2. Find Solution using Gauss Forward formula
| x | f(x) |
| 310 | 2.4914 |
| 320 | 2.5052 |
| 330 | 2.5185 |
| 340 | 2.5315 |
| 350 | 2.5441 |
| 360 | 2.5563 |
x = 337.5
Solution:
The value of table for `x` and `y`
| x | 310 | 320 | 330 | 340 | 350 | 360 |
|---|
| y | 2.4914 | 2.5052 | 2.5185 | 2.5315 | 2.5441 | 2.5563 |
|---|
Gauss's forward method to find solution
`h=320-310=10`
Taking `x_0=330` then `p=(x-x_0)/h=(x-330)/10`
Now the central difference table is
| `x` | `p=(x-330)/10` | `y` | `Deltay` | `Delta^2y` | `Delta^3y` | `Delta^4y` | `Delta^5y` |
| 310 | -2 | 2.4914 | | | | | |
| | | 0.0138 | | | | |
| 320 | -1 | 2.5052 | | -0.0005 | | | |
| | | 0.0133 | | 0.0002 | | |
| 330 | 0 | 2.5185 | | -0.0003 | | -0.0003 | |
| | | 0.013 | | -0.0001 | | 0.0004 |
| 340 | 1 | 2.5315 | | -0.0004 | | 0.0001 | |
| | | 0.0126 | | 0 | | |
| 350 | 2 | 2.5441 | | -0.0004 | | | |
| | | 0.0122 | | | | |
| 360 | 3 | 2.5563 | | | | | |
`x = 337.5`
`p = (x - x_0)/h = (337.5 - 330)/10 = 0.75`
`y_0=2.5185, Delta y_0=0.013,Delta^2y_(-1)=-0.0003,Delta^3y_(-1)=-0.0001,Delta^4y_(-2)=-0.0003,Delta^5y_(-2)=0.0004`
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) + ((p + 2)(p + 1)p(p - 1)(p - 2))/(5!) * Delta^5y_(-2)`
`y_(0.75) = 2.5185 + (0.75)(0.013) + ((0.75)(0.75 - 1))/(2) * (-0.0003) + ((0.75 + 1)(0.75)(0.75 - 1))/(6) * (-0.0001) + ((0.75 + 1)(0.75)(0.75 - 1)(0.75 - 2))/(24) * (-0.0003) + ((0.75 + 2)(0.75 + 1)(0.75)(0.75 - 1)(0.75 - 2))/(120) * (0.0004)`
`y_(0.75)=2.5185 +0.00975 +0.000028125 +0.0000054687 -0.000005127 +0.0000037598`
`y_(0.75)=2.52828`
Solution of Gauss's forward interpolation is `y(337.5) = 2.52828`
This material is intended as a summary. Use your textbook for detail explanation.
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