#Comparison
#As mentioned in Holmes, cubic B-splines reduce the problem down to having to solve for (approximately) n unknowns.
#The advantage of B-splines is that they are easier to code.
#This procedure is faster than the algorithm presented in the pre-class work, and requires
#minimal storage, which means finding the coefficients for a cubic spline is
#fairly easy even for a large number of interpolation points.
#They are also very useful for least squares fitting of data,
#as well when solving differential equations numerically [Holmes, 20016].
#The only problem that can present is that if the matrix A is not invertible, the algorithm will not converge.
#The algorithm presented in the pre class work is easy to understand since is based on the primary conditions
#and assumptions of the theory. However, this hard-coded algorithm is just for input vectors of size 3.
#Time and space complexity will depend on the functions to solve.

function k= B_cubic(x0, y0)

    #We will use the same assumptions presented in the Assignment, also considering that h=1.
    #We will conside the form: Ck = z

  C = []; # coefficients of k
  z = []; #

  n  = size(x0)(1);

  for i = 1:(n-2)
    disp(i);

    #d: equation derived equation for the unknown ks in terms of the known ys.
    Y = [Y; 6.* (y0(i+2)-2.* y0(i+1) - y0(i))];

    #appending the rows
    c = zeros(1, n);
    c(i) = 1;
    c(i+1) = 4;
    c(i+2) = 1;

    C = [C; c];

  endfor

  #as mentioned before, we are considering the natural spline conditions k1=kn=0
  Y = [0; Y; 0];

  c = zeros(1,n);
  c(1) = 1;

  C = [c; C];
  c = zeros(1,n);
  c(n) = 1;

  C = [C; c];

  #Obtaining the values of ks
  k = C^(-1)*z;

  return
endfunction