Derive the newton's forword difference
formula wsing the operator
erelations.
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NEWTON'S FORWARD DIFFERENCE FORMULA
Making use of forward difference operator and forward difference table ( will be defined a little later) this scheme simplifies the calculations involved in the polynomial approximation of fuctons which are known at equally spaced data points.
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).
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