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:
- 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.
- 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.
- 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.
- 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.
- 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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