Formula

Gauss Backward formula

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

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Examples
1. Find Solution using Gauss Backward formula

x	f(x)
1940	17
1950	20
1960	27
1970	32
1980	36
1990	38

x = 1976

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

x	1940	1950	1960	1970	1980	1990
y	17	20	27	32	36	38

Gauss's backward difference interpolation method to find solution

`h=1950-1940=10`

Taking `x_0=1970` then `p=(x-x_0)/h=(x-1970)/10`

Now the central difference table is

`x`	`p=(x-1970)/10`	`y`	`Deltay`	`Delta^2y`	`Delta^3y`	`Delta^4y`	`Delta^5y`
1940	-3	17
			3
1950	-2	20		4
			7		-6
1960	-1	27		-2		7
			5		1		-9
1970	0	32		-1		-2
			4		-1
1980	1	36		-2
			2
1990	2	38

`x = 1976`

`p = (x - x_0)/h = (1976 - 1970)/10 = 0.6`

`y_0=32, Delta y_(-1)=5,Delta^2y_(-1)=-1,Delta^3y_(-2)=1,Delta^4y_(-2)=-2,Delta^5y_(-3)=-9`

Gauss's backward interpolation formula is
`y_p=y_0+p Delta y_(-1) + ((p + 1)p)/(2!) * Delta^2y_(-1) + ((p + 1)p(p - 1))/(3!) * Delta^3y_(-2) + ((p + 2)(p + 1)p(p - 1))/(4!) * Delta^4y_(-2) + ((p + 2)(p + 1)p(p - 1)(p - 2))/(5!) * Delta^5y_(-3)`

`y_(0.6) = 32 + (0.6)(5) + ((0.6 + 1)(0.6))/(2) * (-1) + ((0.6 + 1)(0.6)(0.6 - 1))/(6) * (1) + ((0.6 + 2)(0.6 + 1)(0.6)(0.6 - 1))/(24) * (-2) + ((0.6 + 2)(0.6 + 1)(0.6)(0.6 - 1)(0.6 - 2))/(120) * (-9)`

`y_(0.6)=32 +3 -0.48 -0.064 +0.0832 -0.104832`

`y_(0.6)=34.43437`


Solution of Gauss's backward interpolation is `y(1976) = 34.43437`

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