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Gauss's central difference formula for equal intervals.

 Gauss's central difference formula for equal intervals:

We shall develop central difference formulae which are best suitable for interpolation near the middle of the tabulated set (table).
x:             x₋₂  x₋₁  x₀  x₁  x₂
y=f(x):    y₋₂  y₋₁  y₀  y₁  y₂
Difference table
Difference table.


Gauss's forward interpolation formula for equal intervals:

f(x) = y₀+[u/1!]∆y₀+{[u(u-1)]/2!}∆²y₋₁+{[(u+1)(u)(u-1)]/3!}∆³y₋₁+{[(u+1)u(u-1)(u-2)]/4!}∆⁴y₋₂ +......., Where u= (x-x₀)/h
Remark:
This formula is applicable when u lies between 0 and 1 i.e (0<u<1).
Example: Using Gauss's forward formula to evaluate y₃₀ given that y₂₁=18.4708, y₂₅=17.8144, y₂₉=17.1070, y₃₃=16.3432 and y₃₇=15.5154.
Solution. The difference table is
Forward difference table
Forward difference table.

To find y=f(x) at x=30, i.e f(30):
Taking x₀ = 29, h=4, x=30, then u= (x-x₀)/h = (30-29)/4 = 0.25
Using Gauss's forward difference formula
f(x) = y₀+[u/1!]∆y₀+{[u(u-1)]/2!}∆²y₋₁+{[(u+1)(u)(u-1)]/3!}∆³y₋₁+[(u+1)u(u-1)(u-2)/4!]∆⁴y₋₂ +......
=> f(30) = y₃₀ = 17.1070+(0.25)(-0.7638) + [(0.25)(-0.75)]/2×(-0.564) + [(1.25)(.25)(-.75)]/6×(-0.0076) + [(1.25)(.25)(-0.75)(-1.75)]/24× (-0.0022)
=> f(30) = y₃₀ = 17.1070-0.19095+0.00529+0.0003-0.00004
=>  y₃₀ = 16.92 Ans.

Gauss's backward interpolation formula:

f(x) = y₀+[u/1!]∆y₋₁+{[u(u+1)]/2!}∆²y₋₁+{[(u+1)(u)(u-1)]/3!}∆³y₋₂+[(u+2)(u+1)u(u-1)/4!]∆⁴y₋₂ +......., Where u= (x-x₀)/h
Remark:
This formula is applicable when u lies between -1 and 0.
Example: Find by Gauss's backward formula the sales of a concern for the year 1936, given that
Year:                                                1901   1911   1921    1931   1941   1951
Sale( in thousands rupees):        12       15   20         27         39         52
Solution. Taking the origin at 1941 and h=10,
u= (x-x₀)/h= (1936-1941)/10 = -0.5.   [∵-1<u<0]
The central table is:
Forward difference table
Forward difference table.

Using Gauss backward formula:
f(x) = y₀+[u/1!]∆y₋₁+{[u(u+1)]/2!}∆²y₋₁+{[(u+1)(u)(u-1)]/3!}∆³y₋₂+.......
=> f(1936)= 39 + (-.5)×(12)+[(.5)(-.5)]/2 × [(.5)(-.5)(-1.5)]/6 ×(-4)
=> f(1936)= 39-6-0.125-0.25 = 32.625 thousands Ans.

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