Formula
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Gauss Forward formula
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`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 = 3.5
Solution:
The value of table for `x` and `y`
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`
This material is intended as a summary. Use your textbook for detail explanation.
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