Exercise: Gauss-Newton Method
The unknown, i.e., the x vector, is:
X1 x = X2
The right-hand side is just the error:
[0.5 b = e = y - (exp(xia) + xz) = 0.5 √5j
Now evaluate the gradient of e:
Vei = [∂e/∂a1 exp(x1a1) ∂e/∂a2]T
Which gives A as:
∂e/∂a1 = exp(x1a1) - 1.5255
∂e/∂a2 = exp(x1a2) - 3.9841
∂e/∂a3 = exp(x1a3) - 145.8125
We now have all the pieces of information to calculate the update. Apply the formula:
xk+1 = tk (AT A)-1(ATb) = [42] - [0 0.5]
Where the value of u should be determined by a selected line search algorithm. (But in this case, the solution is x = [1.2, 4.5]T).