# Lagrange Interpolating Polynomial!

• February 16th 2006, 01:43 AM
BanderHM
Lagrange Interpolating Polynomial!
Can any one explain this formula "Lagrange Interpolating Polynomial" to me in plain English.
Is there any program around to calc this formaula?

Thanks.
• February 16th 2006, 04:20 AM
CaptainBlack
Quote:

Originally Posted by BanderHM
Can any one explain this formula "Lagrange Interpolating Polynomial" to me in plain English.
Is there any program around to calc this formaula?

Thanks.

The Lagrangeian interpolating polynomial for a set of data ${(x_i,y_i),\ i=1,\ ..\ n}$
is a polynomial of degree $n$ (the number of data points) which passes exactly
through each of the data points. It can be written explicitly in terms of the
data as shown below:

Consider the polynomials:

$
P_i(x)=\prod_{j=1,\ j \ne i}^n \frac{x-x_j}{x_i-x_j}
$
,

This equals $1$ at $x=x_i$, and equals $0$ when $x=x_j\ ,j \ne i$.

Hence:

$
P(x)=\sum_{i=1}^n\ y_i\ P_i(x)
$

is a polynomial such that $P(x_i)=y_i, \ \mbox{i=1,\ ..\ n}$.

This is the Lagrange interpolating polynomial for the data $(x_i,y_i),\ i=1,\ ..\ n$.

There is one more thing to say about Lagrangian interpolation, and that is
it is of more importance to theory than practice. If you actualy want to
interpolate on real data there are almost always better ways of doing so
than Lagrangian interpolation.

RonL
• February 16th 2006, 01:54 PM
BanderHM
How?
Quote:

Originally Posted by CaptainBlack
If you actualy want to
interpolate on real data there are almost always better ways of doing so
than Lagrangian interpolation.
RonL

How? What are they?

Thanks.
• February 16th 2006, 02:17 PM
CaptainBlack
Quote:

Originally Posted by BanderHM
How? What are they?

Thanks.

Splines, othogonal polynomials, rational functions (Pade approximations)...

the list is almost endless

RonL