Computing square root of a number using False Position method

This is a Java program that uses the False Position method to find the square root of a non-negative number. The False Position method is a numerical method that finds the root of a function by bracketing the root between two points and then iteratively reducing the size of the bracketed interval. In this program, we are solving the equation $x^{2} - A = 0.$ to find the square root of A.

Getting Started

Prerequisites

To run this program, you will need:

• Java Development Kit (JDK) 8 or higher installed on your machine
• A Java IDE or a text editor to write and run the program

Usage

To run the program, compile the Main.java file and run the Main class.

javac Main.java
java Main

The program will prompt the user to enter the number to find the square root of, the initial interval [X0, X1], the epsilon value, and the maximum number of iterations. The program will then compute the square root of the number using the false position method and print out the result.

Algorithm

The false position method(Reqular Falsi) works as follows:

1. Choose initial estimate $x_0$ and $x_1$ that bracket the root. Such that $f(x_0) * f(x_1) < 0$.

2. Check if $f(x_0) * f(x_1) < 0$. If not, then choose another interval

3. Compute $x_2$ as $x$ intercept of the tangent(secant) line passing thethrough $x_0$ and $x_1$

   $x_{i+1} = x_{i} - f(x_{i})\left(\frac{x_{i}-x_{i-1}}{f(x_{i})-f(x_{i-1})} \right)$

4. Check the if $[x_1, x_2]$ bracket the root $f(x_1)f(x_2) < 0$

   • if YES continue to the next interation, $x_{i-1} = x_{1}, x_{i} = x_{2}$
   • if No, $x_{i-1} = x_{0}, x_{i} = x_{2}$

5. Repeat step 3-4 until a stop condition is met:

   • $f(x) < epsilon$
   • $|\frac{x - x_{prev}}{x}| < epsilon$
   • iteration > max_iteration

In this program, we are using the above algorithm to solve the equation $x^{2} - A = 0.$ to find the square root of A.

Example

Here is an example of how to use the program:

Find square root of: 25
Initial interval [X0, X1]: 
X0 = 0
X1 = 10
epsilon = 1.0E-7
Max iteration = 100
Iteration 1: X0 = 0.0, X1 = 10.0, X2 = 2.5
Iteration 2: X0 = 2.5, X1 = 10.0, X2 = 4.0
Iteration 3: X0 = 4.0, X1 = 10.0, X2 = 4.642857
Iteration 4: X0 = 4.642857, X1 = 10.0, X2 = 4.878049
Iteration 5: X0 = 4.878049, X1 = 10.0, X2 = 4.9590163
Iteration 6: X0 = 4.9590163, X1 = 10.0, X2 = 4.9863014
Iteration 7: X0 = 4.9863014, X1 = 10.0, X2 = 4.995429
Iteration 8: X0 = 4.995429, X1 = 10.0, X2 = 4.9984756
Iteration 9: X0 = 4.9984756, X1 = 10.0, X2 = 4.999492
Iteration 10: X0 = 4.999492, X1 = 10.0, X2 = 4.9998307
Iteration 11: X0 = 4.9998307, X1 = 10.0, X2 = 4.9999437
Iteration 12: X0 = 4.9999437, X1 = 10.0, X2 = 4.9999814
Iteration 13: X0 = 4.9999814, X1 = 10.0, X2 = 4.9999933
Iteration 14: X0 = 4.9999933, X1 = 10.0, X2 = 4.999998
Iteration 15: X0 = 4.999998, X1 = 10.0, X2 = 4.999999
Iteration 16: X0 = 4.999999, X1 = 10.0, X2 = 4.9999995
Stop condition 2 : |x - x_prev| / |x| < epsilon
Found square root of 25.0 = 4.9999995

Authors

• Phuwadon Decharam

License

This project is licensed under the MIT License - see the LICENSE.md file for details.

Acknowledgments

• This program is based on the False Position method described in "Numerical Methods for Engineers" by Steven C. Chapra and Raymond P. Canale.