Spline

C++ cubic spline interpolation

This is a lightweight implementation of cubic splines to interpolate points f(x_i) = y_i with the following features.

• available spline types:
  • cubic C² splines: global, twice continuously differentiable
  • cubic Hermite splines: local, continuously differentiable (C¹)
• boundary conditions: first and second order derivatives can be specified, not-a-knot condition, periodic condition is not implemented
• extrapolation
  • linear: if first order derivatives are specified or 2nd order = 0
  • quadratic: if 2nd order derivatives not equal to zero specified
• monotonicity can be enforced (when input is monotonic as well)

Usage

The library is a header-only file with no external dependencies and can be used like this:

#include <vector>
#include "spline.h"
...
    std::vector<double> X, Y;
    ...
    // default cubic spline (C^2) with natural boundary conditions (f''=0)
    tk::spline s(X,Y);			// X needs to be strictly increasing
    double value=s(1.3);		// interpolated value at 1.3
    double deriv=s.deriv(1,1.3);	// 1st order derivative at 1.3
    std::vector<double> solutions = s.solve(0.0);	// solves s(x)=0.0
    ...

The constructor can take more arguments to define the spline, e.g.:

    // cubic Hermite splines (C^1) with enforced monotonicity and
    // left curvature equal to 0.0 and right slope equal 1.0
    tk::spline s(X,Y,tk::spline::cspline_hermite, true,
                 tk::spline::second_deriv, 0.0,
                 tk::spline::first_deriv, 1.0);

This is identical to (must be called in that order):

    tk::spline s;
    s.set_boundary(tk::spline::second_deriv, 0.0,
                   tk::spline::first_deriv, 1.0);
    s.set_points(X,Y);
    s.make_monotonic();

Spline types

Splines are piecewise polynomial functions to interpolate points (x_i, y_i). In particular, cubic splines can be represented as

• f(x) = a_i + b_i (x-x_i) + c_i (x-x_i)² + d_i (x-x_i)³, for all x in [x_i, x_i+1)
• f(x_i)=y_i

The following splines are available.

• tk::spline::cspline: cubic C² spline
  • twice continuously differentiable, e.g. f'(x_i) and f''(x_i) exist
  • this, together with boundary conditions uniquely determines the spline
  • requires solving a sparse equation system
  • is a global spline in the sense that changing an input point will impact the spline everywhere
  • setting first order derivatives at the boundary will break C² at the boundary
• tk::spline::cspline_hermite: cubic Hermite spline
  • once continuously differentiable (C¹)
  • first order derivatives are specified by finite differences, e.g. on a uniform x-grid:
    • f'(x_i) = (y_i+1-y_i-1)/(x_i+1-x_i-1)
  • is a local spline in the sense that changing grid points will only affect the spline around that grid point in a few adjacent segments

A function to enforce monotonicity is available as well:

• tk::spline::make_monotonic(): will make the spline monotonic if input grid points are monotonic
  • this function can only be called after set_points(...) has been called
  • it will break C² if the original spline was C² and not already monotonic
  • it will break boundary conditions if it was not monotonic in the first or last segment

References

https://kluge.in-chemnitz.de/opensource/spline/