Numerical Analysis Algorithms

Description

Current pull request adds 2 new algorithms.

Least Square Approximation

This algorithm is a response to the problem risen by analysis of some data, projected in grafing the data to points (in 2 dimension)
and figuring out a pattern that the data follow or fiding a median for those data.

Least Square Approximation creates a polynomial of any degree with the given set of points, starting from the following idea:
    --> Fiding something in common for those points means, mathematically, to evaluate a distance; since we want to observe, relative to a polynomial, the behaviour of the data, we define a function ( called E ) that describes this idea:

                                     E(a₀, a₁, ... , a_D) = $\sum_{i=0..n}$ (y_i - P(x_i))², where:

      P(..) -> polynomial we want to find; D -> degree of polynomial
      n -> number of points (data)
      y_i -> ordinate of some data
      x_i -> abscissa of some data
      a₀, a₁, ... , a_D -> coefficients of terms of the polynomial

Now, for the idea to really produce a result, the function E shall be as small as possible, fact supported by smaller the sum, closer the values added, meaning y_i and P(x_i) are very close to each other. This now becomes a Calculus 2 problem, more specifically minimising the function E. This is possible by taking the partial derivative of the function "E" with respect to each coefficient a_i and evaluate it to zero, as follows:

                                      $\frac{∂}{∂a_k}$ E(a₀, a₁, ..., a_D) = 0 , for k = 0 .. D

Choosing a degree for P and replacing it's form into the formula for E above, we will have a system of equations, generated by the derivatives above, looking something like:

                                      $\sum_{i=0..D}$ (a_i * $\sum_{j=1..n}$ x_j ^j+k) = $\sum_{i=0..n}$ y_i* x_i ^k , for k = 0 .. D

Decomposing the bigger sum from the left side and exposing all the D + 1 equations, we can observe that the left side can be rewritten as follows:
$\to$ A matrix A which holds, as elements, $\sum_{j=1..n}$ x_j ^j+k ( For each line, the elements will be the sums for the respective derivative (k representing it's index) )
$\to$ A column vector x which holds the coefficients $a_0, a_1, ..., a_D$, in this order

Also, the right side can be expressed as a column vector b, which holds the sums $\sum_{i=0..n}$ y_i* x_i ^k

Finally, the whole algorithm is based on solving the equation: A * x = b for the column vector x; in this implementation, the method used is QR decomposition and ultimately solving it.

Logarithm evaluation

This function calculates the log_basex, for any base > 0, x > 0

For the case base = e and x between 0 and 1, the function uses the MacLaurin Series for ln: ln(x) = $\sum_{n>=1} {(-1)^n * \frac{x^n}{n}}$