Станчо Вълканов Павлов (firstname.lastname@example.org )
Аднан Шараф (email@example.com)
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:
W i( xj ) = 0 if i≠j and Wi ( xi )≠ 0.
Normalizing we get
polynomials of degree n-1.
f(x)=y1w1(x) + y2w2(x) + y3w3(x)
+ ... + ynwn(x)
is the desired polynomial.
At full its beauty the formula is