tensionSplineRBF

SPLINEWITHTENSIONRBF Spline with tension.
 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 0 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 = tensionSplineRBF(r, e)
0002 %SPLINEWITHTENSIONRBF Spline with tension.
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 0 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 fx = -(1/(2*pi*e*e))*(log(r.*e./2)+Ce+besselk(0, r.*e));
0019 fx(r==0) = 0; % Singularity at r == 0
0020 
0021 end
0022