Formula

Bessel's formula

`p = (x - x_0)/h` `y_p=(y_0+y_1)/2+(p-1/2)*Delta y_0 + (p(p-1))/(2!) * (Delta^2y_(-1)+Delta^2y_(0))/2 + ((p-1/2)p(p-1))/(3!) * Delta^3y_(-1) + ((p+1)p(p-1)(p-2))/(4!) * (Delta^4y_(-2)+Delta^4y_(-1))/2 + ((p-1/2)(p+1)p(p-1)(p-2))/(5!) * Delta^5y_(-2) + ...`

---

Examples
1. Find Solution using Bessel's formula

x	f(x)
20	2854
24	3162
28	3544
32	3992

x = 25

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

x	20	24	28	32
y	2854	3162	3544	3992

Bessel's method to find solution

`h=24-20=4`

Taking `x_0=24` then `p=(x-x_0)/h=(x-24)/4`

The difference table is

`x`	`p=(x-24)/4`	`y`	`Deltay`	`Delta^2y`	`Delta^3y`
20	-1	2854
			308
24	0	3162		74
			382		-8
28	1	3544		66
			448
32	2	3992

`x = 25`

`p = (x - x_0)/h = (25 - 24)/4 = 0.25`

`y_0=3162, Delta y_0=382,Delta^2y_(-1)=74,Delta^3y_(-1)=-8`

Bessel's formula is
`y_p=(y_0+y_1)/2+(p-1/2)*Delta y_0 + (p(p-1))/(2!) * (Delta^2y_(-1)+Delta^2y_(0))/2 + ((p-1/2)p(p-1))/(3!) * Delta^3y_(-1)`

`y_(0.25) = (3162+3544)/2 + (0.25-1/2)*(382) + (0.25(0.25-1))/(2)*((74+66))/2 + ((0.25-1/2)0.25(0.25-1))/(6)*(-8)`

`y_(0.25)=3353-95.5 -6.5625 -0.0625`

`y_(0.25)=3250.875`


Solution of Bessel's interpolation is `y(25) = 3250.875`

---

---