cos(3.0 * x));
}

/** Figures out the root */
double bisect(double func(double), double a, double b, double epsilon) {
  double f_a = func(a);
  double f_b = func(b);

  if (f_a * f_b > 0) {
    return 0;
  }
  else if (f_a == 0) {
    return a;
  }
  else if (f_b == 0) {
    return b;
  }

  /* f_a * f_b < 0 */
  double m;
  while ( fabs(b - a) > epsilon) {
    m = (a + b) / 2.0;
    double f_mid = func(m);

    /* Replace a or b? */
    if (f_mid == 0.0) {
      break; /* exit while loop */
    }
    else if (f_mid * f_a < 0.0) {
      b = m;
    }
    else
    /* f_mid * f_a < 0.0 */
    {
      a = m;
    }
  }
  return m; // return the value of the new m which is now the root
}

/** Returns the polynomial 3x^3 - 3x^2 + 2 */
double func_1(double x) {
  return (3.0 * pow(x, 3.0) - 3.0 * pow(x, 2.0) + 0.2);
}

3x^3 - 3x^2 + 2

3x^3 - 3x^2 + 2 = 0 

 3(-0.63)^3 - 3(-0.63)^2 + 2 = zero

3(0)^3 - 3(0)^2 +2 = ?