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 :

Similarly, high order central differences are defined as :

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

Bessel’s Interpolation formula –

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 ? 

Output : 

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)