Partial derivative of a polynomial using Doubly Linked List
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++
// 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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