A simple example of coding and simulating a little problem in Java

  Here we can visually understand the convergence of Cauchy sequence by entering the value of x and N.
  Definition: The function f(x,N,i) of two positive
integers x, i and a rational number N is defined by
          .
  The function g(N,i) of a positive integer i and a
rational number N is defined by
          .
  [•] is Gauss' symbol and r(x) is a positive square root of x. The vertical axis shows a sequence of values on the function of (x,N,i). The horizontal axis shows a sequence of values on the function of (N,i). The function of (x,N,i) has a minimum value in the assigned (i=1) condition and a maximum value in the assigned (i=1000) condition when the domain of i is between 1 and 1000.
  The domain of x is over 0 and below 10^8, and the domain of N is over 0.1^16 and below 10^16. When you enter the numerical values of x and N, please push the enter key each time you enter x or N in the text box.
  If you enter the value of x and N in other domains, the program automatically sets the variables such as x=10^5 and N=10^5.

              source code: CauchySequence.java

import java.math.BigDecimal;
import java.math.BigInteger;
import java.awt.*;
import java.awt.event.*;
import java.applet.Applet;
import javax.swing.JLabel;
import javax.swing.JTextField;

public class CauchySequence extends Applet implements ActionListener{

	private static final long serialVersionUID = -6724959421340457497L;
	JTextField yx = new JTextField("100000");
	JTextField yn = new JTextField("100000");
	JLabel label1 = new JLabel("Convergence of Cauchy sequence",
	JLabel.CENTER);
	JLabel label2 = new JLabel("A natural number, 0<x<10^8",
	JLabel.CENTER);
	JLabel label3 = new JLabel("A rational number, 0.1^16<N<10^16",
	JLabel.CENTER);

	// variable set
	double x = 100000;
	double n = 100000;

	public void init(){
		label1.setPreferredSize(new Dimension(416,24));
		label1.setFont(new Font("Serif",Font.BOLD,14));
		add(label1);
		label2.setPreferredSize(new Dimension(170,23));
		label2.setFont(new Font("Serif",Font.BOLD,11));
		add(label2);
		yx.setPreferredSize(new Dimension(80,23));
		add(yx);
		label3.setPreferredSize(new Dimension(210,23));
		label3.setFont(new Font("Serif",Font.BOLD,11));
		add(label3);
		yn.setPreferredSize(new Dimension(120,23));
		add(yn);
		yx.addActionListener(this);
		yn.addActionListener(this);
	}

	public void actionPerformed(ActionEvent e){
		if(e.getSource()==yx){
			x=Double.valueOf(yx.getText()).doubleValue();
			if(x <= 0 || x >= Math.pow(10.0,8) || x!=(long)x){
				x=100000;
			}
		}
		if(e.getSource()==yn){
			n = Double.valueOf(yn.getText()).doubleValue();
			BigDecimal n0 = new BigDecimal(yn.getText());
			BigDecimal n1 = new BigDecimal(Math.pow(0.1,16));
			BigDecimal n2 = new BigDecimal(Math.pow(10.0,16));
			int d1 = n0.compareTo(n1);
			int d2 = n0.compareTo(n2);
 			if(d1==-1 || d1==0 || d2==0 || d2==1){
				n=100000;
			}
		}
		yx.setText(""+(long)x);
		yn.setText(""+n);
		repaint();
	}

	public void paint(Graphics g){

		// variable set
		int         i;
		double      b, m, p, y, minp, maxp, minb, maxb;
		BigDecimal  a0, a2, b0, n0, p0, p1, x0;
		BigInteger  a1;
		double[] pp = new double[1000];
		double[] bb = new double[1000];
		int[] xxx = new int[1000];
		int[] yyy = new int[1000];

		for(i=1;i<=1000;i++){
			p = Math.exp(10-0.01*i);
			m = n/p;
			p0 = new BigDecimal(p);
			n0 = new BigDecimal(n);
			p1 = n0.divide(p0,30,BigDecimal.ROUND_HALF_EVEN);
			pp[i-1] = Math.log(m);
			y = Math.sqrt(x);
			x0 = new BigDecimal(y);
			a0 = x0.multiply(p1);
			a1 = a0.toBigInteger();
			a2 = new BigDecimal(a1);
			b0 = a2.divide(p1,30,BigDecimal.ROUND_HALF_EVEN);
			b = b0.doubleValue();
			bb[i-1] = b;
		}

		minp = pp[0];
		maxp = pp[0];
		minb = bb[0];
		maxb = bb[0];

		for(i=0;i<=999;i++){
			if(pp[i]>maxp){
				maxp = pp[i];
			}
		}

		for(i=0;i<=999;i++){
			if(pp[i]<minp){
				minp = pp[i];
			}
		}

		for(i=0;i<=999;i++){
			if(bb[i]>maxb){
				maxb = bb[i];
			}
		}

		for(i=0;i<=999;i++){
			if(bb[i]<minb){
				minb = bb[i];
			}
		}

		for (i=0;i<=999;i++){
			double xx = maxp-minp;
			double yy = maxb-minb;
			xxx[i] = (int)((pp[i]-minp)*(351/xx))+33;
			yyy[i] = 441-(int)((bb[i]-minb)*(351/yy));
		}

		Graphics2D g2 = (Graphics2D)g;
		GradientPaint gp1 = new GradientPaint(0, 0,
		new Color(154,181,228), 0,470,new Color(225,232,245), true);
		g2.setPaint(gp1);
		g2.fillRect(0,0,416,470);
		super.paint(g);
		GradientPaint gp2 = new GradientPaint(0, 33,
		new Color(225,232,245), 0,351,new Color(154,181,228), true);
		g2.setPaint(gp2);
		g2.fillRect(33,90,354,351);
		g2.setColor (Color.black);
		g2.setFont(new Font ("Serif",Font.PLAIN,11));
		g2.drawString("f",15,95);
		g2.drawString("(x,N,i)",1,108);
		g2.drawString("g(N,i)",365,455);
		for (i=0;i<=998;i++){
  			g.drawLine(xxx[i], yyy[i], xxx[i+1], yyy[i+1]);
		}
	}
}

  I have recently begun to code the examples of numerical computation in Java and hope to make better use of the computational power and the graphical user interface.