Bessel's Interpolation

Interpolation is the technique of estimating the value of a function for any intermediate value of the independent variable, while the process of computing the value of the function outside the given range is called extrapolation.

Central differences : The central difference operator d is defined by the relations :
 

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Similarly, high order central differences are defined as :
 

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Note - The central differences on the same horizontal line have the same suffix
 

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Bessel's Interpolation formula -
 

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It is very useful when u = 1/2. It gives a better estimate when 1/4 < u < 3/4 
Here f(0) is the origin point usually taken to be mid point, since Bessel's is used to interpolate near the center. 
h is called the interval of difference and u = ( x – f(0) ) / h, Here f(0) is term at the origin chosen.

Examples -
Input : Value at 27.4 ? 
 

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Output : 

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Value at 27.4 is 3.64968

Implementation of Bessel's Interpolation - 

Output: 

 4 -0.154 0.0120001 -0.00300002 0.00399971 -0.00699902 
3.846 -0.142 0.00900006 0.000999689 -0.00299931 
3.704 -0.133 0.00999975 -0.00199962 
3.571 -0.123 0.00800014 
3.448 -0.115 
3.333 
Value at 27.4 is 3.64968

Time complexity: O(n²) 
Auxiliary space: O(n*n)