Algorithm OPTIMAL-PARAMETRIZATION. Given F(x, y, z) ∈ Q[x, y, z], an irreducible homogeneous polynomial defining a rational plane curve C, the algorithm computes an optimal rational parametrization of C

4. The following algorithm for approximating roots combines the bisection method with Newton's method and ensures the convergence even in the case where Newton's method fail to converge. Choose an interval [a, b] such that f(a)f(b) ≤ 0. Compute initial approximation xo = (a+b)/2 of the root using the bisection method. For i=1,2,... Compute x₁ using Newton's method and X₁-1. If x₁ [a, b], set x₁ = (a + b)/2 (from bisection). Check for convergence. If f(a) f(xi) ≤0 set b = xi, else set a = xį. (a) Implement this algorithm in a PYTHON function with the following specifications: def findzero (a, b, tol, maxit, f, df) # Input: # a, b = The endpoints of the interval # tol = The required tolerance # maxit = Maximum number of iterations # f, df = The function and its derivative # Output: # star #niter # ierr # # # approximation of root = number of iterations for convergence 0, the method converged 1, df was zero or undefined 2, maximum number of iterations has been reached return xstar, niter,…

W1 a W2 w6 w7 6. -1 4 W3 W5 e W4 Figure 1: Weighted, undirected graph G. Weights and Source Node All problems use weights wi through w7. To determine these weights, look at your 7 digit ID. The weight wi corresponds to the first digit of your ID, the weight w2 corresponds to the second digit of your ID, and so on. For problems 2 and 4, we use a source node or starting node s, determined as follows: find the digit in your ID that has the highest value. If it is the first digit then s = a, if it is the second digit, then s = b, and so on. If there is more than one digit with the same highest value, choose the first occurrence of that digit. For example, if your ID is 4526767, then the highest value is 7, and you choose the first 7, i.e., digit 5, and set s = е.

Algorithm OPTIMAL-PARAMETRIZATION.Given F(x, y, z) ∈ Q[x, y, z], an irreducible homogeneous polynomial defining a rational plane curve C, the algorithm computes an optimal rational parametrization of C