Formula
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Bessel's formula
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`p = (x - x_0)/h`
`y_p=(y_0+y_1)/2+(p-1/2)*Delta y_0 + (p(p-1))/(2!) * (Delta^2y_(-1)+Delta^2y_(0))/2 + ((p-1/2)p(p-1))/(3!) * Delta^3y_(-1) + ((p+1)p(p-1)(p-2))/(4!) * (Delta^4y_(-2)+Delta^4y_(-1))/2 + ((p-1/2)(p+1)p(p-1)(p-2))/(5!) * Delta^5y_(-2) + ...`
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Examples
1. Find Solution using Bessel's formula
| x | f(x) |
| 20 | 2854 |
| 24 | 3162 |
| 28 | 3544 |
| 32 | 3992 |
x = 25
Solution:
The value of table for `x` and `y`
| x | 20 | 24 | 28 | 32 |
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| y | 2854 | 3162 | 3544 | 3992 |
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Bessel's method to find solution
`h=24-20=4`
Taking `x_0=24` then `p=(x-x_0)/h=(x-24)/4`
The difference table is
| `x` | `p=(x-24)/4` | `y` | `Deltay` | `Delta^2y` | `Delta^3y` |
| 20 | -1 | 2854 | | | |
| | | 308 | | |
| 24 | 0 | 3162 | | 74 | |
| | | 382 | | -8 |
| 28 | 1 | 3544 | | 66 | |
| | | 448 | | |
| 32 | 2 | 3992 | | | |
`x = 25`
`p = (x - x_0)/h = (25 - 24)/4 = 0.25`
`y_0=3162, Delta y_0=382,Delta^2y_(-1)=74,Delta^3y_(-1)=-8`
Bessel's formula is
`y_p=(y_0+y_1)/2+(p-1/2)*Delta y_0 + (p(p-1))/(2!) * (Delta^2y_(-1)+Delta^2y_(0))/2 + ((p-1/2)p(p-1))/(3!) * Delta^3y_(-1)`
`y_(0.25) = (3162+3544)/2 + (0.25-1/2)*(382) + (0.25(0.25-1))/(2)*((74+66))/2 + ((0.25-1/2)0.25(0.25-1))/(6)*(-8)`
`y_(0.25)=3353-95.5 -6.5625 -0.0625`
`y_(0.25)=3250.875`
Solution of Bessel's interpolation is `y(25) = 3250.875`
This material is intended as a summary. Use your textbook for detail explanation.
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