
Divided differences
Hi,
my book starts to explain divided difference by saying that :
Suppose that $\displaystyle P_n(x)$ is the nth Lagrange polynomial that agrees with the function $\displaystyle f $at distinct numbers $\displaystyle x_0,x_1,\cdots,x_n$. The divided differences of $\displaystyle f$ with respect to $\displaystyle x_0,x_1,\cdots,x_n$ are used to express $\displaystyle P_n(x)$ in the form
$\displaystyle P_n(x) = a_0+a_1(xx_0)+a_2(xx_0)(xx_1)+\cdots+a_n(xx_0)(xx_1)\cdots(xx_{n1})$
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,
$\displaystyle P_1(x) = f(x_0) + \frac{f(x_1)f(x_0)}{x_1x_0}(xx_0),$
The literature often states that a convenient form of the 2order polynomial is,
$\displaystyle P_2(x) =b_0 + b_1(xx_0)+b_2(xx_0)(xx_1).$
How did man figure out that this is a convenient form? Trial and error? Thinking really hard?
Thanks.
