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# CubicInterpolation

## NAME

CubicInterpolation −

Cubic interpolation between discrete points.

## SYNOPSIS

#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

## 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.

Possible enhancements

implement missing schemes (FourthOrder and ModifiedParabolic) and missing boundary conditions (Periodic and Lagrange).

Tests

to be adapted from old ones.

## Member Enumeration Documentation

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

## Constructor & Destructor Documentation

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.

## Author

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