ppCoefficientsFromSplineCoefficients

Convert spline coefficients into piecewise-polynomial interval coefficients.

Developer documentation: this item describes internal implementation details.

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Declaration

 [C,tpp,Xtpp] = ppCoefficientsFromSplineCoefficients(options)

Parameters

• options.xi spline coefficients
• options.knotPoints spline knot points
• options.S spline degree
• options.Xtpp (optional) splines at the points tpp

Returns

• C polynomial coefficients to be used in polyval, size(C) = [length(tpp)-1, K]
• tpp piece-wise polynomial intervals, size(tpp) = numel(knotPoints) - 2*S - 1
• Xtpp splines at the points tpp

Discussion

This helper rewrites the spline as local Taylor data over each interval. If u = t - tpp(i) on interval i, the cached representation satisfies

\[f_i(u) = \sum_{m=0}^{S} \frac{c_{i,m}}{m!} u^m,\]

where the rows of C store the coefficients \(c_{i,m}\) in the order expected by polyval after the factorial scaling used in evaluateFromPPCoefficients.

  [C, tpp] = BSpline.ppCoefficientsFromSplineCoefficients(xi=xi, knotPoints=knotPoints, S=3);
  xq = BSpline.evaluateFromPPCoefficients(queryPoints=tQuery, C=C, tpp=tpp);