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Newton’s Divided Difference Interpolation Formula

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  • Difficulty Level : Medium
  • Last Updated : 14 Nov, 2022
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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(xn) be the (n+1) values of the function y=f(x) corresponding to the arguments x=x0, x1, x2…xn, where interval differences are not same Then the first divided difference is given by

f[x_0, x_1]=\frac{f(x_1)-f(x_0)}{x_1-x_0}

The second divided difference is given by

f[x_0, x_1, x_2]=\frac{f[x_1, x_2]-f[x_0, x_1]}{x_2-x_0}

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

f(x)=f(x_0)+(x-x_0)f[x_0, x_1]+(x-x_0)(x-x_1)f[x_0, x_1, x_2]+..........................+(x-x_0)(x-x_1)...(x-x_k_-_1)f[x_0, x_1, x_2...x_k]

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#




// 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) 


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