devxlogo

How to calculate sqrt()

How to calculate sqrt()

Question:
How do you calculate sqrt()?

Answer:
Of course the best way to compute square roots is to useMath.sqrt(),but I assume your question is academic. I’m not sure of the bestway to find square roots, but historically, Newton’s Method is themost significant.

Newton’s Method tells us how to approximate an x-interceptof a differentiable curve described by the equation y = f(x). Howdoes this help us to find the square root of 42? Well, thex-intercepts of the parabola y = x^2 – 42 occur at sqrt(42) and-sqrt(42).

The x-intercept of a curve y = f(x) occurs at a point x = a such thatf(a) = 0. So, Newton’s Method iterates a sequence of progressivelybetter guesses until f(guess) is acceptably close to 0:

guess = 1;   while (delta < |f(guess)|)  // delta small      guess = improve(guess);   return guess;
How do we improve a bad guess?Newton’s idea is this:
Draw a line tangent to the curve at the point (guess, f(guess)).
Using calculus, it’s pretty easy to get the equation for this line:
y – f(guess) = slope * (x – guess)
where slope = f'(guess) = the derivative of f at x = guess.

Newton reasoned that this line crudely approximates the curve y = f(x)near the point of tangency, and therefore the x-intercept of this linecrudely approximates the x-intercept of y = f(x).

It’s easy to calculate the x-intercept of this line. Just set y = 0 andsolve for x:

x = guess – f(guess)/f'(guess)
Hence:
improve(guess) = guess – f(guess)/f'(guess)
>From an implementation standpoint, the only complication is computingf'(x). Again we dust off our calculus book, where we discover
f'(x) = limit (f(x + delta) – f(x))/delta as delta tends to 0
Dropping the limit and choosing delta to be suitably small shouldgive us an acceptable approximation of f'(x).
See also  Essential Measures for Safeguarding Your Digital Data

Here’s a Java implementation with a crude GUI. It can be used tofind the n-th root of any real number a where 0 <= a and 0 <= n.

import java.awt.*;// A system for approximating n-th root of aclass Solver {   private int n;   private double a;   private static double delta = 0.00000001; // controls accuracy   private static double initGuess = 1;   // constructs a solver for x^n – a   public Solver(int x, double y) { n = x; a = y; }   // your basic iterator for the improve method   public double solve() {      double guess = initGuess;      while(!goodEnough(guess))         guess = improve(guess);      return guess;   }   // f(x) = x^n – a = the function we’re solving   private double f(double x) {      return Math.pow(x, n) – a;   }   // df(x) = f'(x), approximately   private double df(double x) {      return (f(x + delta) – f(x))/delta;   }   // 0 <= |f(guess)| <= delta?   private boolean goodEnough(double guess) {      return (Math.abs(f(guess)) <= delta);   }   // Newton's Method   private double improve(double guess) {      return guess - f(guess)/df(guess);   }}// a primitive GUIpublic class Main extends Frame {   private TextField root;   private TextField base;   private TextField result;   public Main() {      setTitle("Root Tester");      setLayout(new FlowLayout());      root = new TextField("root", 15);      base = new TextField("base", 15);      result = new TextField("result", 15);      add(root);      add(base);      add(result);   }   public boolean action(Event e, Object o) {      if (e.target == base) {         double d = Double.valueOf(base.getText()).doubleValue();         int i = Integer.valueOf(root.getText()).intValue();         if (0 <= i && 0 <= d) {            Solver s = new Solver(i, d); // make a solver for x^i - d            result.setText("+/- " + String.valueOf(s.solve()));         }         else result.setText("0"); // for now      }      repaint();      return true;   }   public static void main(String args[]) {      Frame m = new Main();      m.resize(300, 200);      m.show();   }} 

devxblackblue

About Our Editorial Process

At DevX, we’re dedicated to tech entrepreneurship. Our team closely follows industry shifts, new products, AI breakthroughs, technology trends, and funding announcements. Articles undergo thorough editing to ensure accuracy and clarity, reflecting DevX’s style and supporting entrepreneurs in the tech sphere.

See our full editorial policy.

About Our Journalist