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An answer here seems to shows the algorithm Mathematica uses to compute:

Interpolation[{{x1,y1},{x2,y2},{x3,y3},{x4,y4},{x5,y5}},Method->"Spline",InterpolationOrder->3].

I used to assume Mathematica found a natural cubic spline interpolation, but it seems Mathematica actually finds a linear combination of BSplineBasis functions that interpolate the data. What are the pros and cons of doing this using BSpline interpolation versus natural cubic spline interpolation? Also, what is special about the interpolation found using BSplineBasis functions?

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