Formula
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Stirling's formula
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`p = (x - x_0)/h`
`y_p=y_0+p*(Delta y_0+Delta y_(-1))/2 + (p^2)/(2!) * Delta^2y_(-1) + (p(p^2 - 1^2))/(3!) * (Delta^3y_(-1)+Delta^3y_(-2))/2 + (p^2(p^2 - 1^2))/(4!) * Delta^4y_(-2) + ...`
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Examples
1. Find Solution using Stirling's formula
| x | f(x) |
| 20 | 49225 |
| 25 | 48316 |
| 30 | 47236 |
| 35 | 45926 |
| 40 | 44306 |
x = 28
Solution:
The value of table for `x` and `y`
| x | 20 | 25 | 30 | 35 | 40 |
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| y | 49225 | 48316 | 47236 | 45926 | 44306 |
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Stirling's method to find solution
`h=25-20=5`
Taking `x_0=30` then `p=(x-x_0)/h=(x-30)/5`
The difference table is
| `x` | `p=(x-30)/5` | `y` | `Deltay` | `Delta^2y` | `Delta^3y` | `Delta^4y` |
| 20 | -2 | 49225 | | | | |
| | | -909 | | | |
| 25 | -1 | 48316 | | -171 | | |
| | | -1080 | | -59 | |
| 30 | 0 | 47236 | | -230 | | -21 |
| | | -1310 | | -80 | |
| 35 | 1 | 45926 | | -310 | | |
| | | -1620 | | | |
| 40 | 2 | 44306 | | | | |
`x = 28`
`p = (x - x_0)/h = (28 - 30)/5 = -0.4`
`y_0=47236, Delta y_0=-1310,Delta^2y_(-1)=-230,Delta^3y_(-1)=-80,Delta^4y_(-2)=-21`
Stirling's formula is
`y_p=y_0+p*(Delta y_0+Delta y_(-1))/2 + (p^2)/(2!) * Delta^2y_(-1) + (p(p^2 - 1^2))/(3!) * (Delta^3y_(-1)+Delta^3y_(-2))/2 + (p^2(p^2 - 1^2))/(4!) * Delta^4y_(-2)`
`y_(-0.4) = 47236 + (-0.4)*((-1310-1080))/2 + ((0.16))/(2)*(-230) + ((-0.4)(0.16 - 1))/(6)*((-80-59))/2 + ((0.16)(0.16 - 1))/(24)*(-21)`
`y_(-0.4)=47236+478 -18.4 -3.892 +0.1176`
`y_(-0.4)=47691.8256`
Solution of Stirling's interpolation is `y(28) = 47691.8256`
This material is intended as a summary. Use your textbook for detail explanation.
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