CubicInterpolation −
Cubic interpolation between discrete points.
#include <ql/math/interpolations/cubicinterpolation.hpp>
Inherits Interpolation.
Inherited by AkimaCubicInterpolation, CubicNaturalSpline, CubicSplineOvershootingMinimization1, CubicSplineOvershootingMinimization2, FritschButlandCubic, KrugerCubic, MonotonicCubicNaturalSpline, MonotonicParabolic, and Parabolic.
Public Types
enum DerivativeApprox { Spline, SplineOM1, SplineOM2, FourthOrder, Parabolic, FritschButland, Akima, Kruger }
enum BoundaryCondition { NotAKnot, FirstDerivative, SecondDerivative, Periodic, Lagrange }
Public Member Functions
template<class I1 , class I2 > CubicInterpolation (const I1 &xBegin, const I1 &xEnd, const I2 &yBegin, CubicInterpolation::DerivativeApprox da, bool monotonic, CubicInterpolation::BoundaryCondition leftCond, Real leftConditionValue, CubicInterpolation::BoundaryCondition rightCond, Real rightConditionValue)
const std::vector< Real > & primitiveConstants () const
const std::vector< Real > & aCoefficients () const
const std::vector< Real > & bCoefficients () const
const std::vector< Real > & cCoefficients () const
const std::vector< bool > & monotonicityAdjustments () const
Cubic interpolation between discrete points.
Cubic interpolation is fully defined when the ${f_i}$ function values at points ${x_i}$ are supplemented with ${f^’_i}$ function derivative values.
Different type of first derivative approximations are implemented, both local and non-local. Local schemes (Fourth-order, Parabolic, Modified Parabolic, Fritsch-Butland, Akima, Kruger) use only $f$ values near $x_i$ to calculate each $f^’_i$. Non-local schemes (Spline with different boundary conditions) use all ${f_i}$ values and obtain ${f^’_i}$ by solving a linear system of equations. Local schemes produce $C^1$ interpolants, while the spline schemes generate $C^2$ interpolants.
Hyman’s monotonicity constraint filter is also implemented: it can be applied to all schemes to ensure that in the regions of local monotoniticity of the input (three successive increasing or decreasing values) the interpolating cubic remains monotonic. If the interpolating cubic is already monotonic, the Hyman filter leaves it unchanged preserving all its original features.
In the case of $C^2$ interpolants the Hyman filter ensures local monotonicity at the expense of the second derivative of the interpolant which will no longer be continuous in the points where the filter has been applied.
While some non-linear schemes (Modified Parabolic, Fritsch-Butland, Kruger) are guaranteed to be locally monotonic in their original approximation, all other schemes must be filtered according to the Hyman criteria at the expense of their linearity.
See R. L. Dougherty, A. Edelman, and J. M. Hyman, ’Nonnegativity-, Monotonicity-, or Convexity-Preserving CubicSpline and
Quintic Hermite Interpolation’ Mathematics Of Computation, v. 52, n. 186, April 1989, pp. 471-494.
Possible enhancements
implement missing schemes (FourthOrder and ModifiedParabolic) and missing boundary conditions (Periodic and Lagrange).
Tests
to be adapted from old ones.
enum DerivativeApprox
Enumerator:
Spline |
Spline approximation (non-local, non-monotonic, linear[?]). Different boundary conditions can be used on the left and right boundaries: see BoundaryCondition. |
SplineOM1
Overshooting minimization 1st derivative.
SplineOM2
Overshooting minimization 2nd derivative.
FourthOrder
Fourth-order approximation (local, non-monotonic, linear)
Parabolic
Parabolic approximation (local, non-monotonic, linear)
FritschButland
Fritsch-Butland approximation (local, monotonic, non-linear)
Akima |
Akima approximation (local, non-monotonic, non-linear) |
||
Kruger |
Kruger approximation (local, monotonic, non-linear) |
enum BoundaryCondition
Enumerator:
NotAKnot
Make second(-last) point an inactive knot.
FirstDerivative
Match value of end-slope.
SecondDerivative
Match value of second derivative at end.
Periodic
Match first and second derivative at either end.
Lagrange
Match end-slope to the slope of the cubic that matches the first four data at the respective end
CubicInterpolation (const I1 &xBegin, const I1 &xEnd, const I2 &yBegin, CubicInterpolation::DerivativeApproxda, boolmonotonic, CubicInterpolation::BoundaryConditionleftCond, RealleftConditionValue, CubicInterpolation::BoundaryConditionrightCond, RealrightConditionValue)
Precondition:
the $ x $ values must be sorted.
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