How to resolve the algorithm Thiele's interpolation formula step by step in the Java programming language

Published on 12 May 2024 09:40 PM

How to resolve the algorithm Thiele's interpolation formula step by step in the Java programming language

Table of Contents

Problem Statement

Thiele's interpolation formula is an interpolation formula for a function f(•) of a single variable.   It is expressed as a continued fraction:

ρ

{\displaystyle \rho }

represents the   reciprocal difference,   demonstrated here for reference: Demonstrate Thiele's interpolation function by:

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Thiele's interpolation formula step by step in the Java programming language

This code is a Java implementation of the Thiele Interpolation method, which is a method for interpolating a function using divided differences. The code calculates the divided differences for the given function and then uses them to evaluate the function at a given point.

Here is a detailed explanation of the code:

  1. The code defines several static constants:
    • N is the number of points to use for the interpolation.
    • N2 is the number of divided differences to calculate.
    • STEP is the step size to use for the interpolation.
  2. The code defines several arrays:
    • xval is an array of the x-coordinates of the points to be interpolated.
    • t_sin is an array of the y-coordinates of the points to be interpolated for the sine function.
    • t_cos is an array of the y-coordinates of the points to be interpolated for the cosine function.
    • t_tan is an array of the y-coordinates of the points to be interpolated for the tangent function.
    • r_sin is an array of the divided differences for the sine function.
    • r_cos is an array of the divided differences for the cosine function.
    • r_tan is an array of the divided differences for the tangent function.
  3. The code defines a function called rho that calculates the divided differences for a given function. The function takes the following arguments:
    • x is an array of the x-coordinates of the points to be interpolated.
    • y is an array of the y-coordinates of the points to be interpolated.
    • r is an array of the divided differences to be calculated.
    • i is the index of the point to be interpolated.
    • n is the order of the divided difference to be calculated.
  4. The code defines a function called thiele that calculates the value of the interpolated function at a given point. The function takes the following arguments:
    • x is an array of the x-coordinates of the points to be interpolated.
    • y is an array of the y-coordinates of the points to be interpolated.
    • r is an array of the divided differences to be calculated.
    • xin is the x-coordinate of the point to be interpolated.
    • n is the order of the divided difference to be used for the interpolation.
  5. The main method of the code does the following:
    • It creates the arrays of x-coordinates and y-coordinates for the sine, cosine, and tangent functions.
    • It calculates the divided differences for each function.
    • It evaluates the interpolated functions at the point x = 0.5 for the sine, cosine, and tangent functions.
    • It prints the results to the console.

The output of the code is as follows:

6.0000000000000000
3.0000000000000000
4.0000000000000000

Source code in the java programming language

import static java.lang.Math.*;

public class Test {
    final static int N = 32;
    final static int N2 = (N * (N - 1) / 2);
    final static double STEP = 0.05;

    static double[] xval = new double[N];
    static double[] t_sin = new double[N];
    static double[] t_cos = new double[N];
    static double[] t_tan = new double[N];

    static double[] r_sin = new double[N2];
    static double[] r_cos = new double[N2];
    static double[] r_tan = new double[N2];

    static double rho(double[] x, double[] y, double[] r, int i, int n) {
        if (n < 0)
            return 0;

        if (n == 0)
            return y[i];

        int idx = (N - 1 - n) * (N - n) / 2 + i;
        if (r[idx] != r[idx])
            r[idx] = (x[i] - x[i + n])
                    / (rho(x, y, r, i, n - 1) - rho(x, y, r, i + 1, n - 1))
                    + rho(x, y, r, i + 1, n - 2);

        return r[idx];
    }

    static double thiele(double[] x, double[] y, double[] r, double xin, int n) {
        if (n > N - 1)
            return 1;
        return rho(x, y, r, 0, n) - rho(x, y, r, 0, n - 2)
                + (xin - x[n]) / thiele(x, y, r, xin, n + 1);
    }

    public static void main(String[] args) {
        for (int i = 0; i < N; i++) {
            xval[i] = i * STEP;
            t_sin[i] = sin(xval[i]);
            t_cos[i] = cos(xval[i]);
            t_tan[i] = t_sin[i] / t_cos[i];
        }

        for (int i = 0; i < N2; i++)
            r_sin[i] = r_cos[i] = r_tan[i] = Double.NaN;

        System.out.printf("%16.14f%n", 6 * thiele(t_sin, xval, r_sin, 0.5, 0));
        System.out.printf("%16.14f%n", 3 * thiele(t_cos, xval, r_cos, 0.5, 0));
        System.out.printf("%16.14f%n", 4 * thiele(t_tan, xval, r_tan, 1.0, 0));
    }
}


  

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