Question

State newton's forward interpolation to find f'(X),f"(X),f'"(X) at x=xo

Answer 1

Consider the equation of the linear interpolation optained in the earlier section :  

f1 - f0  

f0x1 - f1x0

f(x) @ P1(x) = ax-1b =  

x +  

x1 - x0  

x1 - x0

 

=  

      1  

[(x1 - x)f0 + (x-x0)f1]

(x1 - x0)

 

x1 - x  

x - x0

x - x0  

f0 +  

(f1- f0)   +  

f0

x1 - x0  

x1 - x0

x1 - x0  

x - x0

=  f0 +  

(f1- f0)  

x1 - x0

= f0 + r Df0    [ r = (x - x0) / (x1 - x0)  Df0 = f1 - f0 ]

since x1 - x0 is the step lenght h, r can be written as (x - x0)/h and will be between (0, 1).