Numerical interpolation using Stirling's formula calculator

SolutionHelp

1. Using Stirling's interpolation formula, find f(12.2)

x	10	11	12	13	14
f	0.23967	0.28060	0.31788	0.35209	0.38368

2. Using Stirling's interpolation formula, find f(16)

x	0	5	10	15	20	25	30
f	0	0.0875	0.1763	0.2679	0.3640	0.4663	0.5774

Example

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
y	49225	48316	47236	45926	44306

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

$y(28) = 47691.8256$