Lagrange's Interpolation

Станчо Вълканов Павлов ( )      Аднан Шараф (

Let we have n distinct points x1 , x2 , x3 , ... xn on the x-axis.
We are searching for a polynomial of degree n-1 , which passes through n points with coordinates (xi , yi)   i=1..n .
There is only one polynomial with this property.
Let us denote it, the only one of its kind, by f(x). We substitute W(x)=( x-x1 )( x-x2 )( x-x3 )...( x-xn )
Then W( xi ) = 0 for all xi .
We denote by Wi the quotient:     quotient1     W i( xj ) = 0 if i≠j and Wi ( xi )≠ 0.
Normalizing we get quotient2 polynomials of degree n-1.       нормализирани полиноми-w_i
Then:       f(x)=y1w1(x) + y2w2(x) + y3w3(x) + ... + ynwn(x)       is the desired polynomial.
At full its beauty the formula is
Формула на Лагранж-LegrendreFrm