CubicInterpolation Class Reference

Cubic interpolation between discrete points. More...

#include <ql/math/interpolations/cubicinterpolation.hpp>

Inheritance diagram for CubicInterpolation:

Public Types
enum	DerivativeApprox {   Spline , SplineOM1 , SplineOM2 , FourthOrder ,   Parabolic , FritschButland , Akima , Kruger ,   Harmonic }
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
Public Member Functions inherited from Interpolation
bool	empty () const
Real	operator() (Real x, bool allowExtrapolation=false) const
Real	primitive (Real x, bool allowExtrapolation=false) const
Real	derivative (Real x, bool allowExtrapolation=false) const
Real	secondDerivative (Real x, bool allowExtrapolation=false) const
Real	xMin () const
Real	xMax () const
bool	isInRange (Real x) const
void	update ()
Public Member Functions inherited from Extrapolator
void	enableExtrapolation (bool b=true)
	enable extrapolation in subsequent calls
void	disableExtrapolation (bool b=true)
	disable extrapolation in subsequent calls
bool	allowsExtrapolation () const
	tells whether extrapolation is enabled

Additional Inherited Members
Protected Member Functions inherited from Interpolation
void	checkRange (Real x, bool extrapolate) const
Protected Attributes inherited from Interpolation
ext::shared_ptr< Impl >	impl_

Detailed Description

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.

Tests

to be adapted from old ones.

Warning

See the Interpolation class for information about the required lifetime of the underlying data.

Member Enumeration Documentation

DerivativeApprox

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)
Harmonic	Weighted harmonic mean approximation (local, monotonic, non-linear)

BoundaryCondition

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

Constructor & Destructor Documentation

CubicInterpolation()

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 )

Precondition

the \( x \) values must be sorted.