regularizedSplineRBF

REGULARIZEDSPLINERBF Regularized spline 
 Definition of the spline with tension from [2]. You can also find a brief description of its behaviour in [1].
 Note that [1] suggests using a polynomial of degree 1 with this RBF.
 
 INPUT:
   - r: value to evaluate.
   - e: tension parameter.
 
 OUTPUT:
   - fx: value of the RBF at r.
 
 References: 
  [1] https://pro.arcgis.com/en/pro-app/tool-reference/spatial-analyst/how-spline-works.htm
  [2] Mitas, L., and H. Mitasova. 1988. General Variational Approach to the Interpolation Problem. Comput. Math. Applic. Vol. 16. No. 12. pp. 983–992. Great Britain.

0001 function fx = regularizedSplineRBF(r, e)
0002 %REGULARIZEDSPLINERBF Regularized spline
0003 % Definition of the spline with tension from [2]. You can also find a brief description of its behaviour in [1].
0004 % Note that [1] suggests using a polynomial of degree 1 with this RBF.
0005 %
0006 % INPUT:
0007 %   - r: value to evaluate.
0008 %   - e: tension parameter.
0009 %
0010 % OUTPUT:
0011 %   - fx: value of the RBF at r.
0012 %
0013 % References:
0014 %  [1] https://pro.arcgis.com/en/pro-app/tool-reference/spatial-analyst/how-spline-works.htm
0015 %  [2] Mitas, L., and H. Mitasova. 1988. General Variational Approach to the Interpolation Problem. Comput. Math. Applic. Vol. 16. No. 12. pp. 983–992. Great Britain.
0016 
0017 Ce = 0.5772156649015328606065120900824; % Value of the euler constant. To get it according to your computer precision: vpa(eulergamma);
0018 r(r==0) = 1e-15; % Singularity at r == 0
0019 fx = (1/2*pi) * ( (r.*r./4).*(log(r/2*e)+Ce-1) + e*e*(besselk(0, r/e)+Ce+log(r/2*pi)) );
0020 
0021 end
0022