Table of Contents

Special Points (Kink Declaration)

ChebyshevSpline.WithSpecialPoints is the C# entry point for declaring known kinks at construction time. It is equivalent to passing the same values as knots to a regular ChebyshevSpline constructor, but the name makes the intent clearer when the values represent discontinuities, barriers, strikes, or other non-smooth locations.

Why Declare Kinks

Without a kink declaration, spectral methods converge slowly on non-smooth functions. Declaring the kink as a sub-interval boundary restores spectral convergence on each smooth piece.

using ChebyshevSharp;

// Without kink declaration: one global polynomial must approximate the kink.
var bad = new ChebyshevApproximation(
    (x, _) => Math.Abs(x[0]),
    1, new[] { new[] { -1.0, 1.0 } }, new[] { 31 });
bad.Build(verbose: false);

// With kink declaration: this piecewise-linear example reaches machine precision
// with 11 nodes per piece in the regression tests.
var good = ChebyshevSpline.WithSpecialPoints(
    function: (x, _) => Math.Abs(x[0]),
    numDimensions: 1,
    domain: new[] { new[] { -1.0, 1.0 } },
    specialPoints: new[] { new[] { 0.0 } },
    nNodesNested: new[] { new[] { 11, 11 } });
good.Build(verbose: false);

double value = good.Eval(new[] { -0.25 }, new[] { 0 });  // approximately 0.25

Factory vs Constructor

specialPoints is intentionally not a ChebyshevApproximation constructor parameter. Declaring special points changes the representation from one dense tensor to a piecewise spline, so the C# API exposes it through ChebyshevSpline.WithSpecialPoints(...).

Use Adaptive Refinement if you want a heuristic scan for candidate knots instead of declaring known locations yourself.

Validation Rules

  • specialPoints must have one array per dimension.
  • Points must be finite, sorted, unique, and strictly inside the domain; endpoints are already piece boundaries.
  • Exactly one construction mode must be supplied: nNodesNested, nNodes, or errorThreshold.
  • For nNodesNested, each dimension needs specialPoints[d].Length + 1 node counts, one for each sub-interval.

Per-Sub-Interval Node Counts

Pass nested arrays to nNodesNested for per-piece refinement:

using ChebyshevSharp;

var spl = ChebyshevSpline.WithSpecialPoints(
    function: (x, _) => Math.Abs(x[0]) + x[1] * x[1],
    numDimensions: 2,
    domain: new[] { new[] { -1.0, 1.0 }, new[] { -1.0, 1.0 } },
    specialPoints: new[] { new[] { 0.0 }, Array.Empty<double>() },
    nNodesNested: new[] { new[] { 7, 9 }, new[] { 11 } });
// Dim 0: 2 pieces (split at 0.0) with 7 and 9 nodes.
// Dim 1: 1 piece (no kink) with 11 nodes.

Use nested counts when one side of a kink needs more resolution than the other. Use flat nNodes when every sub-interval can use the same node count. Use errorThreshold when you want each piece to choose its node count by the standard error-driven build loop.

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