Implement Newton’s method for systems. Use it to find solutions to the following systems of algebraic equations:

(a) Use initial guesses p0 = [0.1, 1.2, 2.5] and p0 = [1, 0, 1] , and write down the solutions: x+y+z= 3 x^2+y^2+z^2= 5 e^x+xy−xz= 1

(b) Long range navigation (LORAN) was a navigation system used during World War II. At its heart, it involved finding the intersection of two hyperbolic curves like the following: ( x^2/186^2 )− (y^2 / (300^2−186^2)) = 1 (y−500)^2/ 279^2 − (x-300)^2/(500^2−279^2) = 1 There are four intersections between these two curves. Find at least two of them by trying different initial guesses (report both your initial guesses and the solutions).

(c) The amount of pressure p required to sink a disk of radius r into soft soil can be modeled with the equation: p = k1 e^(k2*r)+ k3*r where k1, k2 and k3are constants depending on things like soil depth and composition. For the soil we are working with, we run some tests. We need p = 10 in order to sink a disk with r= 1. We need p = 12 for r = 2, and we need p = 15 for r = 3. Use Newton’s method to find k1, k2,and k3.