Skip to content
Related Articles

Related Articles

Improve Article

Bessel’s Interpolation

  • Last Updated : 25 May, 2021

Interpolation is the technique of estimating the value of a function for any intermediate value of the independent variable, while the process of computing the value of the function outside the given range is called extrapolation.

Central differences : The central difference operator d is defined by the relations :
 

Attention reader! Don’t stop learning now. Get hold of all the important mathematical concepts for competitive programming with the Essential Maths for CP Course at a student-friendly price. To complete your preparation from learning a language to DS Algo and many more,  please refer Complete Interview Preparation Course.

Similarly, high order central differences are defined as :
 



Note – The central differences on the same horizontal line have the same suffix
 

Bessel’s Interpolation formula –
 

It is very useful when u = 1/2. It gives a better estimate when 1/4 < u < 3/4 
Here f(0) is the origin point usually taken to be mid point, since Bessel’s is used to interpolate near the center. 
h is called the interval of difference and u = ( x – f(0) ) / h, Here f(0) is term at the origin chosen.

Examples –
Input : Value at 27.4 ? 
 

Output : 

Value at 27.4 is 3.64968

Implementation of Bessel’s Interpolation – 

C++




// CPP Program to interpolate using Bessel's interpolation
#include <bits/stdc++.h>
using namespace std;
 
// calculating u mentioned in the formula
float ucal(float u, int n)
{
    if (n == 0)
        return 1;
 
    float temp = u;
    for (int i = 1; i <= n / 2; i++)
        temp = temp * (u - i);
 
    for (int i = 1; i < n / 2; i++)
        temp = temp * (u + i);
 
    return temp;
}
 
// calculating factorial of given number n
int fact(int n)
{
    int f = 1;
    for (int i = 2; i <= n; i++)
        f *= i;
 
    return f;
}
 
int main()
{
    // Number of values given
    int n = 6;
    float x[] = { 25, 26, 27, 28, 29, 30 };
 
    // y[][] is used for difference table
    // with y[][0] used for input
    float y[n][n];
    y[0][0] = 4.000;
    y[1][0] = 3.846;
    y[2][0] = 3.704;
    y[3][0] = 3.571;
    y[4][0] = 3.448;
    y[5][0] = 3.333;
 
    // Calculating the central difference table
    for (int i = 1; i < n; i++)
        for (int j = 0; j < n - i; j++)
            y[j][i] = y[j + 1][i - 1] - y[j][i - 1];
 
    // Displaying the central difference table
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n - i; j++)
            cout << setw(4) << y[i][j] << "\t";
        cout << endl;
    }
 
    // value to interpolate at
    float value = 27.4;
 
    // Initializing u and sum
    float sum = (y[2][0] + y[3][0]) / 2;
 
    // k is origin thats is f(0)
    int k;
    if (n % 2) // origin for odd
        k = n / 2;
    else
        k = n / 2 - 1; // origin for even
 
    float u = (value - x[k]) / (x[1] - x[0]);
 
    // Solving using bessel's formula
    for (int i = 1; i < n; i++) {
        if (i % 2)
            sum = sum + ((u - 0.5) *
                  ucal(u, i - 1) * y[k][i]) / fact(i);
        else
            sum = sum + (ucal(u, i) *
                  (y[k][i] + y[--k][i]) / (fact(i) * 2));
    }
 
    cout << "Value at " << value << " is " << sum << endl;
 
    return 0;
}

Java




// Java Program to interpolate using Bessel's interpolation
import java.text.*;
class GFG{
// calculating u mentioned in the formula
static double ucal(double u, int n)
{
    if (n == 0)
        return 1;
 
    double temp = u;
    for (int i = 1; i <= n / 2; i++)
        temp = temp * (u - i);
 
    for (int i = 1; i < n / 2; i++)
        temp = temp * (u + i);
 
    return temp;
}
 
// calculating factorial of given number n
static int fact(int n)
{
    int f = 1;
    for (int i = 2; i <= n; i++)
        f *= i;
 
    return f;
}
 
public static void main(String[] args)
{
    // Number of values given
    int n = 6;
    double x[] = { 25, 26, 27, 28, 29, 30 };
 
    // y[][] is used for difference table
    // with y[][0] used for input
    double[][] y=new double[n][n];
    y[0][0] = 4.000;
    y[1][0] = 3.846;
    y[2][0] = 3.704;
    y[3][0] = 3.571;
    y[4][0] = 3.448;
    y[5][0] = 3.333;
 
    // Calculating the central difference table
    for (int i = 1; i < n; i++)
        for (int j = 0; j < n - i; j++)
            y[j][i] = y[j + 1][i - 1] - y[j][i - 1];
 
    // Displaying the central difference table
    DecimalFormat df = new DecimalFormat("#.########");
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n - i; j++)
            System.out.print(y[i][j]+"\t");
        System.out.println("");
    }
 
    // value to interpolate at
    double value = 27.4;
 
    // Initializing u and sum
    double sum = (y[2][0] + y[3][0]) / 2;
 
    // k is origin thats is f(0)
    int k;
    if ((n % 2)>0) // origin for odd
        k = n / 2;
    else
        k = n / 2 - 1; // origin for even
 
    double u = (value - x[k]) / (x[1] - x[0]);
 
    // Solving using bessel's formula
    for (int i = 1; i < n; i++) {
        if ((i % 2)>0)
            sum = sum + ((u - 0.5) *
                ucal(u, i - 1) * y[k][i]) / fact(i);
        else
            sum = sum + (ucal(u, i) *
                (y[k][i] + y[--k][i]) / (fact(i) * 2));
    }
 
    System.out.printf("Value at "+value+" is %.5f",sum);
 
}
}
// This code is contributed by mits

Python3




# Python3 Program to interpolate
# using Bessel's interpolation
 
# calculating u mentioned in the
# formula
def ucal(u, n):
 
    if (n == 0):
        return 1;
 
    temp = u;
    for i in range(1, int(n / 2 + 1)):
        temp = temp * (u - i);
 
    for i in range(1, int(n / 2)):
        temp = temp * (u + i);
 
    return temp;
 
# calculating factorial of
# given number n
def fact(n):
 
    f = 1;
    for i in range(2, n + 1):
        f *= i;
 
    return f;
 
# Number of values given
n = 6;
x = [25, 26, 27, 28, 29, 30];
 
# y[][] is used for difference
# table with y[][0] used for input
y = [[0 for i in range(n)]
        for j in range(n)];
y[0][0] = 4.000;
y[1][0] = 3.846;
y[2][0] = 3.704;
y[3][0] = 3.571;
y[4][0] = 3.448;
y[5][0] = 3.333;
 
# Calculating the central
# difference table
for i in range(1, n):
    for j in range(n - i):
        y[j][i] = y[j + 1][i - 1] - y[j][i - 1];
 
# Displaying the central
# difference table
for i in range(n):
    for j in range(n - i):
        print(y[i][j], "\t", end = " ");
    print("");
 
# value to interpolate at
value = 27.4;
 
# Initializing u and sum
sum = (y[2][0] + y[3][0]) / 2;
 
# k is origin thats is f(0)
k = 0;
if ((n % 2) > 0): # origin for odd
    k = int(n / 2);
else:
    k = int(n / 2 - 1); # origin for even
 
u = (value - x[k]) / (x[1] - x[0]);
 
# Solving using bessel's formula
for i in range(1, n):
 
    if (i % 2):
        sum = sum + ((u - 0.5) *
                 ucal(u, i - 1) *
              y[k][i]) / fact(i);
    else:
        sum = sum + (ucal(u, i) * (y[k][i] +
                     y[k - 1][i]) / (fact(i) * 2));
        k -= 1;
 
print("Value at", value, "is", round(sum, 5));
 
# This code is contributed by mits

C#




// C# Program to interpolate using Bessel's interpolation
 
class GFG{
// calculating u mentioned in the formula
static double ucal(double u, int n)
{
    if (n == 0)
        return 1;
 
    double temp = u;
    for (int i = 1; i <= n / 2; i++)
        temp = temp * (u - i);
 
    for (int i = 1; i < n / 2; i++)
        temp = temp * (u + i);
 
    return temp;
}
 
// calculating factorial of given number n
static int fact(int n)
{
    int f = 1;
    for (int i = 2; i <= n; i++)
        f *= i;
 
    return f;
}
 
public static void Main()
{
    // Number of values given
    int n = 6;
    double []x = { 25, 26, 27, 28, 29, 30 };
 
    // y[,] is used for difference table
    // with y[,0] used for input
    double[,] y=new double[n,n];
    y[0,0] = 4.000;
    y[1,0] = 3.846;
    y[2,0] = 3.704;
    y[3,0] = 3.571;
    y[4,0] = 3.448;
    y[5,0] = 3.333;
 
    // Calculating the central difference table
    for (int i = 1; i < n; i++)
        for (int j = 0; j < n - i; j++)
            y[j,i] = y[j + 1,i - 1] - y[j,i - 1];
 
    // Displaying the central difference table
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n - i; j++)
            System.Console.Write(y[i,j]+"\t");
        System.Console.WriteLine("");
    }
 
    // value to interpolate at
    double value = 27.4;
 
    // Initializing u and sum
    double sum = (y[2,0] + y[3,0]) / 2;
 
    // k is origin thats is f(0)
    int k;
    if ((n % 2)>0) // origin for odd
        k = n / 2;
    else
        k = n / 2 - 1; // origin for even
 
    double u = (value - x[k]) / (x[1] - x[0]);
 
    // Solving using bessel's formula
    for (int i = 1; i < n; i++) {
        if ((i % 2)>0)
            sum = sum + ((u - 0.5) *
                ucal(u, i - 1) * y[k,i]) / fact(i);
        else
            sum = sum + (ucal(u, i) *
                (y[k,i] + y[--k,i]) / (fact(i) * 2));
    }
 
    System.Console.WriteLine("Value at "+value+" is "+System.Math.Round(sum,5));
 
}
}
// This code is contributed by mits

PHP




<?php
// PHP Program to interpolate
// using Bessel's interpolation
 
// calculating u mentioned
// in the formula
function ucal($u, $n)
{
    if ($n == 0)
        return 1;
 
    $temp = $u;
    for ($i = 1;
         $i <= (int)($n / 2); $i++)
        $temp = $temp *
               ($u - $i);
 
    for ($i = 1;
         $i < (int)($n / 2); $i++)
        $temp = $temp * ($u + $i);
 
    return $temp;
}
 
// calculating factorial
// of given number n
function fact($n)
{
    $f = 1;
    for ($i = 2; $i <= $n; $i++)
        $f *= $i;
 
    return $f;
}
 
// Number of values given
$n = 6;
$x = array(25, 26, 27,
           28, 29, 30);
 
// y[][] is used for difference
// table with y[][0] used for input
$y;
for($i = 0; $i < $n; $i++)
for($j = 0; $j < $n; $j++)
$y[$i][$j] = 0.0;
$y[0][0] = 4.000;
$y[1][0] = 3.846;
$y[2][0] = 3.704;
$y[3][0] = 3.571;
$y[4][0] = 3.448;
$y[5][0] = 3.333;
 
// Calculating the central
// difference table
for ($i = 1; $i < $n; $i++)
    for ($j = 0; $j < $n - $i; $j++)
        $y[$j][$i] = $y[$j + 1][$i - 1] -
                     $y[$j][$i - 1];
 
// Displaying the central
// difference table
for ($i = 0; $i < $n; $i++)
{
    for ($j = 0; $j < $n - $i; $j++)
        echo str_pad($y[$i][$j], 4) . "\t";
    echo "\n";
}
 
// value to interpolate at
$value = 27.4;
 
// Initializing u and sum
$sum = ($y[2][0] +
        $y[3][0]) / 2;
 
// k is origin thats is f(0)
$k;
if ($n % 2) // origin for odd
    $k = $n / 2;
else
    $k = $n / 2 - 1; // origin for even
 
$u = ($value - $x[$k]) /
     ($x[1] - $x[0]);
 
// Solving using
// bessel's formula
for ($i = 1; $i < $n; $i++)
{
    if ($i % 2)
        $sum = $sum + (($u - 0.5) *
                   ucal($u, $i - 1) *
                     $y[$k][$i]) / fact($i);
    else
        $sum = $sum + (ucal($u, $i) *
                      ($y[$k][$i] +
                       $y[--$k][$i]) /
                       (fact($i) * 2));
}
 
echo "Value at " . $value .
     " is " . $sum . "\n";
 
// This code is contributed by mits
?>

Javascript




<script>
 
// Javascript Program to interpolate
// using Bessel's interpolation
 
// Calculating u mentioned in the formula
function ucal(u, n)
{
    if (n == 0)
        return 1;
 
    var temp = u;
    for(var i = 1; i <= n / 2; i++)
        temp = temp * (u - i);
 
    for(var i = 1; i < n / 2; i++)
        temp = temp * (u + i);
 
    return temp;
}
 
// Calculating factorial of given number n
function fact(n)
{
    var f = 1;
    for(var i = 2; i <= n; i++)
        f *= i;
 
    return f;
}
 
// Driver code
 
// Number of values given
var n = 6;
var x = [ 25, 26, 27, 28, 29, 30 ];
 
// y is used for difference table
// with y[0] used for input
var y = Array(n).fill(0.0).map(x => Array(n).fill(0.0));;
y[0][0] = 4.000;
y[1][0] = 3.846;
y[2][0] = 3.704;
y[3][0] = 3.571;
y[4][0] = 3.448;
y[5][0] = 3.333;
 
// Calculating the central difference table
for(var i = 1; i < n; i++)
    for(var j = 0; j < n - i; j++)
        y[j][i] = y[j + 1][i - 1] - y[j][i - 1];
 
// Displaying the central difference table
for(var i = 0; i < n; i++)
{
    for(var j = 0; j < n - i; j++)
        document.write(y[i][j].toFixed(6) +
                      "    ");
                       
    document.write('<br>');
}
 
// Value to interpolate at
var value = 27.4;
 
// Initializing u and sum
var sum = (y[2][0] + y[3][0]) / 2;
 
// k is origin thats is f(0)
var k;
 
// Origin for odd
if ((n % 2) > 0)
    k = n / 2;
else
 
    // Origin for even
    k = n / 2 - 1;
 
var u = (value - x[k]) / (x[1] - x[0]);
 
// Solving using bessel's formula
for(var i = 1; i < n; i++)
{
    if ((i % 2) > 0)
        sum = sum + ((u - 0.5) *
        ucal(u, i - 1) * y[k][i]) / fact(i);
    else
        sum = sum + (ucal(u, i) *
         (y[k][i] + y[--k][i]) / (fact(i) * 2));
}
 
document.write("Value at " + value.toFixed(6) +
                      " is " + sum.toFixed(6));
 
// This code is contributed by Princi Singh
 
</script>

Output: 

    4    -0.154    0.0120001    -0.00300002    0.00399971    -0.00699902    
3.846    -0.142    0.00900006    0.000999689    -0.00299931    
3.704    -0.133    0.00999975    -0.00199962    
3.571    -0.123    0.00800014    
3.448    -0.115    
3.333    
Value at 27.4 is 3.64968

This article is contributed by Shubham Rana. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
 




My Personal Notes arrow_drop_up
Recommended Articles
Page :