Newton's Interpolation, numerical methods question

[albema]

Can anyone tell me what is the equation for First Order of Newton’s Interpolation?

The one of Interpolation I know is Lagrange’s Interpolation such as

p(x)= \frac {x-x1}{x0-x1} f(x0) + \frac {x-x0}{x1-x0} f(x1)

and so on ...

Thanks in advance

 

[wunderboy]

well...

well, i don't have math type on this computer, so you will have to bear with me, but:

the nth term is found by dividing the function evaluated at the n-1th term by the derivative of the function evaluated at the n-1th term and then subtracting that from the nth term.

that's it!

 

[HallsofIvy]

How can you find the nth term by subtracting something from the nth term?

The only Newton Interpolation formula I know is "Newton's divided difference" formula- which does not involve finding derivatives. It's a bit complicated to write out but here is a good explanation from Wolfram:
http://mathworld.wolfram.com/NewtonsForwardDifferenceFormula.html.

As far as "first order" is concerned, it is just f(x)= f(x₀)+ (f(x₁)- f(x₀))(x- x₀)/(x₁-x₀). Which obviously takes on the correct values at x= 0 and x= 1 and linearly interpolates between. If you look closely you will see that it gives exactly the same values as Lagrange's formula: all linear interpolation schemes between two points are the same.

 

[wunderboy]

well, i don't have math type on this computer, so you will have to bear with me, but:

the nth term is found by dividing the function evaluated at the n-1th term by the derivative of the function evaluated at the n-1th term and then subtracting that from the nth term.

that's it!

good point... should be subtracted from the n-1th term. bad typing on my part! thanks for noticing;-)