## Lagrange's Interpolation

Станчо Вълканов Павлов (stancho_pavlov@yahoo.com )
Аднан Шараф (engadnansharaf@yahoo.com)
stancho_pavlov@yahoo.com

Let we have n distinct points
*x*_{1} , x_{2} , x_{3} , ... x_{n}
on the x-axis.

We are searching for a polynomial of degree n-1 , which passes through n points with coordinates
*(x*_{i} , y_{i}) * 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-x*_{1} )( x-x_{2} )( x-x_{3} )...( x-x_{n} )

Then *W( x*_{i} ) = 0 for all *x*_{i} .

We denote by *W*_{i} the quotient:
* W*_{ i}( x_{j} ) = 0 if * i≠j * and *W*_{i }( x_{i} )≠ 0.

Normalizing we get
polynomials of degree n-1.

Then:
*f(x)=y*_{1}w_{1}(x) + y_{2}w_{2}(x) + y_{3}w_{3}(x)
+ ... + y_{n}w_{n}(x)
is the desired polynomial.

At full its beauty the formula is