# Divided differences

• Aug 30th 2010, 03:40 AM
Mollier
Divided differences
Hi,

my book starts to explain divided difference by saying that :

Suppose that $P_n(x)$ is the nth Lagrange polynomial that agrees with the function $f$at distinct numbers $x_0,x_1,\cdots,x_n$. The divided differences of $f$ with respect to $x_0,x_1,\cdots,x_n$ are used to express $P_n(x)$ in the form

$P_n(x) = a_0+a_1(x-x_0)+a_2(x-x_0)(x-x_1)+\cdots+a_n(x-x_0)(x-x_1)\cdots(x-x_{n-1})$

Since I'm pretty slow, I do not see the logic behind this choice. I'm that I'm lacking some basic theory such that this appears like a mystical choice. Could someone point me in the right direction please?

Note:

I do understand why the interpolating polynomial between two points x_0 and x_1 can be written as,

$P_1(x) = f(x_0) + \frac{f(x_1)-f(x_0)}{x_1-x_0}(x-x_0),$

The literature often states that a convenient form of the 2-order polynomial is,

$P_2(x) =b_0 + b_1(x-x_0)+b_2(x-x_0)(x-x_1).$

How did man figure out that this is a convenient form? Trial and error? Thinking really hard?

Thanks.
• Aug 30th 2010, 12:14 PM
Mollier
Never mind guys, I found a great explanation at: 5.3 Newton Polynomials | Department of Electrical and Computer Engineering | University of Waterloo.
I have been spoon-fed and am quite happy :)