Given n-1 equations f_1(x), ... f_{n-1}(x), where x \in R^n (yes, x has 1 more element than number of equations) and given x_0 for which f_i(x_0) = 0, there is a continuous curve for x values for which f_i(x) = 0. To move on the curve we use continuation methods based on Newton Iterations that requires not only the functions but also the jacobian of the functions. The function and its jacobian must be provided in a file in C code (or at least as objcet file) and the object file must be linked to the main process. We will move in this curve starting at x_0. This movement can be made in two directions, we define an orthogonal hyperplane of the form b(x-x_0) = s -s_0, where b,x,x_0 \in R^n and s,s_0 \in R. So that the vector b is horthogonal to the curve. With b we are able to move on the curve for which f_i(x) = 0 and we will get different values of x. While we are moving on the curve we can evaluate some f_n(x) and we can do several things with this value: a.- we can return the x that has minimum value for f_n(x) in the covered path. b.- we can test the 0 value for f_n(x) (the sign cahnges in the step) and stop. So, we are near from the x value that is the root for F(x) (F: R^n -> R^n)
The function has as input: (X_0, b_0, s_0, s_1, action) where: .- x_0 makes f_i(x_0) = 0 for i \in (1,n-1) .- the vector b_0 is simply a vector that will help us stablishing the direction of the movement (it can be one component with value 1 and the rest with value 0). It will be used to get the orthogonal b_h that makes b_h(x-x_0) = s - s_0. .- s_0 is the starting s value .- s_1 is the final s value. These two values indicate the amount of movement on the curve. .- action is the value that tells the action to be done at each step: action = 0 if you just want to move from s_0 to s_1. action = 1 if you are looking for the x that makes f_n(x) be the minimun in the covered path. action = 2 if you are looking for the x value that makes f_n(x) =0.
The output is: (x_e, b_e, s_a, s_e, info) where these values depend on the action defined as input: If action was 0 (go from s_0 to s_1) then info can be: .- info = 1. So, all went right and s_a = s_0, s_e = s_1, x_e and b_e are these that make f_i(x_e) = 0 for i \in (1,n-1) and we have been able to move until b_e(x-x_e) = s -s_e .- info = 2. So it has detected that the curve is a closed curve and we have arrived to the starting x_0. In this case we have x_e = x_0, s_a = s_0, s_e = s_p (where s_p is the length of the period) b_e is the vector orthoghonal to the curve at x_0. .- info = -2. So it was not posible to move on the curve, there was a singularity just at x_0. Return values are x_e = x_0, s_a = s_0, s_e = s_0 and b_e = b_h ( where b_h(x-x_0) = s -s_0). .- info = -1. The process has been able to move on the curve but it has stoped because it could not continue (a singularity in the patha s_s). The return values are the point until it has been able to move to: x_e and f_i(x_1) = 0, b_1 is the orthogonal direction at this point, s_a = s_0 and s_e = s_s a value between s_0 and s_1 (the point untill it was able to go to) If action was 1 (go from s_0 to s_1 and give me the point that makes f_n(x) minimun) .- Info = 10, we arrived to s_1, but the minimum was the starting point, but we return the last point: x_e = x_1, b_e = b_1, s_a = s_0, s_e = s_1. .- info = 11, we arrived to s_1 and the minimum is a point in the midle of the path. return values: x_e = x_min for which f_n(x_min) is the minimum. s_a = s_0, s_e = s_min, b_e = b_min .- info = 12, we arrived to s_1 and the minimum is at the last point. Return values are same as info = 10. --- .- info = 20. The curve is closed and the minimun is for the starting point. returns x_e = x_0, s_a = s_0, s_e = period of the curve, b_e = b_0 but orthogonal. .- info = 21. The curve is closed and the minimun has been reached at s_min. return values x_e = x_min, s_a = s_min, s_b = period of the curve, b_e = b_min --- .- info = -20. It was not posible to move on the curve. return values are x_0, b_0, s_a=s_0, s_b = s_0 --- .- info = -10. The process has been able to move on the curve but it has stoped because it could not continue (a singularity in the path at s_s) and the minimun value has been reached at the starting point. Return values correspond to the singularity point: x_e = x_s, b_e = b_s, s_a = s_0 s_e = s_s. .- info = -11. The process has been able to move on the curve but it has stoped because it could not continue (a singularity in the path at s_s) and the minimun value has been reached at some point between s_0 and the singularity point s_s. Returns the min value: x_e = x_min, b_e = b_min, s_a = s_min, s_e = s_s. .- info = -12. The process has been able to move on the curve but it has stoped because it could not continue (a singularity in the path at s_s) and the minimun value has been reached at the stoping point s_s. Returns the singularity point: x_e = x_s, b_e = b_s, s_a = s_0 s_e = s_s. if action was 2 ( go from s_0 to s_1 but stop when f_n(x) = 0) In this case info values are same as values for action 1. and return values are, when f_n(x) = 0 is reached, the point tha makes F(x) = 0.