Formula
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Gauss Backward formula
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`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 |
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| y | 17 | 20 | 27 | 32 | 36 | 38 |
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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`
This material is intended as a summary. Use your textbook for detail explanation.
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