/usr/include/ql/math/interpolations/cubicinterpolation.hpp is in libquantlib0-dev 1.7.1-1.
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/*
Copyright (C) 2000, 2001, 2002, 2003 RiskMap srl
Copyright (C) 2001, 2002, 2003 Nicolas Di Césaré
Copyright (C) 2004, 2008, 2009, 2011 Ferdinando Ametrano
Copyright (C) 2009 Sylvain Bertrand
Copyright (C) 2013 Peter Caspers
This file is part of QuantLib, a free-software/open-source library
for financial quantitative analysts and developers - http://quantlib.org/
QuantLib is free software: you can redistribute it and/or modify it
under the terms of the QuantLib license. You should have received a
copy of the license along with this program; if not, please email
<quantlib-dev@lists.sf.net>. The license is also available online at
<http://quantlib.org/license.shtml>.
This program is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the license for more details.
*/
/*! \file cubicinterpolation.hpp
\brief cubic interpolation between discrete points
*/
#ifndef quantlib_cubic_interpolation_hpp
#define quantlib_cubic_interpolation_hpp
#include <ql/math/matrix.hpp>
#include <ql/math/interpolation.hpp>
#include <ql/methods/finitedifferences/tridiagonaloperator.hpp>
#include <vector>
namespace QuantLib {
namespace detail {
class CoefficientHolder {
public:
CoefficientHolder(Size n)
: n_(n), primitiveConst_(n-1), a_(n-1), b_(n-1), c_(n-1),
monotonicityAdjustments_(n) {}
virtual ~CoefficientHolder() {}
Size n_;
// P[i](x) = y[i] +
// a[i]*(x-x[i]) +
// b[i]*(x-x[i])^2 +
// c[i]*(x-x[i])^3
std::vector<Real> primitiveConst_, a_, b_, c_;
std::vector<bool> monotonicityAdjustments_;
};
template <class I1, class I2> class CubicInterpolationImpl;
}
//! %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.
\todo implement missing schemes (FourthOrder and ModifiedParabolic) and
missing boundary conditions (Periodic and Lagrange).
\test to be adapted from old ones.
\ingroup interpolations
*/
class CubicInterpolation : public Interpolation {
public:
enum DerivativeApprox {
/*! Spline approximation (non-local, non-monotonic, linear[?]).
Different boundary conditions can be used on the left and right
boundaries: see BoundaryCondition.
*/
Spline,
//! Overshooting minimization 1st derivative
SplineOM1,
//! Overshooting minimization 2nd derivative
SplineOM2,
//! Fourth-order approximation (local, non-monotonic, linear)
FourthOrder,
//! Parabolic approximation (local, non-monotonic, linear)
Parabolic,
//! Fritsch-Butland approximation (local, monotonic, non-linear)
FritschButland,
//! Akima approximation (local, non-monotonic, non-linear)
Akima,
//! Kruger approximation (local, monotonic, non-linear)
Kruger
};
enum BoundaryCondition {
//! Make second(-last) point an inactive knot
NotAKnot,
//! Match value of end-slope
FirstDerivative,
//! Match value of second derivative at end
SecondDerivative,
//! Match first and second derivative at either end
Periodic,
/*! Match end-slope to the slope of the cubic that matches
the first four data at the respective end
*/
Lagrange
};
/*! \pre the \f$ x \f$ values must be sorted. */
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) {
impl_ = boost::shared_ptr<Interpolation::Impl>(new
detail::CubicInterpolationImpl<I1,I2>(xBegin, xEnd, yBegin,
da,
monotonic,
leftCond,
leftConditionValue,
rightCond,
rightConditionValue));
impl_->update();
coeffs_ =
boost::dynamic_pointer_cast<detail::CoefficientHolder>(impl_);
}
const std::vector<Real>& primitiveConstants() const {
return coeffs_->primitiveConst_;
}
const std::vector<Real>& aCoefficients() const { return coeffs_->a_; }
const std::vector<Real>& bCoefficients() const { return coeffs_->b_; }
const std::vector<Real>& cCoefficients() const { return coeffs_->c_; }
const std::vector<bool>& monotonicityAdjustments() const {
return coeffs_->monotonicityAdjustments_;
}
private:
boost::shared_ptr<detail::CoefficientHolder> coeffs_;
};
// convenience classes
class CubicNaturalSpline : public CubicInterpolation {
public:
/*! \pre the \f$ x \f$ values must be sorted. */
template <class I1, class I2>
CubicNaturalSpline(const I1& xBegin,
const I1& xEnd,
const I2& yBegin)
: CubicInterpolation(xBegin, xEnd, yBegin,
Spline, false,
SecondDerivative, 0.0,
SecondDerivative, 0.0) {}
};
class MonotonicCubicNaturalSpline : public CubicInterpolation {
public:
/*! \pre the \f$ x \f$ values must be sorted. */
template <class I1, class I2>
MonotonicCubicNaturalSpline(const I1& xBegin,
const I1& xEnd,
const I2& yBegin)
: CubicInterpolation(xBegin, xEnd, yBegin,
Spline, true,
SecondDerivative, 0.0,
SecondDerivative, 0.0) {}
};
class CubicSplineOvershootingMinimization1 : public CubicInterpolation {
public:
/*! \pre the \f$ x \f$ values must be sorted. */
template <class I1, class I2>
CubicSplineOvershootingMinimization1 (const I1& xBegin,
const I1& xEnd,
const I2& yBegin)
: CubicInterpolation(xBegin, xEnd, yBegin,
SplineOM1, false,
SecondDerivative, 0.0,
SecondDerivative, 0.0) {}
};
class CubicSplineOvershootingMinimization2 : public CubicInterpolation {
public:
/*! \pre the \f$ x \f$ values must be sorted. */
template <class I1, class I2>
CubicSplineOvershootingMinimization2 (const I1& xBegin,
const I1& xEnd,
const I2& yBegin)
: CubicInterpolation(xBegin, xEnd, yBegin,
SplineOM2, false,
SecondDerivative, 0.0,
SecondDerivative, 0.0) {}
};
class AkimaCubicInterpolation : public CubicInterpolation {
public:
/*! \pre the \f$ x \f$ values must be sorted. */
template <class I1, class I2>
AkimaCubicInterpolation(const I1& xBegin,
const I1& xEnd,
const I2& yBegin)
: CubicInterpolation(xBegin, xEnd, yBegin,
Akima, false,
SecondDerivative, 0.0,
SecondDerivative, 0.0) {}
};
class KrugerCubic : public CubicInterpolation {
public:
/*! \pre the \f$ x \f$ values must be sorted. */
template <class I1, class I2>
KrugerCubic(const I1& xBegin,
const I1& xEnd,
const I2& yBegin)
: CubicInterpolation(xBegin, xEnd, yBegin,
Kruger, false,
SecondDerivative, 0.0,
SecondDerivative, 0.0) {}
};
class FritschButlandCubic : public CubicInterpolation {
public:
/*! \pre the \f$ x \f$ values must be sorted. */
template <class I1, class I2>
FritschButlandCubic(const I1& xBegin,
const I1& xEnd,
const I2& yBegin)
: CubicInterpolation(xBegin, xEnd, yBegin,
FritschButland, false,
SecondDerivative, 0.0,
SecondDerivative, 0.0) {}
};
class Parabolic : public CubicInterpolation {
public:
/*! \pre the \f$ x \f$ values must be sorted. */
template <class I1, class I2>
Parabolic(const I1& xBegin,
const I1& xEnd,
const I2& yBegin)
: CubicInterpolation(xBegin, xEnd, yBegin,
CubicInterpolation::Parabolic, false,
SecondDerivative, 0.0,
SecondDerivative, 0.0) {}
};
class MonotonicParabolic : public CubicInterpolation {
public:
/*! \pre the \f$ x \f$ values must be sorted. */
template <class I1, class I2>
MonotonicParabolic(const I1& xBegin,
const I1& xEnd,
const I2& yBegin)
: CubicInterpolation(xBegin, xEnd, yBegin,
Parabolic, true,
SecondDerivative, 0.0,
SecondDerivative, 0.0) {}
};
//! %Cubic interpolation factory and traits
/*! \ingroup interpolations */
class Cubic {
public:
Cubic(CubicInterpolation::DerivativeApprox da
= CubicInterpolation::Kruger,
bool monotonic = false,
CubicInterpolation::BoundaryCondition leftCondition
= CubicInterpolation::SecondDerivative,
Real leftConditionValue = 0.0,
CubicInterpolation::BoundaryCondition rightCondition
= CubicInterpolation::SecondDerivative,
Real rightConditionValue = 0.0)
: da_(da), monotonic_(monotonic),
leftType_(leftCondition), rightType_(rightCondition),
leftValue_(leftConditionValue), rightValue_(rightConditionValue) {}
template <class I1, class I2>
Interpolation interpolate(const I1& xBegin,
const I1& xEnd,
const I2& yBegin) const {
return CubicInterpolation(xBegin, xEnd, yBegin,
da_, monotonic_,
leftType_, leftValue_,
rightType_, rightValue_);
}
static const bool global = true;
static const Size requiredPoints = 2;
private:
CubicInterpolation::DerivativeApprox da_;
bool monotonic_;
CubicInterpolation::BoundaryCondition leftType_, rightType_;
Real leftValue_, rightValue_;
};
namespace detail {
template <class I1, class I2>
class CubicInterpolationImpl : public CoefficientHolder,
public Interpolation::templateImpl<I1,I2> {
public:
CubicInterpolationImpl(const I1& xBegin,
const I1& xEnd,
const I2& yBegin,
CubicInterpolation::DerivativeApprox da,
bool monotonic,
CubicInterpolation::BoundaryCondition leftCondition,
Real leftConditionValue,
CubicInterpolation::BoundaryCondition rightCondition,
Real rightConditionValue)
: CoefficientHolder(xEnd-xBegin),
Interpolation::templateImpl<I1,I2>(xBegin, xEnd, yBegin,
Cubic::requiredPoints),
da_(da),
monotonic_(monotonic),
leftType_(leftCondition), rightType_(rightCondition),
leftValue_(leftConditionValue),
rightValue_(rightConditionValue),
tmp_(n_), dx_(n_-1), S_(n_-1), L_(n_) {
if (leftType_ == CubicInterpolation::Lagrange
|| rightType_ == CubicInterpolation::Lagrange) {
QL_REQUIRE((xEnd-xBegin) >= 4,
"Lagrange boundary condition requires at least "
"4 points (" << (xEnd-xBegin) << " are given)");
}
}
void update() {
for (Size i=0; i<n_-1; ++i) {
dx_[i] = this->xBegin_[i+1] - this->xBegin_[i];
S_[i] = (this->yBegin_[i+1] - this->yBegin_[i])/dx_[i];
}
// first derivative approximation
if (da_==CubicInterpolation::Spline) {
for (Size i=1; i<n_-1; ++i) {
L_.setMidRow(i, dx_[i], 2.0*(dx_[i]+dx_[i-1]), dx_[i-1]);
tmp_[i] = 3.0*(dx_[i]*S_[i-1] + dx_[i-1]*S_[i]);
}
// left boundary condition
switch (leftType_) {
case CubicInterpolation::NotAKnot:
// ignoring end condition value
L_.setFirstRow(dx_[1]*(dx_[1]+dx_[0]),
(dx_[0]+dx_[1])*(dx_[0]+dx_[1]));
tmp_[0] = S_[0]*dx_[1]*(2.0*dx_[1]+3.0*dx_[0]) +
S_[1]*dx_[0]*dx_[0];
break;
case CubicInterpolation::FirstDerivative:
L_.setFirstRow(1.0, 0.0);
tmp_[0] = leftValue_;
break;
case CubicInterpolation::SecondDerivative:
L_.setFirstRow(2.0, 1.0);
tmp_[0] = 3.0*S_[0] - leftValue_*dx_[0]/2.0;
break;
case CubicInterpolation::Periodic:
QL_FAIL("this end condition is not implemented yet");
case CubicInterpolation::Lagrange:
L_.setFirstRow(1.0, 0.0);
tmp_[0] = cubicInterpolatingPolynomialDerivative(
this->xBegin_[0],this->xBegin_[1],
this->xBegin_[2],this->xBegin_[3],
this->yBegin_[0],this->yBegin_[1],
this->yBegin_[2],this->yBegin_[3],
this->xBegin_[0]);
break;
default:
QL_FAIL("unknown end condition");
}
// right boundary condition
switch (rightType_) {
case CubicInterpolation::NotAKnot:
// ignoring end condition value
L_.setLastRow(-(dx_[n_-2]+dx_[n_-3])*(dx_[n_-2]+dx_[n_-3]),
-dx_[n_-3]*(dx_[n_-3]+dx_[n_-2]));
tmp_[n_-1] = -S_[n_-3]*dx_[n_-2]*dx_[n_-2] -
S_[n_-2]*dx_[n_-3]*(3.0*dx_[n_-2]+2.0*dx_[n_-3]);
break;
case CubicInterpolation::FirstDerivative:
L_.setLastRow(0.0, 1.0);
tmp_[n_-1] = rightValue_;
break;
case CubicInterpolation::SecondDerivative:
L_.setLastRow(1.0, 2.0);
tmp_[n_-1] = 3.0*S_[n_-2] + rightValue_*dx_[n_-2]/2.0;
break;
case CubicInterpolation::Periodic:
QL_FAIL("this end condition is not implemented yet");
case CubicInterpolation::Lagrange:
L_.setLastRow(0.0,1.0);
tmp_[n_-1] = cubicInterpolatingPolynomialDerivative(
this->xBegin_[n_-4],this->xBegin_[n_-3],
this->xBegin_[n_-2],this->xBegin_[n_-1],
this->yBegin_[n_-4],this->yBegin_[n_-3],
this->yBegin_[n_-2],this->yBegin_[n_-1],
this->xBegin_[n_-1]);
break;
default:
QL_FAIL("unknown end condition");
}
// solve the system
L_.solveFor(tmp_, tmp_);
} else if (da_==CubicInterpolation::SplineOM1) {
Matrix T_(n_-2, n_, 0.0);
for (Size i=0; i<n_-2; ++i) {
T_[i][i]=dx_[i]/6.0;
T_[i][i+1]=(dx_[i+1]+dx_[i])/3.0;
T_[i][i+2]=dx_[i+1]/6.0;
}
Matrix S_(n_-2, n_, 0.0);
for (Size i=0; i<n_-2; ++i) {
S_[i][i]=1.0/dx_[i];
S_[i][i+1]=-(1.0/dx_[i+1]+1.0/dx_[i]);
S_[i][i+2]=1.0/dx_[i+1];
}
Matrix Up_(n_, 2, 0.0);
Up_[0][0]=1;
Up_[n_-1][1]=1;
Matrix Us_(n_, n_-2, 0.0);
for (Size i=0; i<n_-2; ++i)
Us_[i+1][i]=1;
Matrix Z_ = Us_*inverse(T_*Us_);
Matrix I_(n_, n_, 0.0);
for (Size i=0; i<n_; ++i)
I_[i][i]=1;
Matrix V_ = (I_-Z_*T_)*Up_;
Matrix W_ = Z_*S_;
Matrix Q_(n_, n_, 0.0);
Q_[0][0]=1.0/(n_-1)*dx_[0]*dx_[0]*dx_[0];
Q_[0][1]=7.0/8*1.0/(n_-1)*dx_[0]*dx_[0]*dx_[0];
for (Size i=1; i<n_-1; ++i) {
Q_[i][i-1]=7.0/8*1.0/(n_-1)*dx_[i-1]*dx_[i-1]*dx_[i-1];
Q_[i][i]=1.0/(n_-1)*dx_[i]*dx_[i]*dx_[i]+1.0/(n_-1)*dx_[i-1]*dx_[i-1]*dx_[i-1];
Q_[i][i+1]=7.0/8*1.0/(n_-1)*dx_[i]*dx_[i]*dx_[i];
}
Q_[n_-1][n_-2]=7.0/8*1.0/(n_-1)*dx_[n_-2]*dx_[n_-2]*dx_[n_-2];
Q_[n_-1][n_-1]=1.0/(n_-1)*dx_[n_-2]*dx_[n_-2]*dx_[n_-2];
Matrix J_ = (I_-V_*inverse(transpose(V_)*Q_*V_)*transpose(V_)*Q_)*W_;
Array Y_(n_);
for (Size i=0; i<n_; ++i)
Y_[i]=this->yBegin_[i];
Array D_ = J_*Y_;
for (Size i=0; i<n_-1; ++i)
tmp_[i]=(Y_[i+1]-Y_[i])/dx_[i]-(2.0*D_[i]+D_[i+1])*dx_[i]/6.0;
tmp_[n_-1]=tmp_[n_-2]+D_[n_-2]*dx_[n_-2]+(D_[n_-1]-D_[n_-2])*dx_[n_-2]/2.0;
} else if (da_==CubicInterpolation::SplineOM2) {
Matrix T_(n_-2, n_, 0.0);
for (Size i=0; i<n_-2; ++i) {
T_[i][i]=dx_[i]/6.0;
T_[i][i+1]=(dx_[i]+dx_[i+1])/3.0;
T_[i][i+2]=dx_[i+1]/6.0;
}
Matrix S_(n_-2, n_, 0.0);
for (Size i=0; i<n_-2; ++i) {
S_[i][i]=1.0/dx_[i];
S_[i][i+1]=-(1.0/dx_[i+1]+1.0/dx_[i]);
S_[i][i+2]=1.0/dx_[i+1];
}
Matrix Up_(n_, 2, 0.0);
Up_[0][0]=1;
Up_[n_-1][1]=1;
Matrix Us_(n_, n_-2, 0.0);
for (Size i=0; i<n_-2; ++i)
Us_[i+1][i]=1;
Matrix Z_ = Us_*inverse(T_*Us_);
Matrix I_(n_, n_, 0.0);
for (Size i=0; i<n_; ++i)
I_[i][i]=1;
Matrix V_ = (I_-Z_*T_)*Up_;
Matrix W_ = Z_*S_;
Matrix Q_(n_, n_, 0.0);
Q_[0][0]=1.0/(n_-1)*dx_[0];
Q_[0][1]=1.0/2*1.0/(n_-1)*dx_[0];
for (Size i=1; i<n_-1; ++i) {
Q_[i][i-1]=1.0/2*1.0/(n_-1)*dx_[i-1];
Q_[i][i]=1.0/(n_-1)*dx_[i]+1.0/(n_-1)*dx_[i-1];
Q_[i][i+1]=1.0/2*1.0/(n_-1)*dx_[i];
}
Q_[n_-1][n_-2]=1.0/2*1.0/(n_-1)*dx_[n_-2];
Q_[n_-1][n_-1]=1.0/(n_-1)*dx_[n_-2];
Matrix J_ = (I_-V_*inverse(transpose(V_)*Q_*V_)*transpose(V_)*Q_)*W_;
Array Y_(n_);
for (Size i=0; i<n_; ++i)
Y_[i]=this->yBegin_[i];
Array D_ = J_*Y_;
for (Size i=0; i<n_-1; ++i)
tmp_[i]=(Y_[i+1]-Y_[i])/dx_[i]-(2.0*D_[i]+D_[i+1])*dx_[i]/6.0;
tmp_[n_-1]=tmp_[n_-2]+D_[n_-2]*dx_[n_-2]+(D_[n_-1]-D_[n_-2])*dx_[n_-2]/2.0;
} else { // local schemes
if (n_==2)
tmp_[0] = tmp_[1] = S_[0];
else {
switch (da_) {
case CubicInterpolation::FourthOrder:
QL_FAIL("FourthOrder not implemented yet");
break;
case CubicInterpolation::Parabolic:
// intermediate points
for (Size i=1; i<n_-1; ++i)
tmp_[i] = (dx_[i-1]*S_[i]+dx_[i]*S_[i-1])/(dx_[i]+dx_[i-1]);
// end points
tmp_[0] = ((2.0*dx_[ 0]+dx_[ 1])*S_[ 0] - dx_[ 0]*S_[ 1]) / (dx_[ 0]+dx_[ 1]);
tmp_[n_-1] = ((2.0*dx_[n_-2]+dx_[n_-3])*S_[n_-2] - dx_[n_-2]*S_[n_-3]) / (dx_[n_-2]+dx_[n_-3]);
break;
case CubicInterpolation::FritschButland:
// intermediate points
for (Size i=1; i<n_-1; ++i) {
Real Smin = std::min(S_[i-1], S_[i]);
Real Smax = std::max(S_[i-1], S_[i]);
tmp_[i] = 3.0*Smin*Smax/(Smax+2.0*Smin);
}
// end points
tmp_[0] = ((2.0*dx_[ 0]+dx_[ 1])*S_[ 0] - dx_[ 0]*S_[ 1]) / (dx_[ 0]+dx_[ 1]);
tmp_[n_-1] = ((2.0*dx_[n_-2]+dx_[n_-3])*S_[n_-2] - dx_[n_-2]*S_[n_-3]) / (dx_[n_-2]+dx_[n_-3]);
break;
case CubicInterpolation::Akima:
tmp_[0] = (std::abs(S_[1]-S_[0])*2*S_[0]*S_[1]+std::abs(2*S_[0]*S_[1]-4*S_[0]*S_[0]*S_[1])*S_[0])/(std::abs(S_[1]-S_[0])+std::abs(2*S_[0]*S_[1]-4*S_[0]*S_[0]*S_[1]));
tmp_[1] = (std::abs(S_[2]-S_[1])*S_[0]+std::abs(S_[0]-2*S_[0]*S_[1])*S_[1])/(std::abs(S_[2]-S_[1])+std::abs(S_[0]-2*S_[0]*S_[1]));
for (Size i=2; i<n_-2; ++i) {
if ((S_[i-2]==S_[i-1]) && (S_[i]!=S_[i+1]))
tmp_[i] = S_[i-1];
else if ((S_[i-2]!=S_[i-1]) && (S_[i]==S_[i+1]))
tmp_[i] = S_[i];
else if (S_[i]==S_[i-1])
tmp_[i] = S_[i];
else if ((S_[i-2]==S_[i-1]) && (S_[i-1]!=S_[i]) && (S_[i]==S_[i+1]))
tmp_[i] = (S_[i-1]+S_[i])/2.0;
else
tmp_[i] = (std::abs(S_[i+1]-S_[i])*S_[i-1]+std::abs(S_[i-1]-S_[i-2])*S_[i])/(std::abs(S_[i+1]-S_[i])+std::abs(S_[i-1]-S_[i-2]));
}
tmp_[n_-2] = (std::abs(2*S_[n_-2]*S_[n_-3]-S_[n_-2])*S_[n_-3]+std::abs(S_[n_-3]-S_[n_-4])*S_[n_-2])/(std::abs(2*S_[n_-2]*S_[n_-3]-S_[n_-2])+std::abs(S_[n_-3]-S_[n_-4]));
tmp_[n_-1] = (std::abs(4*S_[n_-2]*S_[n_-2]*S_[n_-3]-2*S_[n_-2]*S_[n_-3])*S_[n_-2]+std::abs(S_[n_-2]-S_[n_-3])*2*S_[n_-2]*S_[n_-3])/(std::abs(4*S_[n_-2]*S_[n_-2]*S_[n_-3]-2*S_[n_-2]*S_[n_-3])+std::abs(S_[n_-2]-S_[n_-3]));
break;
case CubicInterpolation::Kruger:
// intermediate points
for (Size i=1; i<n_-1; ++i) {
if (S_[i-1]*S_[i]<0.0)
// slope changes sign at point
tmp_[i] = 0.0;
else
// slope will be between the slopes of the adjacent
// straight lines and should approach zero if the
// slope of either line approaches zero
tmp_[i] = 2.0/(1.0/S_[i-1]+1.0/S_[i]);
}
// end points
tmp_[0] = (3.0*S_[0]-tmp_[1])/2.0;
tmp_[n_-1] = (3.0*S_[n_-2]-tmp_[n_-2])/2.0;
break;
default:
QL_FAIL("unknown scheme");
}
}
}
std::fill(monotonicityAdjustments_.begin(),
monotonicityAdjustments_.end(), false);
// Hyman monotonicity constrained filter
if (monotonic_) {
Real correction;
Real pm, pu, pd, M;
for (Size i=0; i<n_; ++i) {
if (i==0) {
if (tmp_[i]*S_[0]>0.0) {
correction = tmp_[i]/std::fabs(tmp_[i]) *
std::min<Real>(std::fabs(tmp_[i]),
std::fabs(3.0*S_[0]));
} else {
correction = 0.0;
}
if (correction!=tmp_[i]) {
tmp_[i] = correction;
monotonicityAdjustments_[i] = true;
}
} else if (i==n_-1) {
if (tmp_[i]*S_[n_-2]>0.0) {
correction = tmp_[i]/std::fabs(tmp_[i]) *
std::min<Real>(std::fabs(tmp_[i]),
std::fabs(3.0*S_[n_-2]));
} else {
correction = 0.0;
}
if (correction!=tmp_[i]) {
tmp_[i] = correction;
monotonicityAdjustments_[i] = true;
}
} else {
pm=(S_[i-1]*dx_[i]+S_[i]*dx_[i-1])/
(dx_[i-1]+dx_[i]);
M = 3.0 * std::min(std::min(std::fabs(S_[i-1]),
std::fabs(S_[i])),
std::fabs(pm));
if (i>1) {
if ((S_[i-1]-S_[i-2])*(S_[i]-S_[i-1])>0.0) {
pd=(S_[i-1]*(2.0*dx_[i-1]+dx_[i-2])
-S_[i-2]*dx_[i-1])/
(dx_[i-2]+dx_[i-1]);
if (pm*pd>0.0 && pm*(S_[i-1]-S_[i-2])>0.0) {
M = std::max<Real>(M, 1.5*std::min(
std::fabs(pm),std::fabs(pd)));
}
}
}
if (i<n_-2) {
if ((S_[i]-S_[i-1])*(S_[i+1]-S_[i])>0.0) {
pu=(S_[i]*(2.0*dx_[i]+dx_[i+1])-S_[i+1]*dx_[i])/
(dx_[i]+dx_[i+1]);
if (pm*pu>0.0 && -pm*(S_[i]-S_[i-1])>0.0) {
M = std::max<Real>(M, 1.5*std::min(
std::fabs(pm),std::fabs(pu)));
}
}
}
if (tmp_[i]*pm>0.0) {
correction = tmp_[i]/std::fabs(tmp_[i]) *
std::min(std::fabs(tmp_[i]), M);
} else {
correction = 0.0;
}
if (correction!=tmp_[i]) {
tmp_[i] = correction;
monotonicityAdjustments_[i] = true;
}
}
}
}
// cubic coefficients
for (Size i=0; i<n_-1; ++i) {
a_[i] = tmp_[i];
b_[i] = (3.0*S_[i] - tmp_[i+1] - 2.0*tmp_[i])/dx_[i];
c_[i] = (tmp_[i+1] + tmp_[i] - 2.0*S_[i])/(dx_[i]*dx_[i]);
}
primitiveConst_[0] = 0.0;
for (Size i=1; i<n_-1; ++i) {
primitiveConst_[i] = primitiveConst_[i-1]
+ dx_[i-1] *
(this->yBegin_[i-1] + dx_[i-1] *
(a_[i-1]/2.0 + dx_[i-1] *
(b_[i-1]/3.0 + dx_[i-1] * c_[i-1]/4.0)));
}
}
Real value(Real x) const {
Size j = this->locate(x);
Real dx_ = x-this->xBegin_[j];
return this->yBegin_[j] + dx_*(a_[j] + dx_*(b_[j] + dx_*c_[j]));
}
Real primitive(Real x) const {
Size j = this->locate(x);
Real dx_ = x-this->xBegin_[j];
return primitiveConst_[j]
+ dx_*(this->yBegin_[j] + dx_*(a_[j]/2.0
+ dx_*(b_[j]/3.0 + dx_*c_[j]/4.0)));
}
Real derivative(Real x) const {
Size j = this->locate(x);
Real dx_ = x-this->xBegin_[j];
return a_[j] + (2.0*b_[j] + 3.0*c_[j]*dx_)*dx_;
}
Real secondDerivative(Real x) const {
Size j = this->locate(x);
Real dx_ = x-this->xBegin_[j];
return 2.0*b_[j] + 6.0*c_[j]*dx_;
}
private:
CubicInterpolation::DerivativeApprox da_;
bool monotonic_;
CubicInterpolation::BoundaryCondition leftType_, rightType_;
Real leftValue_, rightValue_;
mutable Array tmp_;
mutable std::vector<Real> dx_, S_;
mutable TridiagonalOperator L_;
inline Real cubicInterpolatingPolynomialDerivative(
Real a, Real b, Real c, Real d,
Real u, Real v, Real w, Real z, Real x) const {
return (-((((a-c)*(b-c)*(c-x)*z-(a-d)*(b-d)*(d-x)*w)*(a-x+b-x)
+((a-c)*(b-c)*z-(a-d)*(b-d)*w)*(a-x)*(b-x))*(a-b)+
((a-c)*(a-d)*v-(b-c)*(b-d)*u)*(c-d)*(c-x)*(d-x)
+((a-c)*(a-d)*(a-x)*v-(b-c)*(b-d)*(b-x)*u)
*(c-x+d-x)*(c-d)))/
((a-b)*(a-c)*(a-d)*(b-c)*(b-d)*(c-d));
}
};
}
}
#endif
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