Editing Polynomial interpolation (section)

==== Theorem ====
For a polynomial <math>p_n</math> of degree less than or equal to <math>n</math>, that interpolates <math>f</math> at the nodes <math>x_i</math> where <math>i = 0,1,2,3,\cdots,n</math>. Let <math>p_{n+1}</math> be the polynomial of degree less than or equal to <math>n+1</math> that interpolates <math>f</math> at the nodes <math>x_i</math> where <math>i = 0,1,2,3,\cdots,n, n+1</math>. Then <math>p_{n+1}</math> is given by:<math display="block">p_{n+1}(x) = p_n(x) +a_{n+1}w_n(x) </math>where <math display="inline">w_n(x) := \prod_{i=0}^n (x-x_i)  </math> also known as Newton basis and <math display="inline">a_{n+1} :={f(x_{n+1})-p_n(x_{n+1}) \over w_n(x_{n+1})}    </math>.

'''Proof:'''

This can be shown for the case where <math>i = 0,1,2,3,\cdots,n</math>:<math display="block">p_{n+1}(x_i) = p_n(x_i) +a_{n+1}\prod_{j=0}^n (x_i-x_j) = p_n(x_i) </math>and when <math>i = n+1</math>:<math display="block">p_{n+1}(x_{n+1}) = p_n(x_{n+1}) +{f(x_{n+1})-p_n(x_{n+1}) \over w_n(x_{n+1})}   w_n(x_{n+1}) = f(x_{n+1})  </math>By the uniqueness of interpolated polynomials of degree less than <math>n+1</math>, <math display="inline">p_{n+1}(x) = p_n(x) +a_{n+1}w_n(x) </math> is the required polynomial interpolation. The function can thus be expressed as:

<math display="inline">p_{n}(x) = a_0+a_1(x-x_0)+a_2(x-x_0)(x-x_1)+\cdots + a_n(x-x_0)\cdots(x-x_{n-1}) .</math>