Software Project in Optimization

The project consists of two phases. In the first phase, you are required to write a program that can calculate all the roots, including the complex ones, of a polynomial Pn(x) of positive integer degree n and the coefficient a0 of xn is positive. The solution method is recursive, which means it will break down the problem into smaller sub-problems until it can solve them easily. The recursive solution method has the following steps: Step 1: If the degree of the polynomial Pn(x) is less than or equal to 4, use the solve_p4c.c program to find the roots. Step 2: If the degree of the polynomial Pn(x) is odd, find a segment (a,b) where Pn(a)Pn(b) < 0. You can calculate Pn(0) to determine which value to use for a and b. Then use the bisection method or the bisection method with regula falsi to find a real root x. After finding x*, do a long division of Pn(x) by (x-x*) and recursively call the algorithm on Qn-1(x). Step 3: If the degree of the polynomial Pn(x) is even, use Newton's complex method to find a root x* which may be complex x*=a*+bi. If x turns out to be real (b in absolute value is less than epsilon), do a long division of Pn(x) by (x-a*) and recursively call the algorithm on Qn-1(x). If x* turns out to be complex, do a long division of Pn(x) by (x2-2ax2+(a)2+(b*)2) and recursively call the algorithm on Qn-2(x). While discovering each root, it must be accumulated in a data structure and if it is complex, its conjugate must also be added. In the second phase of the project, you are required to write a program that can find the global minimum of an even degree polynomial. The program must follow these steps: Step 1: Calculate the derived polynomial. Step 2: Find the roots of the derived polynomial using the program from phase one. Step 3: Extract the real roots from the derived polynomial. Step 4: Evaluate the polynomial at each real root to find the minimum value. Step 5: The minimum value of the polynomial at these points is the global minimum. Overall, this project requires you to have a good understanding of polynomials, roots, and recursion. The project also requires you to implement various numerical methods, such as bisection method, regula falsi, and Newton's method.