I want to maximize the following equation using Newton Rapson Method in MATLAB:
z= -(x1 - sqrt(5))2 - (x2-pi)2-10
Initial Approximation:
X0=[6.597, 5.891]T
I take it that you wish to maximize z, which then is the same to minimize the function
f(x) = [x_1 - sqrt(5)]^2 + [x_2 - pi]^2.
Newton-Raphson will in this case solve the problem immediately, since this is a straitly convex quadratic function. Starting from the point you provided (or ANY other starting point, for that matter), the next point will be defined by the point which sets the gradient of the second-order approximation to the zero vector. Since the function f is quadratic, the approximation is exact, and moreover since the function f is separable over the variables, the first iteration will move from x_0 to the optimal solution (sqrt(5), pi) directly.
This analysis is great, but I worry that we may be missing something in the original notation? Note that the starting solution is quite far away from the exact solution ... and there is a "-10" (typo)?
Anyway, if the problem is as Michael phrases it, we can get a pair of equations for dz/dx_1 = 0 , and dz/dx_ 2 = 0, and solve immediately.
2 (x_1 - sqrt(5)) = 0, implies x_1 = sqrt(5), and 2(x_2 - pi) = 0, implies x_2 = pi
Are we missing something?
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