Which Class Should I Use?

ChebyshevSharp has four approximation families. Pick the simplest class that matches the function shape and dimensionality before tuning node counts or ranks.

Decision Map

Is the function smooth on one rectangular domain?
  yes, and dimension is small or moderate -> ChebyshevApproximation
  no, but singularities or kinks are known -> ChebyshevSpline
  no, and the function changes by regime -> split the domain, then use ChebyshevSpline

Is the function high-dimensional?
  mostly additive or grouped by variables -> ChebyshevSlider
  coupled across many variables -> ChebyshevTT

Class Comparison

Class	Best fit	Build cost	Derivatives	Main trade-off
ChebyshevApproximation	Smooth 1D-5D functions on one box	Dense grid, prod(n_i)	Analytical spectral derivatives	Dense grid grows exponentially
ChebyshevSpline	Kinks, jumps, barriers, or piecewise-smooth functions	Pieces times dense grid	Analytical within each piece	You must choose useful knots
ChebyshevSlider	High-dimensional functions with weak cross-group coupling	Sum of group grids	Uses each group interpolant	Accuracy depends on grouping and pivot
ChebyshevTT	High-dimensional functions with general coupling	TT-Cross, about O(d * n * r^2)	Finite-difference derivatives	Rank and seed affect compression and accuracy

Practical Defaults

• Start with ChebyshevApproximation for smooth functions up to about five dimensions.
• Use ChebyshevSpline when the error estimate stays high near a known kink or discontinuity.
• Use ChebyshevSlider when variables can be partitioned into nearly independent groups.
• Use ChebyshevTT when cross-variable coupling matters and the dense grid is too large.
• For TT work, avoid ToDense() unless the dense grid is intentionally small; dense-grid materialization is guarded and will throw for oversized tensors.

Validation Checklist

After choosing a class:

1. Compare a held-out sample of true function values against interpolated values.
2. Increase node counts or TT rank and check whether the error decreases.
3. For splines, inspect each piece separately; one bad interval can dominate the global error.
4. For TT, run with a fixed seed first, then try a few seeds when rank is near the cap.
5. Save and reload one model in tests when the interpolant is used outside the build process.