Newton’s Divided Difference Interpolation Formula

Interpolation is an estimation of a value within two known values in a sequence of values. Newton’s divided difference interpolation formula is an interpolation technique used when the interval difference is not same for all sequence of values. Suppose f(x₀), f(x₁), f(x₂)………f(x_n) be the (n+1) values of the function y=f(x) corresponding to the arguments x=x₀, x₁, x₂…x_n, where interval differences are not same Then the first divided difference is given by

The second divided difference is given by

and so on… Divided differences are symmetric with respect to the arguments i.e independent of the order of arguments. so, f[x₀, x₁]=f[x₁, x₀] f[x₀, x₁, x₂]=f[x₂, x₁, x₀]=f[x₁, x₂, x₀] By using first divided difference, second divided difference as so on .A table is formed which is called the divided difference table. Divided difference table:

Advantages of NEWTON’S DIVIDED DIFFERENCE INTERPOLATION FORMULA

These are useful for interpolation.
Through difference table, we can find out the differences in higher order.
Differences at each stage in each of the columns are easily measured by subtracting the previous value from its immediately succeeding value.
The differences are found out successively between the two adjacent values of the y variable till the ultimate difference vanishes or become a constant.

NEWTON’S DIVIDED DIFFERENCE INTERPOLATION FORMULA

Examples:

Input: Value at 7
Output: Value at 7 is 13.47

Below is the implementation of Newton’s divided difference interpolation method.

C++

// CPP program for implementing
// Newton divided difference formula
#include <bits/stdc++.h>

using namespace std;
 
// Function to find the product term

float proterm(int i, float value, float x[])
{

    float pro = 1;

    for (int j = 0; j < i; j++) {

        pro = pro * (value - x[j]);

    }

    return pro;
}
 
// Function for calculating
// divided difference table

void dividedDiffTable(float x[], float y[][10], int n)
{

    for (int i = 1; i < n; i++) {

        for (int j = 0; j < n - i; j++) {

            y[j][i] = (y[j][i - 1] - y[j + 1]

                         [i - 1]) / (x[j] - x[i + j]);

        }

    }
}
 
// Function for applying Newton's
// divided difference formula

float applyFormula(float value, float x[],

                   float y[][10], int n)
{

    float sum = y[0][0];
 
    for (int i = 1; i < n; i++) {

      sum = sum + (proterm(i, value, x) * y[0][i]);

    }

    return sum;
}
 
// Function for displaying 
// divided difference table

void printDiffTable(float y[][10],int n)
{

    for (int i = 0; i < n; i++) {

        for (int j = 0; j < n - i; j++) {

            cout << setprecision(4) << 

                                 y[i][j] << "\t ";

        }

        cout << "\n";

    }
}
 
// Driver Function

int main()
{

    // number of inputs given

    int n = 4;

    float value, sum, y[10][10];

    float x[] = { 5, 6, 9, 11 };
 
    // y[][] is used for divided difference

    // table where y[][0] is used for input

    y[0][0] = 12;

    y[1][0] = 13;

    y[2][0] = 14;

    y[3][0] = 16;
 
    // calculating divided difference table

    dividedDiffTable(x, y, n);
 
    // displaying divided difference table

    printDiffTable(y,n);
 
    // value to be interpolated

    value = 7;
 
    // printing the value

    cout << "\nValue at " << value << " is "

               << applyFormula(value, x, y, n) << endl;

    return 0;
}

Java

// Java program for implementing
// Newton divided difference formula

import java.text.*;

import java.math.*;
 
class GFG{
// Function to find the product term

static float proterm(int i, float value, float x[])
{

    float pro = 1;

    for (int j = 0; j < i; j++) {

        pro = pro * (value - x[j]);

    }

    return pro;
}
 
// Function for calculating
// divided difference table

static void dividedDiffTable(float x[], float y[][], int n)
{

    for (int i = 1; i < n; i++) {

        for (int j = 0; j < n - i; j++) {

            y[j][i] = (y[j][i - 1] - y[j + 1]

                        [i - 1]) / (x[j] - x[i + j]);

        }

    }
}
 
// Function for applying Newton's
// divided difference formula

static float applyFormula(float value, float x[],

                float y[][], int n)
{

    float sum = y[0][0];
 
    for (int i = 1; i < n; i++) {

    sum = sum + (proterm(i, value, x) * y[0][i]);

    }

    return sum;
}
 
// Function for displaying 
// divided difference table

static void printDiffTable(float y[][],int n)
{

    DecimalFormat df = new DecimalFormat("#.####");

    df.setRoundingMode(RoundingMode.HALF_UP);

    for (int i = 0; i < n; i++) {

        for (int j = 0; j < n - i; j++) {

            String str1 = df.format(y[i][j]);

            System.out.print(str1+"\t ");

        }

        System.out.println("");

    }
}
 
// Driver Function

public static void main(String[] args)
{

    // number of inputs given

    int n = 4;

    float value, sum;

    float y[][]=new float[10][10];

    float x[] = { 5, 6, 9, 11 };
 
    // y[][] is used for divided difference

    // table where y[][0] is used for input

    y[0][0] = 12;

    y[1][0] = 13;

    y[2][0] = 14;

    y[3][0] = 16;
 
    // calculating divided difference table

    dividedDiffTable(x, y, n);
 
    // displaying divided difference table

    printDiffTable(y,n);
 
    // value to be interpolated

    value = 7;
 
    // printing the value

    DecimalFormat df = new DecimalFormat("#.##");

    df.setRoundingMode(RoundingMode.HALF_UP);

    System.out.println("\nValue at "+df.format(value)+" is "

            +df.format(applyFormula(value, x, y, n)));
}
}
// This code is contributed by mits

Python3

# Python3 program for implementing 
# Newton divided difference formula 
 
# Function to find the product term 

def proterm(i, value, x): 

    pro = 1; 

    for j in range(i): 

        pro = pro * (value - x[j]); 

    return pro; 
 
# Function for calculating 
# divided difference table 

def dividedDiffTable(x, y, n):
 
    for i in range(1, n): 

        for j in range(n - i): 

            y[j][i] = ((y[j][i - 1] - y[j + 1][i - 1]) /

                                     (x[j] - x[i + j]));

    return y;
 
# Function for applying Newton's 
# divided difference formula 

def applyFormula(value, x, y, n): 
 
    sum = y[0][0]; 
 
    for i in range(1, n):

        sum = sum + (proterm(i, value, x) * y[0][i]); 

    return sum; 
 
# Function for displaying divided 
# difference table 

def printDiffTable(y, n): 
 
    for i in range(n): 

        for j in range(n - i): 

            print(round(y[i][j], 4), "\t", 

                               end = " "); 
 
        print(""); 
 
# Driver Code
 
# number of inputs given 

n = 4; 

y = [[0 for i in range(10)] 

        for j in range(10)]; 

x = [ 5, 6, 9, 11 ]; 
 
# y[][] is used for divided difference 
# table where y[][0] is used for input 

y[0][0] = 12; 

y[1][0] = 13; 

y[2][0] = 14; 

y[3][0] = 16; 
 
# calculating divided difference table 

y=dividedDiffTable(x, y, n); 
 
# displaying divided difference table 
printDiffTable(y, n); 
 
# value to be interpolated 

value = 7; 
 
# printing the value 

print("\nValue at", value, "is",

        round(applyFormula(value, x, y, n), 2))
 
# This code is contributed by mits

// C# program for implementing 
// Newton divided difference formula 

using System;
 
class GFG{ 
// Function to find the product term 

static float proterm(int i, float value, float[] x) 
{ 

    float pro = 1; 

    for (int j = 0; j < i; j++) { 

        pro = pro * (value - x[j]); 

    } 

    return pro; 
} 
 
// Function for calculating 
// divided difference table 

static void dividedDiffTable(float[] x, float[,] y, int n) 
{ 

    for (int i = 1; i < n; i++) { 

        for (int j = 0; j < n - i; j++) { 

            y[j,i] = (y[j,i - 1] - y[j + 1,i - 1]) / (x[j] - x[i + j]); 

        } 

    } 
} 
 
// Function for applying Newton's 
// divided difference formula 

static float applyFormula(float value, float[] x, 

                float[,] y, int n) 
{ 

    float sum = y[0,0]; 
 
    for (int i = 1; i < n; i++) { 

    sum = sum + (proterm(i, value, x) * y[0,i]); 

    } 

    return sum; 
} 
 
// Function for displaying 
// divided difference table 

static void printDiffTable(float[,] y,int n) 
{ 

    for (int i = 0; i < n; i++) { 

        for (int j = 0; j < n - i; j++) { 

            Console.Write(Math.Round(y[i,j],4)+"\t "); 

        } 

        Console.WriteLine(""); 

    } 
} 
 
// Driver Function 

public static void Main() 
{ 

    // number of inputs given 

    int n = 4; 

    float value; 

    float[,] y=new float[10,10]; 

    float[] x = { 5, 6, 9, 11 }; 
 
    // y[][] is used for divided difference 

    // table where y[][0] is used for input 

    y[0,0] = 12; 

    y[1,0] = 13; 

    y[2,0] = 14; 

    y[3,0] = 16; 
 
    // calculating divided difference table 

    dividedDiffTable(x, y, n); 
 
    // displaying divided difference table 

    printDiffTable(y,n); 
 
    // value to be interpolated 

    value = 7; 
 
    // printing the value 

    Console.WriteLine("\nValue at "+(value)+" is "

            +Math.Round(applyFormula(value, x, y, n),2)); 
} 
} 
// This code is contributed by mits

PHP

<?php
// PHP program for implementing 
// Newton divided difference formula 
 
// Function to find the product term 

function proterm($i, $value, $x) 
{ 

    $pro = 1; 

    for ($j = 0; $j < $i; $j++) 

    { 

        $pro = $pro * ($value - $x[$j]); 

    } 

    return $pro; 
} 
 
// Function for calculating 
// divided difference table 

function dividedDiffTable($x, &$y, $n) 
{ 

    for ($i = 1; $i < $n; $i++) 

    { 

        for ($j = 0; $j < $n - $i; $j++) 

        { 

            $y[$j][$i] = ($y[$j][$i - 1] - 

                          $y[$j + 1][$i - 1]) / 

                         ($x[$j] - $x[$i + $j]); 

        } 

    } 
} 
 
// Function for applying Newton's 
// divided difference formula 

function applyFormula($value, $x, $y,$n) 
{ 

    $sum = $y[0][0]; 
 
    for ($i = 1; $i < $n; $i++) 

    { 

        $sum = $sum + (proterm($i, $value, $x) * 

                                   $y[0][$i]); 

    } 

    return $sum; 
} 
 
// Function for displaying 
// divided difference table 

function printDiffTable($y, $n) 
{ 

    for ($i = 0; $i < $n; $i++) 

    { 

        for ($j = 0; $j < $n - $i; $j++) 

        { 

            echo round($y[$i][$j], 4) . "\t "; 

        } 

        echo "\n"; 

    } 
} 
 
// Driver Code
 
// number of inputs given 

$n = 4; 

$y = array_fill(0, 10, array_fill(0, 10, 0)); 

$x = array( 5, 6, 9, 11 ); 
 
// y[][] is used for divided difference 
// table where y[][0] is used for input 

$y[0][0] = 12; 

$y[1][0] = 13; 

$y[2][0] = 14; 

$y[3][0] = 16; 
 
// calculating divided difference table 

dividedDiffTable($x, $y, $n); 
 
// displaying divided difference table 

printDiffTable($y, $n); 
 
// value to be interpolated 

$value = 7; 
 
// printing the value 

echo "\nValue at " . $value . " is " . 

      round(applyFormula($value, $x, 

                         $y, $n), 2) . "\n"
 
// This code is contributed by mits
?>

Javascript

// JavaScript program for implementing
// Newton divided difference formula
 
// Function to find the product term

function proterm(i, value, x)
{

    let pro = 1;

    for (var j = 0; j < i; j++) {

        pro = pro * (value - x[j]);

    }

    return pro;
}
 
// Function for calculating
// divided difference table

function dividedDiffTable(x, y, n)
{

    for (var i = 1; i < n; i++) {

        for (var j = 0; j < n - i; j++) {

            y[j][i] = (y[j][i - 1] - y[j + 1]

                         [i - 1]) / (x[j] - x[i + j]);

        }

    }
}
 
// Function for applying Newton's
// divided difference formula

function applyFormula(value, x, y, n)

{

    let sum = y[0][0];
 
    for (var i = 1; i < n; i++) {

      sum = sum + (proterm(i, value, x) * y[0][i]);

    }

    return sum;
}
 
// Function for displaying 
// divided difference table

function printDiffTable(y, n)
{

    for (var i = 0; i < n; i++) {

        for (var j = 0; j < n - i; j++) {

            process.stdout.write(y[i][j].toFixed(4) + "\t ");

        }

        process.stdout.write("\n");

    }
}
 
// Driver Function
 
// number of inputs given
let n = 4;
let value, sum;
let x = [ 5, 6, 9, 11 ];
 
let y = [];

for (var i = 0; i < 10; i++)

    y.push(new Array(10));
 
// y[][] is used for divided difference
// table where y[][0] is used for input
y[0][0] = 12;
y[1][0] = 13;
y[2][0] = 14;
y[3][0] = 16;
 
// calculating divided difference table
dividedDiffTable(x, y, n);
 
// displaying divided difference table
printDiffTable(y,n);
 
// value to be interpolated
value = 7;
 
// printing the value

console.log("\nValue at " + value + " is " + applyFormula(value, x, y, n));
 
// This code is contributed by phasing17

Output

12     1     -0.1667     0.05     
13     0.3333     0.1333     
14     1     
16     

Value at 7 is 13.47

Time complexity: O(n²)
Auxiliary space: O(1)

Article Tags :

DSA

Mathematical