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 a 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:

NEWTON’S DIVIDED DIFFERENCE INTERPOLATION FORMULA


 f(x)=f(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)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 for Newton’s divided difference interpolation method.

C++

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// 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;
}

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Java

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// 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

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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#

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// 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 

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PHP

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<?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
?>

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Output:

12     1     -0.1667     0.05     
13     0.3333     0.1333     
14     1     
16     

Value at 7 is 13.47


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