optimization project

Optimization In this project, there are two stages: Stage 1: You need to write a program that takes a polynomial Pn(x) of a positive integer degree n and a positive coefficient a0 of xn and computes all its roots, including the complex ones. The solution method is by recursion in the following form: a. If the degree n is less than or equal to 4, use the solve_p4c.c program to find the roots. b. Otherwise, if the degree n of the polynomial Pn(x) is even, find a segment (a,b) where Pn(a)* Pn(b) < 0. This is always possible because this polynomial tends to −∞ when x → −∞ and tends to +∞ when x → +∞. You can simply calculate Pn(0), if it is negative, take a=0, look for a value m=2k where Pn(m) is positive and take b=m. If Pn(0) is positive, just do the opposite: set b=0, look for a value m=-2k where Pn(m) is negative and take a=m. Then look for the real root x* of the polynomial (which always exists) by the bisection method or the bisection method with "regula falsi". After finding x*, perform long division of Pn(x) by (x-x*) (i.e., get Qn-1(x) = Pn (x)/(x-x*)) and recursively call the algorithm on Qn-1(x). c. Otherwise, i.e., if the degree n of the polynomial Pn(x) is odd, use the complex Newton-Raphson method to find a root x* that may be complex, x* = a* + bi. If x is real (b in its absolute value is smaller than epsilon), perform long division of Pn(x) by (x-a*) (i.e., get Qn-1(x) = Pn (x)/(x-a*)) and recursively call the algorithm on Qn-1(x). If x* is complex, perform long division of Pn(x) by (x2-2a*x2+(a*)2+(b*)2) (Qn-2(x) = Pn(x)/ (x2-2a*x2+(a*)2+(b*)2)), and recursively call the algorithm on the resulting polynomial Qn-2(x). While discovering each root, it should be stored in a data structure, and if it is complex, its conjugate should also be included. ...... .... .....