алгоритм – Calculating the square root without library methods

Question:

How can you calculate the square root without using sqrt(n) and n^0.5 ?

Answer:

The question actually has many solutions.

The most banal is the method of half division.

double l = 0;
double r = 1e100; //большое число
double m;
while (r - l > 1e-8){ //точность
    m = l + (r - l)/2;
    if (m*m > n) l = m;
            else r = m; 
}
//ответ в l

There are more original ways, for example, simulating a calculation in a column (here is an example , I will not give the code)

The method is more for C, but I think it can be used in Java as well. Explanation

float Q_rsqrt( float number )
{
    long i;
    float x2, y;
    const float threehalfs = 1.5F;

    x2 = number * 0.5F;
    y  = number;
    i  = * ( long * ) &y;                       
    i  = 0x5f3759df - ( i >> 1 );                
    y  = * ( float * ) &i;
    y  = y * ( threehalfs - ( x2 * y * y ) );   // 1 итерация
//  y  = y * ( threehalfs - ( x2 * y * y ) );   // 2 итерация, можно удалить
    return 1/y;
}

You can use logarithms

return Math.exp( Math.log(n) / 2);

You can use numerical methods, such as Newton's method

double x = 1;
for (;;) {
   double nx = (x + n / x) / 2;
   if (abs (x - nx) < 1e-10)  break; //точность
   x = nx;
}

There are many other ways, it all depends on the specific requirements.