Formula

Gauss Forward formula

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

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Examples
1. Find Solution using Gauss Forward formula

x	f(x)
1	1
2	-1
3	1
4	-1
5	1

x = 3.5

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

x	1	2	3	4	5
y	1	-1	1	-1	1

Gauss's forward method to find solution

`h=2-1=1`

Taking `x_0=3` then `p=(x-x_0)/h=(x-3)/1`

Now the central difference table is

`x`	`p=(x-3)/1`	`y`	`Deltay`	`Delta^2y`	`Delta^3y`	`Delta^4y`
1	-2	1
			-2
2	-1	-1		4
			2		-8
3	0	1		-4		16
			-2		8
4	1	-1		4
			2
5	2	1

`x = 3.5`

`p = (x - x_0)/h = (3.5 - 3)/1 = 0.5`

`y_0=1, Delta y_0=-2,Delta^2y_(-1)=-4,Delta^3y_(-1)=8,Delta^4y_(-2)=16`

Gauss's forward interpolation formula is
`y_p=y_0+p Delta y_0 + (p(p - 1))/(2!) * Delta^2y_(-1) + ((p + 1)p(p - 1))/(3!) * Delta^3y_(-1) + ((p + 1)p(p - 1)(p - 2))/(4!) * Delta^4y_(-2)`

`y_(0.5) = 1 + (0.5)(-2) + ((0.5)(0.5 - 1))/(2) * (-4) + ((0.5 + 1)(0.5)(0.5 - 1))/(6) * (8) + ((0.5 + 1)(0.5)(0.5 - 1)(0.5 - 2))/(24) * (16)`

`y_(0.5)=1 -1 +0.5 -0.5 +0.375`

`y_(0.5)=0.375`


Solution of Gauss's forward interpolation is `y(3.5) = 0.375`

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