# 牛顿法求平方根的时间复杂度

sqrt方法的Big O符号是什么？

``````/**Find square root of a number using Newton's method**/
/**Specify number of correct precision required in a square root**/
/**Also specify maxIterations limit so that program won't go into into infinity loop**/
import java.util.*;
public class SqrtNewton{
public static void main(String[] args){
try{
long startTime = System.nanoTime();
Scanner scanner = new Scanner(System.in);
//Number for which square root has to be found
System.out.println("Enter number - ");
long number = scanner.nextLong();
//Maximum no of iterations if program does not found Square root untill then
int maxIterations = 40;
//precision value to untill correct square root is required
int precision = 3;
double x = 1;
//Negative numbers do not have square roots
if (number < 0) throw new IllegalArgumentException("Provided value is invalid");
//iteration start
int itr = 0;
//epsilon value to check equality of double value untill given precision
double epsilon = Math.pow(10,-precision);
double squareRoot = sqrt(number,maxIterations,x,itr,epsilon);
System.out.println("Square Root Of "+number+" With correct precision "+precision+" is :- "+squareRoot);
System.out.printf("Square Root Of %d With correct precision %d is :- %."+precision+"f",number,precision,squareRoot);
System.out.println();
long endTime = System.nanoTime();
System.out.println("Total Running Time - "+(endTime - startTime));
}catch(Exception e){
//e.printStackTrace();
System.err.println("Exception - "+e.getMessage());
}
}
private static double sqrt(long number,int maxIterations,double x,int itr,double epsilon) throws MaxIterationsReachedException{
if(itr >= maxIterations){
throw new MaxIterationsReachedException(maxIterations);
}else{
double x1 = (x + (number/x))/2;
/**To check equality of double number untill given precision**/
/**This will check 1.1333334 - 1.1333334 < 0.000001(if precision is 6)**/
if(Math.abs(x1 - x) <=  epsilon){
System.out.println("Total Iterations - "+itr);
return x1;
}
else
return sqrt(number,maxIterations,x1,++itr,epsilon);
}
}
}

class MaxIterationsReachedException extends Exception{
MaxIterationsReachedException(int maxIterations){
super("Maximum iterations limit "+maxIterations+" reached Increase maxIterations limit if required");
}
}
``````

``````private static double sqrt2(long number, int maxIterations, double x, int itr, double epsilon)
throws MaxIterationsReachedException {
double x1 = (x + (number / x)) / 2;
while (Math.abs(x1 - x) > epsilon) {
if (itr >= maxIterations) {
throw new MaxIterationsReachedException(maxIterations);
}
x = x1;
x1 = (x + (number / x)) / 2;
itr++;
}
System.out.println("Total Iterations - " + itr);
return x1;
}
``````

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