Complete the sequence generated by a polynomial

Given a sequence with some of its term, we need to calculate next K term of this sequence. It is given that sequence is generated by some polynomial, however complex that polynomial can be. Notice polynomial is an expression of the following form:
P(x) = a0 + a1 x +a2 x^2 + a3 x^3 …….. + an x^n

The given sequence can always be described by a number of polynomials, among these polynomial we need to find polynomial with lowest degree and generate next terms using this polynomial only.
Examples:

If given sequence is 1, 2, 3, 4, 5 then its next term will be 6, 7, 8, etc
and this correspond to a trivial polynomial.
If given sequence is 1, 4, 7, 10 then its next term will be 13, 16, etc.

We can solve this problem using a technique called difference of differences method, which is derivable from lagrange polynomial.

The technique is simple, we take the difference between the consecutive terms, if difference are equal then we stop and build up next term of the sequence otherwise we again take the difference between these differences until they become constant.

The technique is explained in below diagram with an example, given sequence is 8, 11, 16, 23 and we are suppose to find next 3 terms of this sequence.

generate-polynomial

In below code same technique is implemented, first we loop until we get a constant difference keeping first number of each difference sequence in a separate vector for rebuilding the sequence again. Then we add K instance of same constant difference to our array for generating new K term of sequence and we follow same procedure in reverse order to rebuild the sequence.

See below code for better understanding.

C++

// C++ code to generate next terms of a given polynomial 
// sequence 
#include <bits/stdc++.h> 
using namespace std; 
  
//  method to print next terms term of sequence 
void nextTermsInSequence(int sequence[], int N, int terms) 
{ 
    int diff[N + terms]; 
  
    //  first copy the sequence itself into diff array 
    for (int i = 0; i < N; i++) 
        diff[i] = sequence[i]; 
  
    bool more = false; 
    vector<int> first; 
    int len = N; 
  
    // loop untill one difference remains or all 
    // difference become constant 
    while (len > 1) 
    { 
        // keeping the first term of sequence for 
        // later rebuilding 
        first.push_back(diff[0]); 
        len--; 
  
        // converting the difference to difference 
        // of differences 
        for (int i = 0; i < len; i++) 
            diff[i] = diff[i + 1] - diff[i]; 
  
        // checking if all difference values are 
        // same or not 
        int i; 
        for (i = 1; i < len; i++) 
            if (diff[i] != diff[i - 1]) 
                break; 
  
        // If some difference values were not same 
        if (i != len) 
           break; 
    } 
  
    int iteration = N - len; 
  
    //  padding terms instance of constant difference 
    // at the end of array 
    for (int i = len; i < len + terms; i++) 
        diff[i] = diff[i - 1]; 
    len += terms; 
  
    //  iterating to get actual sequence back 
    for (int i = 0; i < iteration; i++) 
    { 
        len++; 
  
        //  shifting all difference by one place 
        for (int j = len - 1; j > 0; j--) 
            diff[j] = diff[j - 1]; 
  
        // copying actual first element 
        diff[0] = first[first.size() - i - 1]; 
  
        // converting difference of differences to 
        // difference array 
        for (int j = 1; j < len; j++) 
            diff[j] = diff[j - 1] + diff[j]; 
    } 
  
    //  printing the result 
    for (int i = 0; i < len; i++) 
        cout << diff[i] << " "; 
    cout << endl; 
} 
  
//  Driver code to test above method 
int main() 
{ 
    int sequence[] = {8, 11, 16, 23}; 
    int N = sizeof(sequence) / sizeof(int); 
  
    int terms = 3; 
    nextTermsInSequence(sequence, N, terms); 
  
    return 0; 
} 

Python3

# Python3 code to generate next terms 
# of a given polynomial sequence 
  
# Method to print next terms term of sequence 
def nextTermsInSequence(sequence, N, terms): 
  
    diff = [0] * (N + terms) 
  
    # First copy the sequence itself  
    # into diff array 
    for i in range(N): 
        diff[i] = sequence[i] 
  
    more = False
    first = [] 
    length = N 
  
    # Loop untill one difference remains 
    # or all difference become constant 
    while (length > 1): 
      
        # Keeping the first term of sequence  
        # for later rebuilding 
        first.append(diff[0]) 
        length -= 1
  
        # Converting the difference to difference 
        # of differences 
        for i in range(length): 
            diff[i] = diff[i + 1] - diff[i] 
  
        # Checking if all difference values are 
        # same or not 
        for i in range(1, length): 
            if (diff[i] != diff[i - 1]): 
                break
  
        # If some difference values  
        # were not same 
        if (i != length): 
            break
  
    iteration = N - length 
  
    # Padding terms instance of constant  
    # difference at the end of array 
    for i in range(length, length + terms): 
        diff[i] = diff[i - 1] 
          
    length += terms 
  
    # Iterating to get actual sequence back 
    for i in range(iteration): 
        length += 1
  
        # Shifting all difference by one place 
        for j in range(length - 1, -1, -1): 
            diff[j] = diff[j - 1] 
  
        # Copying actual first element 
        diff[0] = first[len(first) - i - 1] 
  
        # Converting difference of differences to 
        # difference array 
        for j in range(1, length): 
            diff[j] = diff[j - 1] + diff[j] 
  
    # Printing the result 
    for i in range(length): 
        print(diff[i], end = " ") 
          
    print () 
  
# Driver code  
if __name__ == "__main__": 
  
    sequence = [ 8, 11, 16, 23 ] 
    N = len(sequence) 
    terms = 3
      
    nextTermsInSequence(sequence, N, terms) 
  
# This code is contributed by chitranayal 

Output:

8 11 16 23 30 37 44

This article is contributed by Utkarsh Trivedi. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
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My Personal Notes

C++

Python3

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