Divided Differences
Divided difference : The divided difference formula is used to determine the value of constant a0 of nth Lagrange polynomial.
Suppose that Pn(x) is the nth Lagrange polynomial that agrees with the function f at the distinct numbers x0, x1, . . . , xn. Although this polynomial is unique, there are alternate algebraic representations that are useful in certain situations. The divided differences of f with respect to x0, x1, . . . , xn are used
to express Pn(x) in the form
Pn(x) = a0 a1(x−x0) a2(x−x0)(x−x1) · · · an(x−x0) · · · (x−xn-1)
for appropriate constants a0, a1, . . . , an.
To determine the first of these constants, a0, note that if Pn(x) is written in the form of the above equation, then evaluating Pn(x) at x0 leaves only the constant term a0 that is,
a0 = Pn(x0) = f (x0)
Expression : Let the data, (xi, f(xi)), i = 0, 1, 2,…, n, be given. We define the divided differences as follows. First divided difference Consider any two consecutive data values (xi, f(xi)), (xi 1, f(xi 1)).
Then,we define the first divided difference as
....................1.1
Therefore,
Note that
We say that the divided differences are symmetrical about their arguments. Second divided difference Consider any three consecutive data values (xi, f(xi)), (xi 1, f(xi 1)), (xi 2, f(xi 2)). Then, we define the second divided difference as
....................1.2
Therefore,
We can express the divided differences in terms of the ordinates. We have
Notice that the denominators are same as the denominators of the Lagrange fundamental polynomials. In general, we have the second divided difference as
