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Partial derivative of a polynomial using Doubly Linked List
  • Difficulty Level : Expert
  • Last Updated : 23 Nov, 2020

Given a 2-variable polynomial represented by a doubly linked list, the task is to find the partial derivative of a polynomial stored in the doubly-linked list.

Examples:

Input: P(x, y) = 2(x^3 y^4) + 3(x^5 y^7) + 1(x^2 y^6)
Output:
Partial derivatives w.r.t. x: 6(x^2 y^4) + 15(x^4 y^7) + 2(x^1 y^6)
Partial derivatives w.r.t. y: 24(x^2 y^3) + 105(x^4 y^6) + 12(x^1 y^5)
Partial derivatives w.r.t. x and y: 144(x^1 y^2) + 2520(x^3 y^5) + 60(x^0 y^4)

Input: P(x, y) = 3(x^2 y^1) + 4(x^2 y^3) + 2(x^4 y^7)
Output:
Partial derivatives w.r.t. x: 6(x^1 y^1) + 8(x^1 y^3) + 8(x^3 y^7)
Partial derivatives w.r.t. y: 6(x^1 y^0) + 24(x^1 y^2) + 56(x^3 y^6)
Partial derivatives w.r.t. x and y: 48(x^0 y^1) + 1008(x^2 y^5)

Approach: Follow the steps belo to solve this problem:



  • Declare a class or structure to store the contents of a node, i.e. data representing the coefficient, power1 representing the power to which x is raised, power2 representing the power to which y is raised, and the pointers to its next and previous node.
  • Declare functions to calculate derivatives with respect to x, derivative with respect to y, and derivative with respect to x and y.
  • Calculate and print the derivaties obtained.

Below is the implementation of the above approach:

C++

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// C++ program for the above approach
  
#include <bits/stdc++.h>
using namespace std;
  
// Structure of a node
struct node {
    node* link1 = NULL;
    node* link2 = NULL;
    int data = 0;
    int pow1 = 0;
    int pow2 = 0;
};
  
// Function to generate Doubly Linked
// List from given parametrs
void input_equation(node*& head, int d,
                    int p1, int p2)
{
    node* temp = head;
  
    // If list is empty
    if (head == NULL) {
  
        // Create new node
        node* ptr = new node();
        ptr->data = d;
        ptr->pow1 = p1;
        ptr->pow2 = p2;
  
        // Set it as the head
        // of the linked list
        head = ptr;
    }
  
    // If list is not empty
    else {
  
        // Temporarily store
        // address of the head node
        temp = head;
  
        // Traverse the linked list
        while (temp->link2 != NULL) {
  
            // Link to next node
            temp = temp->link2;
        }
  
        // Create new node
        node* ptr = new node();
        ptr->data = d;
        ptr->pow1 = p1;
        ptr->pow2 = p2;
  
        // Connect the nodes
        ptr->link1 = temp;
        temp->link2 = ptr;
    }
}
  
// Function to calculate partial
// derivative w.r.t. X
void derivation_with_x(node*& head)
{
    cout << "Partial derivatives"
         << " w.r.t. x: ";
  
    node* temp = head;
  
    // Traverse the Linked List
    while (temp != NULL) {
  
        if (temp->pow1 != 0) {
            temp->data = (temp->data)
                         * (temp->pow1);
            temp->pow1 = temp->pow1 - 1;
        }
        else {
            temp->data = 0;
            temp->pow1 = 0;
            temp->pow2 = 0;
        }
  
        temp = temp->link2;
    }
  
    temp = head;
  
    cout << " " << temp->data
         << "(x^" << temp->pow1
         << " y^" << temp->pow2
         << ")";
    temp = temp->link2;
  
    while (temp != NULL) {
        cout << " + "
             << temp->data << "(x^"
             << temp->pow1 << " y^"
             << temp->pow2 << ")";
        temp = temp->link2;
    }
  
    cout << "\n";
}
  
// Function to calculate partial
// derivative w.r.t. Y
void derivation_with_y(node*& head)
{
    cout << "Partial derivatives"
         << " w.r.t. y: ";
  
    node* temp = head;
  
    // Traverse the Linked List
    while (temp != NULL) {
  
        if (temp->pow2 != 0) {
            temp->data = (temp->data)
                         * (temp->pow2);
            temp->pow2 = temp->pow2 - 1;
        }
        else {
            temp->data = 0;
            temp->pow1 = 0;
            temp->pow2 = 0;
        }
  
        temp = temp->link2;
    }
  
    temp = head;
    cout << " "
         << temp->data
         << "(x^" << temp->pow1
         << " y^"
         << temp->pow2 << ")";
    temp = temp->link2;
  
    while (temp != NULL) {
        cout << " + "
             << temp->data << "(x^"
             << temp->pow1 << " y^"
             << temp->pow2 << ")";
        temp = temp->link2;
    }
    cout << "\n";
}
  
// Function to calculate partial
// derivative w.r.t. XY
void derivation_with_x_y(node*& head)
{
    cout << "Partial derivatives"
         << " w.r.t. x and y: ";
  
    node* temp = head;
  
    // Derivative with respect to
    // the first variable
    while (temp != NULL) {
        if (temp->pow1 != 0) {
  
            temp->data = (temp->data)
                         * (temp->pow1);
            temp->pow1 = temp->pow1 - 1;
        }
  
        else {
            temp->data = 0;
            temp->pow1 = 0;
            temp->pow2 = 0;
        }
  
        temp = temp->link2;
    }
    temp = head;
  
    // Derivative with respect to
    // the second variable
    while (temp != NULL) {
  
        if (temp->pow2 != 0) {
            temp->data = (temp->data)
                         * (temp->pow2);
            temp->pow2 = temp->pow2 - 1;
        }
  
        else {
            temp->data = 0;
            temp->pow1 = 0;
            temp->pow2 = 0;
        }
  
        temp = temp->link2;
    }
  
    temp = head;
    cout << " "
         << temp->data << "(x^"
         << temp->pow1 << " y^"
         << temp->pow2 << ")";
  
    temp = temp->link2;
  
    // Print the list after the
    // calculating the derivative
    while (temp != NULL) {
  
        cout << " + "
             << temp->data << "(x^"
             << temp->pow1 << " y^"
             << temp->pow2 << ")";
        temp = temp->link2;
    }
    cout << "\n";
}
  
// Driver Code
int main()
{
    node* head1 = NULL;
  
    // Creating doubly-linked list
    input_equation(head1, 2, 3, 4);
    input_equation(head1, 3, 5, 7);
    input_equation(head1, 1, 2, 6);
  
    // Function Call
    derivation_with_x(head1);
    derivation_with_y(head1);
    derivation_with_x_y(head1);
  
    return 0;
}

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Output:

Partial derivatives w.r.t. x:  6(x^2 y^4) + 15(x^4 y^7) + 2(x^1 y^6)
Partial derivatives w.r.t. y:  24(x^2 y^3) + 105(x^4 y^6) + 12(x^1 y^5)
Partial derivatives w.r.t. x and y:  144(x^1 y^2) + 2520(x^3 y^5) + 60(x^0 y^4)

Time Complexity: O(N)
Auxiliary Space: O(1)

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