Obtuse Paclet SymbolTutorials »|See Also »|More About »

InterpolationCoefficients

InterpolationCoefficients[interpolationpoint,controlpointlist,distmat,conlistslist,interpolationorder,(options)]
gives a vector of interpolation coefficients for the point interpolationpoint for Method→"ObtuseAngle". The scalar product between this vector and the vector of function values in the distinct points in the controlpointlist gives the interpolated value in the point interpolationpoint. As last value in the function value list, the CutoffValue should be appended. For zeroth order the interpolated value is the average of the surrounding points. The argument distmat should give the distance matrix for the control points. The distance matrix does not need to be symmetric. The argument conlistslist should be a list of connection lists with the option NeighborLevel running from 1 to interpolationorder. If interpolationorder is zero, NeighborLevel should be 1. The argument interpolationorder should be 0, 1, 2, or 3. If this sounds too complicated, use the Interpolation command with Method → "ObtuseAngle", with a simpler interface. Then this command is called indirectly. But please check the interactive examples below!
  • To use InterpolationCoefficients, you first need to load the Obtuse Package using Needs["Obtuse`"].
  • The following options can be given:
CutoffRadiusInfinitypoints separated by twice the CutoffRadius or more are not connected
DistanceFunctionAutomaticdistance function to use in calculation of the distance matrix and the calculation of the distance from the interpolation point to the control points. Note, that the function here is expected to return the square of the distance.
TypeDirectedthe connection graph can be Directed or Undirected
EXAMPLESCLOSE ALL
Basic Examples   (1)
In[1]:=
Click for copyable input
Some 2D points:
In[2]:=
Click for copyable input
Add function values in the 2D points:
In[3]:=
Click for copyable input
Test interpolation in one point:
In[4]:=
Click for copyable input
Out[4]=
Calculate the interpolation coefficients
In[5]:=
Click for copyable input
Out[5]=
Make a list of the function values, with the CutoffValue at the end.
In[6]:=
Click for copyable input
Out[6]=
If we now take the scalar products of these two vectors, we should obtain the interpolated value again.
In[7]:=
Click for copyable input
Out[7]=
InterpolationCoefficients is a part of the algorithm for Interpolation with Method→"ObtuseAngle". However, the same vector, apart from the ending CutoffValue value, can also be obtained by entering the IdentityMatrix as a list of function values. Note that the function values for Interpolation does not need to be scalars, they can also be vectors or other objects. With this method, the interpolation coefficients could be found for any of the Interpolation methods.
In[8]:=
Click for copyable input
Out[8]=
Here is another way to see the interpolation coefficients: Let the function values be f1, f2, ..., f10
In[9]:=
Click for copyable input
Out[9]=
If we call the function Interpolation with Method→"ObtuseAngle" without specifying the point, we get back a function containing an InterpolationCoefficients function call
In[10]:=
Click for copyable input
Out[10]=
If you are curious, you might inspect various part of the output above.