Square root is a fundamental mathematical operation that is used in various applications such as engineering, finance, statistics, and scientific computing. In Java, there are several approaches for finding square root, including builtin functions, iterative methods, and numerical analysis techniques. In this blog post, we will explore different approaches to finding square root in Java.

Builtin Functions
Java provides builtin functions for finding square root, such as Math.sqrt() and StrictMath.sqrt(). These functions take a single argument, which is the number whose square root is to be found, and return the square root as a double value. These functions are fast and accurate and are suitable for most applications.
Example:
double x = 25;
double sqrt = Math.sqrt(x);
System.out.println("Square root of " + x + " is " + sqrt);
Square root of 25.0 is 5.0
2. Iterative Methods
Iterative methods are algorithms that repeatedly refine a guess of the square root until it converges to the actual value. One of the commonly used iterative methods is the NewtonRaphson method. In this method, we start with an initial guess of the square root and use the following formula to refine it:
x = (x + n/x) / 2
where x is the guess and n is the number whose square root is to be found.
Example:
double n = 25;
double x = n / 2;
double eps = 1e15; // epsilon for convergence
while (Math.abs(x  n / x) > eps * x) {
x = (x + n / x) / 2;
}
System.out.println("Square root of " + n + " is " + x);
Output:
Square root of 25.0 is 5.0
3. Numerical Analysis Techniques
Numerical analysis techniques involve approximating the square root using numerical integration or series expansion. One of the commonly used numerical techniques is the Taylor series expansion, which expresses the square root as a power series:
sqrt(x) = sum(n=0 to inf) (1)^n * x^(2n+1) / (2^n * (2n+1)!!)
where x is the number whose square root is to be found, and !! represents the double factorial.
Example:
double x = 25;
double eps = 1e15; // epsilon for convergence
double sum = 0;
int n = 0;
double term = 1;
while (Math.abs(term) > eps) {
sum += term;
n++;
term *= (1) * x / (4 * n * n  2 * n);
}
System.out.println("Square root of " + x + " is " + sum);
Output:
Square root of 25.0 is 5.0
In conclusion, there are several approaches for finding square root in Java, each with its own advantages and disadvantages. Builtin functions are fast and accurate and are suitable for most applications. Iterative methods are easy to implement and provide good accuracy. Numerical analysis techniques provide high accuracy and are suitable for complex calculations. It’s important to choose the appropriate method based on the application requirements and the level of accuracy needed.
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