How to Determine the Best Interpolation in Newton Forward Difference Method?

[angel23]

in Newton forward differece method.
how can i know that i reached the best interpolation?

for example in a function like sqrt(x) for Xi=1,1.05,1.10,1.15,1.20,1.25,1.3

the best interpolation is at P3(x) why?how can i know?
this really makes me conused

if anyone helped me i will be grateful

 

[dashing]

in Newton forward differece method.

Asalam o Alikum

Mr ,
Value of f(x) at that point define you where the best interpolation between the point is exsist

 

[xhatemx]

simply you construct the table and you will find for this example that after certain iteration the numbers in a certain column will be the same or of accuracy better than that required by the question. this is when you stop . .

 

[HallsofIvy]

What do you mean by "best"? There exist an infinite number of, say, cubic polynomials that interpolate the points you give. One possible definition of "best" is that \Sigma |f(x_i)- y_i| be a minimum. Another is Max |f(x_i)- y_i| be a minimum and yet another is that \sqrt{\int (f(x_i)- y_i)^2 dx} be a minimum. Each of those has applications.

 

[ssd]

What do you mean by "best"? There exist an infinite number of, say, cubic polynomials that interpolate the points you give. One possible definition of "best" is that \Sigma |f(x_i)- y_i| be a minimum. Another is Max |f(x_i)- y_i| be a minimum and yet another is that \sqrt{\int (f(x_i)- y_i)^2 dx} be a minimum. Each of those has applications.

Very true sir, I was just going to mention the same.

 

[angel23]

:) it is too late, sir i got my answer once i posted the question.(it is too late all)

any way thanks.