2. Find Solution of an equation x^3-x+1 using Newton's Divided Difference Interpolation formula
x1 = 2 and x2 = 4
x = 3.8
Step value (h) = 0.5
Finding f(2)

Solution:
Equation is `f(x)=x^3-x+1`.

The value of table for `x` and `y`

x	2	2.5	3	3.5	4
y	7	14.125	25	40.375	61

Numerical divided differences method to find solution

Newton's divided difference table is

x	y	`1^(st)` order	`2^(nd)` order	`3^(rd)` order	`4^(th)` order
2	7
		14.25
2.5	14.125		7.5
		21.75		1
3	25		9		0
		30.75		1
3.5	40.375		10.5
		41.25
4	61

The value of `x` at you want to find the `f(x) : x = 3.8`

Newton's divided difference interpolation formula is
`f(x)=y_0 +(x-x_0) f[x_0, x_1]+(x-x_0)(x-x_1) f[x_0, x_1, x_2]+(x-x_0)(x-x_1)(x-x_2) f[x_0, x_1, x_2, x_3]+(x-x_0)(x-x_1)(x-x_2)(x-x_3) f[x_0, x_1, x_2, x_3, x_4]`

`y(3.8) = 7 + (3.8 -2) xx 14.25 + (3.8 -2)(3.8 -2.5) xx 7.5 + (3.8 -2)(3.8 -2.5)(3.8 -3) xx 1 + (3.8 -2)(3.8 -2.5)(3.8 -3)(3.8 -3.5) xx 0`

`y(3.8) = 7 + (1.8) xx 14.25 + (1.8)(1.3) xx 7.5 + (1.8)(1.3)(0.8) xx 1 + (1.8)(1.3)(0.8)(0.3) xx 0`

`y(3.8) = 7 +25.65 +17.55 +1.872 +0`

`y(3.8) = 52.072`


Solution of divided difference interpolation method `y(3.8) = 52.072`

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