Formula
1. For `x=x_0`
`[(dy)/(dx)]_(x=x_0) = 1/h * [1/2 * (Delta y_0 + Delta y_(-1)) - 1/12 * (Delta^3 y_(-1) + Delta^3 y_(-2)) + 1/60 * (Delta^5 y_(-2) + Delta^5 y_(-3)) + ...]`
`[(d^2y)/(dx^2)]_(x=x_0) = 1/h^2 * [Delta^2 y_(-1) - 1/12 Delta^4 y_(-2) + 1/90 * Delta^6 y_(-3) + ...]`
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Examples
1. Using Stirling's formula to find solution
| x | f(x) |
| 7.47 | 0.193 |
| 7.48 | 0.195 |
| 7.49 | 0.198 |
| 7.50 | 0.201 |
| 7.51 | 0.203 |
| 7.52 | 0.206 |
| 7.53 | 0.208 |
x = 7.5
Solution:
Stirling's formula (central difference formula).
The value of table for `x` and `y`
| x | 7.47 | 7.48 | 7.49 | 7.5 | 7.51 | 7.52 | 7.53 |
|---|
| y | 0.193 | 0.195 | 0.198 | 0.201 | 0.203 | 0.206 | 0.208 |
|---|
Difference table is
| x | y | `Deltay` | `Delta^2y` | `Delta^3y` | `Delta^4y` | `Delta^5y` | `Delta^6y` |
| 7.47 | 0.193 | | | | | | |
| | 0.002 | | | | | |
| 7.48 | 0.195 | | 0.001 | | | | |
| | 0.003 | | -0.001 | | | |
| 7.49 | 0.198 | | 0 | | 0 | | |
| | 0.003 | | -0.001 | | 0.003 | |
| 7.5 | 0.201 | | -0.001 | | 0.003 | | -0.01 |
| | 0.002 | | 0.002 | | -0.007 | |
| 7.51 | 0.203 | | 0.001 | | -0.004 | | |
| | 0.003 | | -0.002 | | | |
| 7.52 | 0.206 | | -0.001 | | | | |
| | 0.002 | | | | | |
| 7.53 | 0.208 | | | | | | |
The value of `x` at you want to find `f(x) : x_0 = 7.5`
`h = x_1 - x_0 = 7.48 - 7.47 = 0.01`
Stirling's Formula is
`[(dy)/(dx)]_(x=x_0) = 1/h * [1/2 * (Delta y_0 + Delta y_(-1)) - 1/12 * (Delta^3 y_(-1) + Delta^3 y_(-2)) + 1/60 * (Delta^5 y_(-2) + Delta^5 y_(-3)) + ...]`
`:.[(dy)/(dx)]_(x=7.5) = 1/0.01 * [1/2 * (0.002 +0.003) - 1/12 * (0.002 -0.001)+ 1/60 * (-0.007 +0.003)]`
`:.[(dy)/(dx)]_(x=7.5) = 1/0.01 * [0.0025-0.0000833333-0.0000666667]`
`:.[(dy)/(dx)]_(x=7.5) = 0.235`
`[(d^2y)/(dx^2)]_(x=x_0) = 1/h^2 * [Delta^2 y_(-1) - 1/12 Delta^4 y_(-2) + 1/90 * Delta^6 y_(-3) + ...]`
`:.[(d^2y)/(dx^2)]_(x=7.5) = 1/0.0001 * [-0.001 - 1/12 * 0.003+ 1/90 * -0.01]`
`:.[(d^2y)/(dx^2)]_(x=7.5) = 1/0.0001 * [-0.001-0.00025-0.0001111111]`
`:.[(d^2y)/(dx^2)]_(x=7.5) = -13.61111`
This material is intended as a summary. Use your textbook for detail explanation.
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