Question

Write three functions that compute the square root of an argument using three different methods. The methods are increasingly sophisticated, and increasingly efficient. The square root of a real number is the values such that x . For us, the values will be double precision variables and so may not be perfectly accurate. Also, for us, assume that is in the range 0.0 to 100.0 You program should have a main() that asks the user for x and an accuracy bound, and then prints out the approximate square root for each of the three methods. If x is negative write an error message and exit. One. Write a pure function double sqrt(x, accuracy) that computes the square root of x using Linear Search Do this in a loop that starts at 0.0 and bestError at x. Each iteration of the loop computes error = x-gº . If error bestError, set bestError to error, and bests to the current s. Then increments by accuracy and move on. At the end of the loop, return bests. This method is linear search, it searches along the number line for the best s that is within ine of the true value. To compute the absolute value of a double, use the function double Labs (double) that is in the math library. Two. Write a pure function double sortes (x, accuracy) that computes the square root of using Binary Search. (This method is also called bisection.) Start a variable low at 0.0 and a variable high at 10.0. This assumes that x will be no larger than 100.0 In a while loop do this: compute the midpoint mid (low high) /2. If Ix-midemid S accuracy return sid. If midomid < x set low-mid. Else set high mid Three. Write a pure function double sgr , accuracy) that computes the square root of x using Newton's Method. Recall the method: Start out an initial guess s=1.0 (or any value). Now repeatedly improve the guess: s (s + x/s)/2 When the guess meets the ending condition Ix-sel is s bound return s. M\ClassApps>ce-im squareRoot.c M:\ClassApps.la Enter x = 47.5 Enter error bound --> le-5 sortLS (47.500)- 6.892020; 33-47.499940 sortBS (47.500) 6.8920241 -47.499999 get (47.500) -6.8920241 -47.500001

Answer 1

/* One : Square root of a number using linear search */

#include<stdio.h>
#include<math.h>
double sqartLs(double x, double accuracy )   
{

if (x == 0.0 || x == 1.0)
return x;
  
   double s = 1.0, result = 1.0,Error=0.0,bestS,bestError;
bestError=x;   
while (result <= x)
{
s=s+0.1;
result = s * s;
if(result<=accuracy)
   {
   Error=x-s*s;
   printf("\nError is:%lf\n",Error);
   if(Error<bestError)
   {
       Error=bestError;
   }
}
}
bestS=s-0.1;
return fabs(bestS);
}
  

int main()
{
double x,accuracy;
   printf("Enter the value of x");
   scanf("%lf",&x);
   if(x>=0.0 && x<=100.0)
   {
       accuracy=x;
printf("\n Square root is :%lf",sqartLs(x,accuracy)) ;
}
else
{
   printf("Range is not satisfied");
   }
return 0;
}
/* Pure function of square root with linear search ( above program part)*/

double sqartLs(double x, double accuracy )   
{
   if (x == 0.0 || x == 1.0)
return x;
    double s = 1.0, result = 1.0,Error=0.0,bestS,bestError;
bestError=x;   
while (result <= x)
{
s=s+0.1;
result = s * s;
if(result<=accuracy)
   {
   Error=x-s*s;
   printf("\nError is:%lf\n",Error);
   if(Error<bestError)
   {
       Error=bestError;
   }
}
}
bestS=s-0.1;
return fabs(bestS);
}

output(linear):

/* Two: Square root of a number using Binary search */

#include<stdio.h>
#include<math.h>
double sqartBs(double x, double accuracy )
{
if (x == 0.0 || x == 1.0)
return x;
  
double low =1.0, high = 100.0, ans=0.0;
while (low <= high)
{   
double mid = (low + high) / 2;
  
if ((mid*mid - x)<=accuracy)
{
return mid;  
       }


if (mid*mid < x)
{
low = mid + 10;
ans = mid;
}
else // If mid*mid is greater than x
high = mid-1.0;   
}
return ans;
}

int main()
{
double x,accuracy;
   printf("Enter the value of x");
   scanf("%lf",&x);
   if(x>=0.0 && x<=100.0)
   {
       accuracy=x;
printf("\n Square root is :%lf",sqartBs(x,accuracy)) ;
}
else
{
   printf("Range is not satisfied");
   }
return 0;
}

/* Pure function of square root with binary search ( from above program part)*/

double sqartBs(double x, double accuracy )
{
if (x == 0.0 || x == 1.0)
return x;
  
double low =1.0, high = 100.0, ans=0.0;
while (low <= high)
{   
double mid = (low + high) / 2;
  
if ((mid*mid - x)<=accuracy)
{
return mid;  
       }


if (mid*mid < x)
{
low = mid + 10;
ans = mid;
}
else // If mid*mid is greater than x
high = mid-1.0;   
}
return ans;
}
output(binary search):

/* Three: Newton method to calculate square root of a number */

#include <stdio.h>
double fabsolute(double number )
{   
if(number < 0){
number = -number;
}
return number;
}

double sqartNt(double x, double accuracy )
{
const double guess = 0.00001;
double s= 1.0;
while(fabsolute(s * s - x) >= guess)
{
   s = (x/s + s)/2.0;
}
return s;
}

int main()
{
double x,accuracy;
   printf("Enter the value of x");
   scanf("%lf",&x);
   if(x>=0.0 && x<=100.0)
   {
       accuracy=x;
printf("\n Square root is :%lf",sqartNt(x,accuracy)) ;
}
else
{
   printf("Range is not satisfied");
   }
return 0;   
}

/* Pure function of square root with Newton method ( from above program part)*/

double sqartNt(double x, double accuracy )
{
const double guess = 0.00001;
double s= 1.0;
while(fabsolute(s * s - x) >= guess)
{
   s = (x/s + s)/2.0;
}
return s;
}

output(Newton method)