Optimisation functions in R

a) 2D.newton.rapson

Employs finite differences to calculate the divergence and Hessian matrix of a two dimentional function. It employs the Newton-Rapson method to find a maximal point. This functions takes as input:

1. float Function. Function to minimise
2. float x0. Starting point (x coordinate)
3. float y0. Starting point (y coordinate)
4. float hx. Finite difference (to calculate derivatives along x coordinate). Smaller values provide more precise derivatives
5. float ** hy**. Finite difference (to calculate derivatives along y coordinate). Smaller values provide more precise derivatives
6. int Niter. Number of iterations for the Newton-Rapson algorithm
7. float zero_level. The iterations stop when the gradient is below this threshold.

It returns a list with two elements c(xmin, ymin).

b) gradient.descent

Employs finite differences to calculate the divergence of a two dimentional function. Finds a minima trough gradient-descent. This functions takes as input:

1. float Function. Function to minimise
2. N*float initial_param_vector. Vector of starting points (dimention N)
3. float h. Finite difference. Smaller values provide more precise derivatives
4. int Niter. Number of iterations for the Newton-Rapson algorithm
5. float learn_rate. Step size at each iteration
6. float zero_level. The iterations stop when the gradient is below this threshold.

It returns the minimun as a vector with N elements.