my book starts to explain divided difference by saying that :
Suppose that is the nth Lagrange polynomial that agrees with the function at distinct numbers . The divided differences of with respect to are used to express in the form
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?
I do understand why the interpolating polynomial between two points x_0 and x_1 can be written as,
The literature often states that a convenient form of the 2-order polynomial is,
How did man figure out that this is a convenient form? Trial and error? Thinking really hard?