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A free/open-source library for quantitative finance
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CubicInterpolation Class Reference

Cubic interpolation between discrete points. More...

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

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Public Types

enum  DerivativeApprox {
  Spline, SplineOM1, SplineOM2, FourthOrder,
  Parabolic, FritschButland, Akima, Kruger,
  Harmonic
}
 
enum  BoundaryCondition {
  NotAKnot, FirstDerivative, SecondDerivative, Periodic,
  Lagrange
}
 
- Public Types inherited from Interpolation
typedef Real argument_type
 
typedef Real result_type
 

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
 Interpolation ()=default
 
 ~Interpolation () override=default
 
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
 Extrapolator ()=default
 
virtual ~Extrapolator ()=default
 
void enableExtrapolation (bool b=true)
 enable extrapolation in subsequent calls More...
 
void disableExtrapolation (bool b=true)
 disable extrapolation in subsequent calls More...
 
bool allowsExtrapolation () const
 tells whether extrapolation is enabled More...
 

Private Attributes

ext::shared_ptr< detail::CoefficientHoldercoeffs_
 

Additional Inherited Members

- Protected Member Functions inherited from Interpolation
void checkRange (Real x, bool extrapolate) const
 
- Protected Attributes inherited from Interpolation
ext::shared_ptr< Implimpl_
 

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.

Definition at line 105 of file cubicinterpolation.hpp.

Member Enumeration Documentation

◆ 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)

Definition at line 107 of file cubicinterpolation.hpp.

◆ 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

Definition at line 138 of file cubicinterpolation.hpp.

Constructor & Destructor Documentation

◆ CubicInterpolation()

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.

Definition at line 158 of file cubicinterpolation.hpp.

Member Function Documentation

◆ primitiveConstants()

const std::vector<Real>& primitiveConstants ( ) const

Definition at line 179 of file cubicinterpolation.hpp.

◆ aCoefficients()

const std::vector<Real>& aCoefficients ( ) const

Definition at line 182 of file cubicinterpolation.hpp.

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◆ bCoefficients()

const std::vector<Real>& bCoefficients ( ) const

Definition at line 183 of file cubicinterpolation.hpp.

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◆ cCoefficients()

const std::vector<Real>& cCoefficients ( ) const

Definition at line 184 of file cubicinterpolation.hpp.

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◆ monotonicityAdjustments()

const std::vector<bool>& monotonicityAdjustments ( ) const

Definition at line 185 of file cubicinterpolation.hpp.

Member Data Documentation

◆ coeffs_

ext::shared_ptr<detail::CoefficientHolder> coeffs_
private

Definition at line 189 of file cubicinterpolation.hpp.