//! identity instance
public IOperator identity(int size)
{
TridiagonalOperator I = new TridiagonalOperator(new Vector(size - 1, 0.0), // lower diagonal
new Vector(size, 1.0), // diagonal
new Vector(size - 1, 0.0)); // upper diagonal
return(I);
}
public IOperator multiply(double a, IOperator o)
{
TridiagonalOperator D = o as TridiagonalOperator;
Vector low = D.lowerDiagonal_ * a,
mid = D.diagonal_ * a,
high = D.upperDiagonal_ * a;
TridiagonalOperator result = new TridiagonalOperator(low, mid, high);
return(result);
}
public IOperator subtract(IOperator A, IOperator B)
{
TridiagonalOperator D1 = A as TridiagonalOperator;
TridiagonalOperator D2 = B as TridiagonalOperator;
Vector low = D1.lowerDiagonal_ - D2.lowerDiagonal_,
mid = D1.diagonal_ - D2.diagonal_,
high = D1.upperDiagonal_ - D2.upperDiagonal_;
TridiagonalOperator result = new TridiagonalOperator(low, mid, high);
return(result);
}
public void generateOperator(double t, TransformedGrid tg, TridiagonalOperator L)
{
for (int i = 1; i < tg.size() - 1; i++)
{
double sigma = diffusion(t, tg.grid(i));
double nu = drift(t, tg.grid(i));
double r = discount(t, tg.grid(i));
double sigma2 = sigma * sigma;
double pd = -(sigma2 / tg.dxm(i) - nu) / tg.dx(i);
double pu = -(sigma2 / tg.dxp(i) + nu) / tg.dx(i);
double pm = sigma2 / (tg.dxm(i) * tg.dxp(i)) + r;
L.setMidRow(i, pd, pm, pu);
}
}
// interface
public override void applyBeforeApplying(IOperator o)
{
TridiagonalOperator L = o as TridiagonalOperator;
switch (side_)
{
case Side.Lower:
L.setFirstRow(-1.0, 1.0);
break;
case Side.Upper:
L.setLastRow(-1.0, 1.0);
break;
default:
throw new ArgumentException("unknown side for Neumann boundary condition");
}
}
public override void applyBeforeSolving(IOperator o, Vector rhs)
{
TridiagonalOperator L = o as TridiagonalOperator;
switch (side_)
{
case Side.Lower:
L.setFirstRow(-1.0, 1.0);
rhs[0] = value_;
break;
case Side.Upper:
L.setLastRow(-1.0, 1.0);
rhs[rhs.size() - 1] = value_;
break;
default:
throw new ArgumentException("unknown side for Neumann boundary condition");
}
}
protected void initializeOperator()
{
finiteDifferenceOperator_ = OperatorFactory.getOperator(process_, intrinsicValues_.grid(),
getResidualTime(), timeDependent_);
}
public override void calculate(IPricingEngineResults r)
{
OneAssetOption.Results results = r as OneAssetOption.Results;
setGridLimits();
initializeInitialCondition();
initializeOperator();
initializeBoundaryConditions();
initializeStepCondition();
List <IOperator> operatorSet = new List <IOperator>();
List <Vector> arraySet = new List <Vector>();
BoundaryConditionSet bcSet = new BoundaryConditionSet();
StepConditionSet <Vector> conditionSet = new StepConditionSet <Vector>();
prices_ = (SampledCurve)intrinsicValues_.Clone();
controlPrices_ = (SampledCurve)intrinsicValues_.Clone();
controlOperator_ = (TridiagonalOperator)finiteDifferenceOperator_.Clone();
controlBCs_[0] = BCs_[0];
controlBCs_[1] = BCs_[1];
operatorSet.Add(finiteDifferenceOperator_);
operatorSet.Add(controlOperator_);
arraySet.Add(prices_.values());
arraySet.Add(controlPrices_.values());
bcSet.Add(BCs_);
bcSet.Add(controlBCs_);
conditionSet.Add(stepCondition_);
conditionSet.Add(new NullCondition <Vector>());
var model = new FiniteDifferenceModel <ParallelEvolver <CrankNicolson <TridiagonalOperator> > >(operatorSet, bcSet);
object temp = arraySet;
model.rollback(ref temp, getResidualTime(), 0.0, timeSteps_, conditionSet);
arraySet = (List <Vector>)temp;
prices_.setValues(arraySet[0]);
controlPrices_.setValues(arraySet[1]);
StrikedTypePayoff striked_payoff = payoff_ as StrikedTypePayoff;
Utils.QL_REQUIRE(striked_payoff != null, () => "non-striked payoff given");
double variance = process_.blackVolatility().link.blackVariance(exerciseDate_, striked_payoff.strike());
double dividendDiscount = process_.dividendYield().link.discount(exerciseDate_);
double riskFreeDiscount = process_.riskFreeRate().link.discount(exerciseDate_);
double spot = process_.stateVariable().link.value();
double forwardPrice = spot * dividendDiscount / riskFreeDiscount;
BlackCalculator black = new BlackCalculator(striked_payoff, forwardPrice, Math.Sqrt(variance), riskFreeDiscount);
results.value = prices_.valueAtCenter()
- controlPrices_.valueAtCenter()
+ black.value();
results.delta = prices_.firstDerivativeAtCenter()
- controlPrices_.firstDerivativeAtCenter()
+ black.delta(spot);
results.gamma = prices_.secondDerivativeAtCenter()
- controlPrices_.secondDerivativeAtCenter()
+ black.gamma(spot);
results.additionalResults["priceCurve"] = prices_;
}
public IOperator subtract(IOperator A, IOperator B)
{
TridiagonalOperator D1 = A as TridiagonalOperator;
TridiagonalOperator D2 = B as TridiagonalOperator;
Vector low = D1.lowerDiagonal_-D2.lowerDiagonal_,
mid = D1.diagonal_-D2.diagonal_,
high = D1.upperDiagonal_-D2.upperDiagonal_;
TridiagonalOperator result = new TridiagonalOperator(low,mid,high);
return result;
}
public IOperator multiply(double a, IOperator o)
{
TridiagonalOperator D = o as TridiagonalOperator;
Vector low = D.lowerDiagonal_*a,
mid = D.diagonal_*a,
high = D.upperDiagonal_*a;
TridiagonalOperator result = new TridiagonalOperator(low,mid,high);
return result;
}
public override void calculate(IPricingEngineResults r)
{
OneAssetOption.Results results = r as OneAssetOption.Results;
setGridLimits();
initializeInitialCondition();
initializeOperator();
initializeBoundaryConditions();
initializeStepCondition();
// typedef StandardSystemFiniteDifferenceModel model_type;
List<IOperator> operatorSet = new List<IOperator>();
List<Vector> arraySet = new List<Vector>();
BoundaryConditionSet bcSet = new BoundaryConditionSet();
StepConditionSet<Vector> conditionSet = new StepConditionSet<Vector>();
prices_ = (SampledCurve)intrinsicValues_.Clone();
controlPrices_ = (SampledCurve)intrinsicValues_.Clone();
controlOperator_ = (TridiagonalOperator)finiteDifferenceOperator_.Clone();
controlBCs_[0] = BCs_[0];
controlBCs_[1] = BCs_[1];
operatorSet.Add(finiteDifferenceOperator_);
operatorSet.Add(controlOperator_);
arraySet.Add(prices_.values());
arraySet.Add(controlPrices_.values());
bcSet.Add(BCs_);
bcSet.Add(controlBCs_);
conditionSet.Add(stepCondition_);
conditionSet.Add(new NullCondition<Vector>());
var model = new FiniteDifferenceModel<ParallelEvolver<CrankNicolson<TridiagonalOperator>>>(operatorSet, bcSet);
object temp = arraySet;
model.rollback(ref temp, getResidualTime(), 0.0, timeSteps_, conditionSet);
arraySet = (List<Vector>)temp;
prices_.setValues(arraySet[0]);
controlPrices_.setValues(arraySet[1]);
StrikedTypePayoff striked_payoff = payoff_ as StrikedTypePayoff;
if (striked_payoff == null)
throw new ApplicationException("non-striked payoff given");
double variance = process_.blackVolatility().link.blackVariance(exerciseDate_, striked_payoff.strike());
double dividendDiscount = process_.dividendYield().link.discount(exerciseDate_);
double riskFreeDiscount = process_.riskFreeRate().link.discount(exerciseDate_);
double spot = process_.stateVariable().link.value();
double forwardPrice = spot * dividendDiscount / riskFreeDiscount;
BlackCalculator black = new BlackCalculator(striked_payoff, forwardPrice, Math.Sqrt(variance), riskFreeDiscount);
results.value = prices_.valueAtCenter()
- controlPrices_.valueAtCenter()
+ black.value();
results.delta = prices_.firstDerivativeAtCenter()
- controlPrices_.firstDerivativeAtCenter()
+ black.delta(spot);
results.gamma = prices_.secondDerivativeAtCenter()
- controlPrices_.secondDerivativeAtCenter()
+ black.gamma(spot);
results.additionalResults.Add("priceCurve", prices_);
}
public override void update()
{
TridiagonalOperator L = new TridiagonalOperator(cH_.n_);
Array<double> tmp = new Array<double>(cH_.n_);
var dx = new Array<double>(cH_.n_ - 1);
var S = new Array<double>(cH_.n_ - 1);
int i=0;
dx[i] = this.xBegin_[i+1] - this.xBegin_[i];
S[i] = (this.yBegin_[i+1] - this.yBegin_[i])/dx[i];
for (i=1; i<cH_.n_-1; ++i) {
dx[i] = this.xBegin_[i+1] - this.xBegin_[i];
S[i] = (this.yBegin_[i+1] - this.yBegin_[i])/dx[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]);
}
/**** BOUNDARY CONDITIONS ****/
// left condition
switch (leftType_)
{
case CubicSplineInterpolation.BoundaryCondition.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 CubicSplineInterpolation.BoundaryCondition.FirstDerivative:
L.setFirstRow(1.0, 0.0);
tmp[0] = leftValue_;
break;
case CubicSplineInterpolation.BoundaryCondition.SecondDerivative:
L.setFirstRow(2.0, 1.0);
tmp[0] = 3.0*S[0] - leftValue_*dx[0]/2.0;
break;
case CubicSplineInterpolation.BoundaryCondition.Periodic:
case CubicSplineInterpolation.BoundaryCondition.Lagrange:
// ignoring end condition value
throw new ApplicationException("this end condition is not implemented yet");
default:
throw new ApplicationException("unknown end condition");
}
// right condition
switch (rightType_)
{
case CubicSplineInterpolation.BoundaryCondition.NotAKnot:
// ignoring end condition value
L.setLastRow(-(dx[cH_.n_-2]+dx[cH_.n_-3])*(dx[cH_.n_-2]+dx[cH_.n_-3]),
-dx[cH_.n_-3]*(dx[cH_.n_-3]+dx[cH_.n_-2]));
tmp[cH_.n_-1] = -S[cH_.n_-3]*dx[cH_.n_-2]*dx[cH_.n_-2] -
S[cH_.n_-2]*dx[cH_.n_-3]*(3.0*dx[cH_.n_-2]+2.0*dx[cH_.n_-3]);
break;
case CubicSplineInterpolation.BoundaryCondition.FirstDerivative:
L.setLastRow(0.0, 1.0);
tmp[cH_.n_-1] = rightValue_;
break;
case CubicSplineInterpolation.BoundaryCondition.SecondDerivative:
L.setLastRow(1.0, 2.0);
tmp[cH_.n_-1] = 3.0*S[cH_.n_-2] + rightValue_*dx[cH_.n_-2]/2.0;
break;
case CubicSplineInterpolation.BoundaryCondition.Periodic:
case CubicSplineInterpolation.BoundaryCondition.Lagrange:
// ignoring end condition value
throw new ApplicationException("this end condition is not implemented yet");
default:
throw new ApplicationException("unknown end condition");
}
// solve the system
tmp = L.solveFor(tmp);
for (int j = 0; i < cH_.monotonicityAdjustments_.Count; j++ )
cH_.monotonicityAdjustments_[j] = false;
if (constrained_)
{
double correction;
double pm, pu, pd, M;
for (i=0; i<cH_.n_; ++i) {
if (i==0) {
if (tmp[i]*S[0]>0.0)
{
correction = tmp[i]/Math.Abs(tmp[i]) *
Math.Min(Math.Abs(tmp[i]),
Math.Abs(3.0*S[0]));
}
else
{
correction = 0.0;
}
if (correction!=tmp[i])
{
tmp[i] = correction;
cH_.monotonicityAdjustments_[i] = true;
}
}
else if (i==cH_.n_-1)
{
if (tmp[i]*S[cH_.n_-2]>0.0)
{
correction = tmp[i]/Math.Abs(tmp[i]) *
Math.Min(Math.Abs(tmp[i]),
Math.Abs(3.0*S[cH_.n_-2]));
}
else
{
correction = 0.0;
}
if (correction!=tmp[i])
{
tmp[i] = correction;
cH_.monotonicityAdjustments_[i] = true;
}
}
else
{
pm=(S[i-1]*dx[i]+S[i]*dx[i-1])/
(dx[i-1]+dx[i]);
M = 3.0 * Math.Min(Math.Min(Math.Abs(S[i-1]),
Math.Abs(S[i])),
Math.Abs(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 = Math.Max(M, 1.5*Math.Min(
Math.Abs(pm),Math.Abs(pd)));
}
}
}
if (i<cH_.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 = Math.Max(M, 1.5*Math.Min(
Math.Abs(pm),Math.Abs(pu)));
}
}
}
if (tmp[i]*pm>0.0)
{
correction = tmp[i]/Math.Abs(tmp[i]) *
Math.Min(Math.Abs(tmp[i]), M);
}
else
{
correction = 0.0;
}
if (correction!=tmp[i])
{
tmp[i] = correction;
cH_.monotonicityAdjustments_[i] = true;
}
}
}
}
for (i=0; i<cH_.n_-1; ++i)
{
cH_.a_[i] = tmp[i];
cH_.b_[i] = (3.0*S[i] - tmp[i+1] - 2.0*tmp[i])/dx[i];
cH_.c_[i] = (tmp[i+1] + tmp[i] - 2.0*S[i])/(dx[i]*dx[i]);
}
cH_.primitiveConst_[0] = 0.0;
for (i=1; i<cH_.n_-1; ++i)
{
cH_.primitiveConst_[i] = cH_.primitiveConst_[i-1]
+ dx[i-1] *
(this.yBegin_[i-1] + dx[i-1] *
(cH_.a_[i-1]/2.0 + dx[i-1] *
(cH_.b_[i-1]/3.0 + dx[i-1] * cH_.c_[i-1]/4.0)));
}
}
public override void update()
{
Vector tmp = new Vector(size_);
List <double> dx = new InitializedList <double>(size_ - 1),
S = new InitializedList <double>(size_ - 1);
for (int i = 0; i < size_ - 1; ++i)
{
dx[i] = xBegin_[i + 1] - xBegin_[i];
S[i] = (yBegin_[i + 1] - yBegin_[i]) / dx[i];
}
// first derivative approximation
if (da_ == CubicInterpolation.DerivativeApprox.Spline)
{
TridiagonalOperator L = new TridiagonalOperator(size_);
for (int i = 1; i < size_ - 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.BoundaryCondition.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.BoundaryCondition.FirstDerivative:
L.setFirstRow(1.0, 0.0);
tmp[0] = leftValue_;
break;
case CubicInterpolation.BoundaryCondition.SecondDerivative:
L.setFirstRow(2.0, 1.0);
tmp[0] = 3.0 * S[0] - leftValue_ * dx[0] / 2.0;
break;
case CubicInterpolation.BoundaryCondition.Periodic:
// ignoring end condition value
throw new NotImplementedException("this end condition is not implemented yet");
case CubicInterpolation.BoundaryCondition.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:
throw new ArgumentException("unknown end condition");
}
// right boundary condition
switch (rightType_)
{
case CubicInterpolation.BoundaryCondition.NotAKnot:
// ignoring end condition value
L.setLastRow(-(dx[size_ - 2] + dx[size_ - 3]) * (dx[size_ - 2] + dx[size_ - 3]),
-dx[size_ - 3] * (dx[size_ - 3] + dx[size_ - 2]));
tmp[size_ - 1] = -S[size_ - 3] * dx[size_ - 2] * dx[size_ - 2] -
S[size_ - 2] * dx[size_ - 3] * (3.0 * dx[size_ - 2] + 2.0 * dx[size_ - 3]);
break;
case CubicInterpolation.BoundaryCondition.FirstDerivative:
L.setLastRow(0.0, 1.0);
tmp[size_ - 1] = rightValue_;
break;
case CubicInterpolation.BoundaryCondition.SecondDerivative:
L.setLastRow(1.0, 2.0);
tmp[size_ - 1] = 3.0 * S[size_ - 2] + rightValue_ * dx[size_ - 2] / 2.0;
break;
case CubicInterpolation.BoundaryCondition.Periodic:
// ignoring end condition value
throw new NotImplementedException("this end condition is not implemented yet");
case CubicInterpolation.BoundaryCondition.Lagrange:
L.setLastRow(0.0, 1.0);
tmp[size_ - 1] = cubicInterpolatingPolynomialDerivative(
this.xBegin_[size_ - 4], this.xBegin_[size_ - 3],
this.xBegin_[size_ - 2], this.xBegin_[size_ - 1],
this.yBegin_[size_ - 4], this.yBegin_[size_ - 3],
this.yBegin_[size_ - 2], this.yBegin_[size_ - 1],
this.xBegin_[size_ - 1]);
break;
default:
throw new ArgumentException("unknown end condition");
}
// solve the system
tmp = L.solveFor(tmp);
}
else // local schemes
{
if (size_ == 2)
{
tmp[0] = tmp[1] = S[0];
}
else
{
switch (da_)
{
case CubicInterpolation.DerivativeApprox.FourthOrder:
throw new NotImplementedException("FourthOrder not implemented yet");
case CubicInterpolation.DerivativeApprox.Parabolic:
// intermediate points
for (int i = 1; i < size_ - 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[size_ - 1] = ((2.0 * dx[size_ - 2] + dx[size_ - 3]) * S[size_ - 2] - dx[size_ - 2] * S[size_ - 3]) / (dx[size_ - 2] + dx[size_ - 3]);
break;
case CubicInterpolation.DerivativeApprox.FritschButland:
// intermediate points
for (int i = 1; i < size_ - 1; ++i)
{
double Smin = Math.Min(S[i - 1], S[i]);
double Smax = Math.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[size_ - 1] = ((2.0 * dx[size_ - 2] + dx[size_ - 3]) * S[size_ - 2] - dx[size_ - 2] * S[size_ - 3]) / (dx[size_ - 2] + dx[size_ - 3]);
break;
case CubicInterpolation.DerivativeApprox.Akima:
tmp[0] = (Math.Abs(S[1] - S[0]) * 2 * S[0] * S[1] + Math.Abs(2 * S[0] * S[1] - 4 * S[0] * S[0] * S[1]) * S[0]) / (Math.Abs(S[1] - S[0]) + Math.Abs(2 * S[0] * S[1] - 4 * S[0] * S[0] * S[1]));
tmp[1] = (Math.Abs(S[2] - S[1]) * S[0] + Math.Abs(S[0] - 2 * S[0] * S[1]) * S[1]) / (Math.Abs(S[2] - S[1]) + Math.Abs(S[0] - 2 * S[0] * S[1]));
for (int i = 2; i < size_ - 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] = (Math.Abs(S[i + 1] - S[i]) * S[i - 1] + Math.Abs(S[i - 1] - S[i - 2]) * S[i]) / (Math.Abs(S[i + 1] - S[i]) + Math.Abs(S[i - 1] - S[i - 2]));
}
}
tmp[size_ - 2] = (Math.Abs(2 * S[size_ - 2] * S[size_ - 3] - S[size_ - 2]) * S[size_ - 3] + Math.Abs(S[size_ - 3] - S[size_ - 4]) * S[size_ - 2]) / (Math.Abs(2 * S[size_ - 2] * S[size_ - 3] - S[size_ - 2]) + Math.Abs(S[size_ - 3] - S[size_ - 4]));
tmp[size_ - 1] = (Math.Abs(4 * S[size_ - 2] * S[size_ - 2] * S[size_ - 3] - 2 * S[size_ - 2] * S[size_ - 3]) * S[size_ - 2] + Math.Abs(S[size_ - 2] - S[size_ - 3]) * 2 * S[size_ - 2] * S[size_ - 3]) / (Math.Abs(4 * S[size_ - 2] * S[size_ - 2] * S[size_ - 3] - 2 * S[size_ - 2] * S[size_ - 3]) + Math.Abs(S[size_ - 2] - S[size_ - 3]));
break;
case CubicInterpolation.DerivativeApprox.Kruger:
// intermediate points
for (int i = 1; i < size_ - 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[size_ - 1] = (3.0 * S[size_ - 2] - tmp[size_ - 2]) / 2.0;
break;
default:
throw new ArgumentException("unknown scheme");
}
}
}
monotonicityAdjustments_.Erase();
// Hyman monotonicity constrained filter
if (monotonic_)
{
double correction;
double pm, pu, pd, M;
for (int i = 0; i < size_; ++i)
{
if (i == 0)
{
if (tmp[i] * S[0] > 0.0)
{
correction = tmp[i] / Math.Abs(tmp[i]) *
Math.Min(Math.Abs(tmp[i]),
Math.Abs(3.0 * S[0]));
}
else
{
correction = 0.0;
}
if (correction != tmp[i])
{
tmp[i] = correction;
monotonicityAdjustments_[i] = true;
}
}
else if (i == size_ - 1)
{
if (tmp[i] * S[size_ - 2] > 0.0)
{
correction = tmp[i] / Math.Abs(tmp[i]) *
Math.Min(Math.Abs(tmp[i]), Math.Abs(3.0 * S[size_ - 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 * Math.Min(Math.Min(Math.Abs(S[i - 1]), Math.Abs(S[i])),
Math.Abs(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 = Math.Max(M, 1.5 * Math.Min(
Math.Abs(pm), Math.Abs(pd)));
}
}
}
if (i < size_ - 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 = Math.Max(M, 1.5 * Math.Min(
Math.Abs(pm), Math.Abs(pu)));
}
}
}
if (tmp[i] * pm > 0.0)
{
correction = tmp[i] / Math.Abs(tmp[i]) *
Math.Min(Math.Abs(tmp[i]), M);
}
else
{
correction = 0.0;
}
if (correction != tmp[i])
{
tmp[i] = correction;
monotonicityAdjustments_[i] = true;
}
}
}
}
// cubic coefficients
for (int i = 0; i < size_ - 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 (int i = 1; i < size_ - 1; ++i)
{
primitiveConst_[i] = primitiveConst_[i - 1]
+ dx[i - 1] *
(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)));
}
}
public override void update()
{
Vector tmp = new Vector(size_);
List <double> dx = new InitializedList <double>(size_ - 1),
S = new InitializedList <double>(size_ - 1);
for (int i = 0; i < size_ - 1; ++i)
{
dx[i] = xBegin_[i + 1] - xBegin_[i];
S[i] = (yBegin_[i + 1] - yBegin_[i]) / dx[i];
}
// first derivative approximation
if (da_ == CubicInterpolation.DerivativeApprox.Spline)
{
TridiagonalOperator L = new TridiagonalOperator(size_);
for (int i = 1; i < size_ - 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.BoundaryCondition.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.BoundaryCondition.FirstDerivative:
L.setFirstRow(1.0, 0.0);
tmp[0] = leftValue_;
break;
case CubicInterpolation.BoundaryCondition.SecondDerivative:
L.setFirstRow(2.0, 1.0);
tmp[0] = 3.0 * S[0] - leftValue_ * dx[0] / 2.0;
break;
case CubicInterpolation.BoundaryCondition.Periodic:
// ignoring end condition value
throw new NotImplementedException("this end condition is not implemented yet");
case CubicInterpolation.BoundaryCondition.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:
throw new ArgumentException("unknown end condition");
}
// right boundary condition
switch (rightType_)
{
case CubicInterpolation.BoundaryCondition.NotAKnot:
// ignoring end condition value
L.setLastRow(-(dx[size_ - 2] + dx[size_ - 3]) * (dx[size_ - 2] + dx[size_ - 3]),
-dx[size_ - 3] * (dx[size_ - 3] + dx[size_ - 2]));
tmp[size_ - 1] = -S[size_ - 3] * dx[size_ - 2] * dx[size_ - 2] -
S[size_ - 2] * dx[size_ - 3] * (3.0 * dx[size_ - 2] + 2.0 * dx[size_ - 3]);
break;
case CubicInterpolation.BoundaryCondition.FirstDerivative:
L.setLastRow(0.0, 1.0);
tmp[size_ - 1] = rightValue_;
break;
case CubicInterpolation.BoundaryCondition.SecondDerivative:
L.setLastRow(1.0, 2.0);
tmp[size_ - 1] = 3.0 * S[size_ - 2] + rightValue_ * dx[size_ - 2] / 2.0;
break;
case CubicInterpolation.BoundaryCondition.Periodic:
// ignoring end condition value
throw new NotImplementedException("this end condition is not implemented yet");
case CubicInterpolation.BoundaryCondition.Lagrange:
L.setLastRow(0.0, 1.0);
tmp[size_ - 1] = cubicInterpolatingPolynomialDerivative(
this.xBegin_[size_ - 4], this.xBegin_[size_ - 3],
this.xBegin_[size_ - 2], this.xBegin_[size_ - 1],
this.yBegin_[size_ - 4], this.yBegin_[size_ - 3],
this.yBegin_[size_ - 2], this.yBegin_[size_ - 1],
this.xBegin_[size_ - 1]);
break;
default:
throw new ArgumentException("unknown end condition");
}
// solve the system
tmp = L.solveFor(tmp);
}
else if (da_ == CubicInterpolation.DerivativeApprox.SplineOM1)
{
Matrix T_ = new Matrix(size_ - 2, size_, 0.0);
for (int i = 0; i < size_ - 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_ = new Matrix(size_ - 2, size_, 0.0);
for (int i = 0; i < size_ - 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_ = new Matrix(size_, 2, 0.0);
Up_[0, 0] = 1;
Up_[size_ - 1, 1] = 1;
Matrix Us_ = new Matrix(size_, size_ - 2, 0.0);
for (int i = 0; i < size_ - 2; ++i)
{
Us_[i + 1, i] = 1;
}
Matrix Z_ = Us_ * Matrix.inverse(T_ * Us_);
Matrix I_ = new Matrix(size_, size_, 0.0);
for (int i = 0; i < size_; ++i)
{
I_[i, i] = 1;
}
Matrix V_ = (I_ - Z_ * T_) * Up_;
Matrix W_ = Z_ * S_;
Matrix Q_ = new Matrix(size_, size_, 0.0);
Q_[0, 0] = 1.0 / (size_ - 1) * dx[0] * dx[0] * dx[0];
Q_[0, 1] = 7.0 / 8 * 1.0 / (size_ - 1) * dx[0] * dx[0] * dx[0];
for (int i = 1; i < size_ - 1; ++i)
{
Q_[i, i - 1] = 7.0 / 8 * 1.0 / (size_ - 1) * dx[i - 1] * dx[i - 1] * dx[i - 1];
Q_[i, i] = 1.0 / (size_ - 1) * dx[i] * dx[i] * dx[i] + 1.0 / (size_ - 1) * dx[i - 1] * dx[i - 1] * dx[i - 1];
Q_[i, i + 1] = 7.0 / 8 * 1.0 / (size_ - 1) * dx[i] * dx[i] * dx[i];
}
Q_[size_ - 1, size_ - 2] = 7.0 / 8 * 1.0 / (size_ - 1) * dx[size_ - 2] * dx[size_ - 2] * dx[size_ - 2];
Q_[size_ - 1, size_ - 1] = 1.0 / (size_ - 1) * dx[size_ - 2] * dx[size_ - 2] * dx[size_ - 2];
Matrix J_ = (I_ - V_ * Matrix.inverse(Matrix.transpose(V_) * Q_ * V_) * Matrix.transpose(V_) * Q_) * W_;
Vector Y_ = new Vector(size_);
for (int i = 0; i < size_; ++i)
{
Y_[i] = this.yBegin_[i];
}
Vector D_ = J_ * Y_;
for (int i = 0; i < size_ - 1; ++i)
{
tmp[i] = (Y_[i + 1] - Y_[i]) / dx[i] - (2.0 * D_[i] + D_[i + 1]) * dx[i] / 6.0;
}
tmp[size_ - 1] = tmp[size_ - 2] + D_[size_ - 2] * dx[size_ - 2] + (D_[size_ - 1] - D_[size_ - 2]) * dx[size_ - 2] / 2.0;
}
else if (da_ == CubicInterpolation.DerivativeApprox.SplineOM2)
{
Matrix T_ = new Matrix(size_ - 2, size_, 0.0);
for (int i = 0; i < size_ - 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_ = new Matrix(size_ - 2, size_, 0.0);
for (int i = 0; i < size_ - 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_ = new Matrix(size_, 2, 0.0);
Up_[0, 0] = 1;
Up_[size_ - 1, 1] = 1;
Matrix Us_ = new Matrix(size_, size_ - 2, 0.0);
for (int i = 0; i < size_ - 2; ++i)
{
Us_[i + 1, i] = 1;
}
Matrix Z_ = Us_ * Matrix.inverse(T_ * Us_);
Matrix I_ = new Matrix(size_, size_, 0.0);
for (int i = 0; i < size_; ++i)
{
I_[i, i] = 1;
}
Matrix V_ = (I_ - Z_ * T_) * Up_;
Matrix W_ = Z_ * S_;
Matrix Q_ = new Matrix(size_, size_, 0.0);
Q_[0, 0] = 1.0 / (size_ - 1) * dx[0];
Q_[0, 1] = 1.0 / 2 * 1.0 / (size_ - 1) * dx[0];
for (int i = 1; i < size_ - 1; ++i)
{
Q_[i, i - 1] = 1.0 / 2 * 1.0 / (size_ - 1) * dx[i - 1];
Q_[i, i] = 1.0 / (size_ - 1) * dx[i] + 1.0 / (size_ - 1) * dx[i - 1];
Q_[i, i + 1] = 1.0 / 2 * 1.0 / (size_ - 1) * dx[i];
}
Q_[size_ - 1, size_ - 2] = 1.0 / 2 * 1.0 / (size_ - 1) * dx[size_ - 2];
Q_[size_ - 1, size_ - 1] = 1.0 / (size_ - 1) * dx[size_ - 2];
Matrix J_ = (I_ - V_ * Matrix.inverse(Matrix.transpose(V_) * Q_ * V_) * Matrix.transpose(V_) * Q_) * W_;
Vector Y_ = new Vector(size_);
for (int i = 0; i < size_; ++i)
{
Y_[i] = this.yBegin_[i];
}
Vector D_ = J_ * Y_;
for (int i = 0; i < size_ - 1; ++i)
{
tmp[i] = (Y_[i + 1] - Y_[i]) / dx[i] - (2.0 * D_[i] + D_[i + 1]) * dx[i] / 6.0;
}
tmp[size_ - 1] = tmp[size_ - 2] + D_[size_ - 2] * dx[size_ - 2] + (D_[size_ - 1] - D_[size_ - 2]) * dx[size_ - 2] / 2.0;
}
else
{
// local schemes
if (size_ == 2)
{
tmp[0] = tmp[1] = S[0];
}
else
{
switch (da_)
{
case CubicInterpolation.DerivativeApprox.FourthOrder:
throw new NotImplementedException("FourthOrder not implemented yet");
case CubicInterpolation.DerivativeApprox.Parabolic:
// intermediate points
for (int i = 1; i < size_ - 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[size_ - 1] = ((2.0 * dx[size_ - 2] + dx[size_ - 3]) * S[size_ - 2] -
dx[size_ - 2] * S[size_ - 3]) / (dx[size_ - 2] + dx[size_ - 3]);
break;
case CubicInterpolation.DerivativeApprox.FritschButland:
// intermediate points
for (int i = 1; i < size_ - 1; ++i)
{
double Smin = Math.Min(S[i - 1], S[i]);
double Smax = Math.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[size_ - 1] = ((2.0 * dx[size_ - 2] + dx[size_ - 3]) * S[size_ - 2] -
dx[size_ - 2] * S[size_ - 3]) / (dx[size_ - 2] + dx[size_ - 3]);
break;
case CubicInterpolation.DerivativeApprox.Akima:
tmp[0] = (Math.Abs(S[1] - S[0]) * 2 * S[0] * S[1] +
Math.Abs(2 * S[0] * S[1] - 4 * S[0] * S[0] * S[1]) * S[0]) /
(Math.Abs(S[1] - S[0]) + Math.Abs(2 * S[0] * S[1] - 4 * S[0] * S[0] * S[1]));
tmp[1] = (Math.Abs(S[2] - S[1]) * S[0] + Math.Abs(S[0] - 2 * S[0] * S[1]) * S[1]) /
(Math.Abs(S[2] - S[1]) + Math.Abs(S[0] - 2 * S[0] * S[1]));
for (int i = 2; i < size_ - 2; ++i)
{
if ((S[i - 2].IsEqual(S[i - 1])) && (S[i].IsNotEqual(S[i + 1])))
{
tmp[i] = S[i - 1];
}
else if ((S[i - 2].IsNotEqual(S[i - 1])) && (S[i].IsEqual(S[i + 1])))
{
tmp[i] = S[i];
}
else if (S[i].IsEqual(S[i - 1]))
{
tmp[i] = S[i];
}
else if ((S[i - 2].IsEqual(S[i - 1])) && (S[i - 1].IsNotEqual(S[i])) && (S[i].IsEqual(S[i + 1])))
{
tmp[i] = (S[i - 1] + S[i]) / 2.0;
}
else
{
tmp[i] = (Math.Abs(S[i + 1] - S[i]) * S[i - 1] + Math.Abs(S[i - 1] - S[i - 2]) * S[i]) /
(Math.Abs(S[i + 1] - S[i]) + Math.Abs(S[i - 1] - S[i - 2]));
}
}
tmp[size_ - 2] = (Math.Abs(2 * S[size_ - 2] * S[size_ - 3] - S[size_ - 2]) * S[size_ - 3] +
Math.Abs(S[size_ - 3] - S[size_ - 4]) * S[size_ - 2]) /
(Math.Abs(2 * S[size_ - 2] * S[size_ - 3] - S[size_ - 2]) +
Math.Abs(S[size_ - 3] - S[size_ - 4]));
tmp[size_ - 1] =
(Math.Abs(4 * S[size_ - 2] * S[size_ - 2] * S[size_ - 3] - 2 * S[size_ - 2] * S[size_ - 3]) *
S[size_ - 2] + Math.Abs(S[size_ - 2] - S[size_ - 3]) * 2 * S[size_ - 2] * S[size_ - 3]) /
(Math.Abs(4 * S[size_ - 2] * S[size_ - 2] * S[size_ - 3] - 2 * S[size_ - 2] * S[size_ - 3]) +
Math.Abs(S[size_ - 2] - S[size_ - 3]));
break;
case CubicInterpolation.DerivativeApprox.Kruger:
// intermediate points
for (int i = 1; i < size_ - 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[size_ - 1] = (3.0 * S[size_ - 2] - tmp[size_ - 2]) / 2.0;
break;
case CubicInterpolation.DerivativeApprox.Harmonic:
// intermediate points
for (int i = 1; i < size_ - 1; ++i)
{
double w1 = 2 * dx[i] + dx[i - 1];
double w2 = dx[i] + 2 * dx[i - 1];
if (S[i - 1] * S[i] <= 0.0)
{
// slope changes sign at point
tmp[i] = 0.0;
}
else
{
// weighted harmonic mean of S[i] and S[i-1] if they
// have the same sign; otherwise 0
tmp[i] = (w1 + w2) / (w1 / S[i - 1] + w2 / S[i]);
}
}
// end points [0]
tmp[0] = ((2 * dx[0] + dx[1]) * S[0] - dx[0] * S[1]) / (dx[1] + dx[0]);
if (tmp[0] * S[0] < 0.0)
{
tmp[0] = 0;
}
else if (S[0] * S[1] < 0)
{
if (Math.Abs(tmp[0]) > Math.Abs(3 * S[0]))
{
tmp[0] = 3 * S[0];
}
}
// end points [n-1]
tmp[size_ - 1] = ((2 * dx[size_ - 2] + dx[size_ - 3]) * S[size_ - 2] - dx[size_ - 2] * S[size_ - 3]) / (dx[size_ - 3] + dx[size_ - 2]);
if (tmp[size_ - 1] * S[size_ - 2] < 0.0)
{
tmp[size_ - 1] = 0;
}
else if (S[size_ - 2] * S[size_ - 3] < 0)
{
if (Math.Abs(tmp[size_ - 1]) > Math.Abs(3 * S[size_ - 2]))
{
tmp[size_ - 1] = 3 * S[size_ - 2];
}
}
break;
default:
throw new ArgumentException("unknown scheme");
}
}
}
monotonicityAdjustments_.Erase();
// Hyman monotonicity constrained filter
if (monotonic_)
{
double correction;
double pm, pu, pd, M;
for (int i = 0; i < size_; ++i)
{
if (i == 0)
{
if (tmp[i] * S[0] > 0.0)
{
correction = tmp[i] / Math.Abs(tmp[i]) *
Math.Min(Math.Abs(tmp[i]),
Math.Abs(3.0 * S[0]));
}
else
{
correction = 0.0;
}
if (correction.IsNotEqual(tmp[i]))
{
tmp[i] = correction;
monotonicityAdjustments_[i] = true;
}
}
else if (i == size_ - 1)
{
if (tmp[i] * S[size_ - 2] > 0.0)
{
correction = tmp[i] / Math.Abs(tmp[i]) *
Math.Min(Math.Abs(tmp[i]), Math.Abs(3.0 * S[size_ - 2]));
}
else
{
correction = 0.0;
}
if (correction.IsNotEqual(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 * Math.Min(Math.Min(Math.Abs(S[i - 1]), Math.Abs(S[i])),
Math.Abs(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 = Math.Max(M, 1.5 * Math.Min(
Math.Abs(pm), Math.Abs(pd)));
}
}
}
if (i < size_ - 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 = Math.Max(M, 1.5 * Math.Min(
Math.Abs(pm), Math.Abs(pu)));
}
}
}
if (tmp[i] * pm > 0.0)
{
correction = tmp[i] / Math.Abs(tmp[i]) *
Math.Min(Math.Abs(tmp[i]), M);
}
else
{
correction = 0.0;
}
if (correction.IsNotEqual(tmp[i]))
{
tmp[i] = correction;
monotonicityAdjustments_[i] = true;
}
}
}
}
// cubic coefficients
for (int i = 0; i < size_ - 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 (int i = 1; i < size_ - 1; ++i)
{
primitiveConst_[i] = primitiveConst_[i - 1]
+ dx[i - 1] *
(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)));
}
}
public override void update()
{
Vector tmp = new Vector(size_);
List<double> dx = new InitializedList<double>(size_ - 1),
S = new InitializedList<double>(size_ - 1);
for (int i = 0; i < size_ - 1; ++i) {
dx[i] = xBegin_[i+1] - xBegin_[i];
S[i] = (yBegin_[i+1] - yBegin_[i])/dx[i];
}
// first derivative approximation
if (da_==CubicInterpolation.DerivativeApprox.Spline) {
TridiagonalOperator L = new TridiagonalOperator(size_);
for (int i = 1; i < size_ - 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.BoundaryCondition.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.BoundaryCondition.FirstDerivative:
L.setFirstRow(1.0, 0.0);
tmp[0] = leftValue_;
break;
case CubicInterpolation.BoundaryCondition.SecondDerivative:
L.setFirstRow(2.0, 1.0);
tmp[0] = 3.0*S[0] - leftValue_*dx[0]/2.0;
break;
case CubicInterpolation.BoundaryCondition.Periodic:
case CubicInterpolation.BoundaryCondition.Lagrange:
// ignoring end condition value
throw new NotImplementedException("this end condition is not implemented yet");
default:
throw new ArgumentException("unknown end condition");
}
// right boundary condition
switch (rightType_) {
case CubicInterpolation.BoundaryCondition.NotAKnot:
// ignoring end condition value
L.setLastRow(-(dx[size_ - 2] + dx[size_ - 3]) * (dx[size_ - 2] + dx[size_ - 3]),
-dx[size_ - 3] * (dx[size_ - 3] + dx[size_ - 2]));
tmp[size_ - 1] = -S[size_ - 3] * dx[size_ - 2] * dx[size_ - 2] -
S[size_ - 2] * dx[size_ - 3] * (3.0 * dx[size_ - 2] + 2.0 * dx[size_ - 3]);
break;
case CubicInterpolation.BoundaryCondition.FirstDerivative:
L.setLastRow(0.0, 1.0);
tmp[size_ - 1] = rightValue_;
break;
case CubicInterpolation.BoundaryCondition.SecondDerivative:
L.setLastRow(1.0, 2.0);
tmp[size_ - 1] = 3.0 * S[size_ - 2] + rightValue_ * dx[size_ - 2] / 2.0;
break;
case CubicInterpolation.BoundaryCondition.Periodic:
case CubicInterpolation.BoundaryCondition.Lagrange:
// ignoring end condition value
throw new NotImplementedException("this end condition is not implemented yet");
default:
throw new ArgumentException("unknown end condition");
}
// solve the system
tmp = L.solveFor(tmp);
} else { // local schemes
if (size_ == 2)
tmp[0] = tmp[1] = S[0];
else {
switch (da_) {
case CubicInterpolation.DerivativeApprox.FourthOrder:
throw new NotImplementedException("FourthOrder not implemented yet");
break;
case CubicInterpolation.DerivativeApprox.Parabolic:
// intermediate points
for (int i = 1; i < size_ - 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[size_ - 1] = ((2.0 * dx[size_ - 2] + dx[size_ - 3]) * S[size_ - 2] - dx[size_ - 2] * S[size_ - 3]) / (dx[size_ - 2] + dx[size_ - 3]);
break;
break;
case CubicInterpolation.DerivativeApprox.FritschButland:
// intermediate points
for (int i=1; i<size_-1; ++i) {
double Smin = Math.Min(S[i-1], S[i]);
double Smax = Math.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[size_-1] = ((2.0*dx[size_-2]+dx[size_-3])*S[size_-2] - dx[size_-2]*S[size_-3]) / (dx[size_-2]+dx[size_-3]);
break;
case CubicInterpolation.DerivativeApprox.Akima:
throw new NotImplementedException("Akima not implemented yet");
// end points
tmp[0] = ((2.0 * dx[0] + dx[1]) * S[0] - dx[0] * S[1]) / (dx[0] + dx[1]);
tmp[size_ - 1] = ((2.0 * dx[size_ - 2] + dx[size_ - 3]) * S[size_ - 2] - dx[size_ - 2] * S[size_ - 3]) / (dx[size_ - 2] + dx[size_ - 3]);
break;
case CubicInterpolation.DerivativeApprox.Kruger:
// intermediate points
for (int i = 1; i < size_ - 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[size_ - 1] = (3.0 * S[size_ - 2] - tmp[size_ - 2]) / 2.0;
break;
default:
throw new ArgumentException("unknown scheme");
}
}
}
monotonicityAdjustments_.Erase();
// Hyman monotonicity constrained filter
if (monotonic_) {
double correction;
double pm, pu, pd, M;
for (int i = 0; i < size_; ++i) {
if (i==0) {
if (tmp[i]*S[0]>0.0) {
correction = tmp[i]/Math.Abs(tmp[i]) *
Math.Min(Math.Abs(tmp[i]),
Math.Abs(3.0*S[0]));
} else {
correction = 0.0;
}
if (correction!=tmp[i]) {
tmp[i] = correction;
monotonicityAdjustments_[i] = true;
}
} else if (i == size_ - 1) {
if (tmp[i] * S[size_ - 2] > 0.0) {
correction = tmp[i]/Math.Abs(tmp[i]) *
Math.Min(Math.Abs(tmp[i]), Math.Abs(3.0 * S[size_ - 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 * Math.Min(Math.Min(Math.Abs(S[i-1]), Math.Abs(S[i])),
Math.Abs(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 = Math.Max(M, 1.5*Math.Min(
Math.Abs(pm),Math.Abs(pd)));
}
}
}
if (i < size_ - 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 = Math.Max(M, 1.5*Math.Min(
Math.Abs(pm),Math.Abs(pu)));
}
}
}
if (tmp[i]*pm>0.0) {
correction = tmp[i]/Math.Abs(tmp[i]) *
Math.Min(Math.Abs(tmp[i]), M);
} else {
correction = 0.0;
}
if (correction!=tmp[i]) {
tmp[i] = correction;
monotonicityAdjustments_[i] = true;
}
}
}
}
// cubic coefficients
for (int i = 0; i < size_ - 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 (int i = 1; i < size_ - 1; ++i) {
primitiveConst_[i] = primitiveConst_[i-1]
+ dx[i-1] *
(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)));
}
}
//! identity instance
public IOperator identity(int size)
{
TridiagonalOperator I = new TridiagonalOperator(new Vector(size - 1, 0.0), // lower diagonal
new Vector(size, 1.0), // diagonal
new Vector(size - 1, 0.0)); // upper diagonal
return I;
}
public override void update()
{
TridiagonalOperator L = new TridiagonalOperator(cH_.n_);
Array <double> tmp = new Array <double>(cH_.n_);
var dx = new Array <double>(cH_.n_ - 1);
var S = new Array <double>(cH_.n_ - 1);
int i = 0;
dx[i] = this.xBegin_[i + 1] - this.xBegin_[i];
S[i] = (this.yBegin_[i + 1] - this.yBegin_[i]) / dx[i];
for (i = 1; i < cH_.n_ - 1; ++i)
{
dx[i] = this.xBegin_[i + 1] - this.xBegin_[i];
S[i] = (this.yBegin_[i + 1] - this.yBegin_[i]) / dx[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]);
}
/**** BOUNDARY CONDITIONS ****/
// left condition
switch (leftType_)
{
case CubicSplineInterpolation.BoundaryCondition.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 CubicSplineInterpolation.BoundaryCondition.FirstDerivative:
L.setFirstRow(1.0, 0.0);
tmp[0] = leftValue_;
break;
case CubicSplineInterpolation.BoundaryCondition.SecondDerivative:
L.setFirstRow(2.0, 1.0);
tmp[0] = 3.0 * S[0] - leftValue_ * dx[0] / 2.0;
break;
case CubicSplineInterpolation.BoundaryCondition.Periodic:
case CubicSplineInterpolation.BoundaryCondition.Lagrange:
// ignoring end condition value
throw new ApplicationException("this end condition is not implemented yet");
default:
throw new ApplicationException("unknown end condition");
}
// right condition
switch (rightType_)
{
case CubicSplineInterpolation.BoundaryCondition.NotAKnot:
// ignoring end condition value
L.setLastRow(-(dx[cH_.n_ - 2] + dx[cH_.n_ - 3]) * (dx[cH_.n_ - 2] + dx[cH_.n_ - 3]),
-dx[cH_.n_ - 3] * (dx[cH_.n_ - 3] + dx[cH_.n_ - 2]));
tmp[cH_.n_ - 1] = -S[cH_.n_ - 3] * dx[cH_.n_ - 2] * dx[cH_.n_ - 2] -
S[cH_.n_ - 2] * dx[cH_.n_ - 3] * (3.0 * dx[cH_.n_ - 2] + 2.0 * dx[cH_.n_ - 3]);
break;
case CubicSplineInterpolation.BoundaryCondition.FirstDerivative:
L.setLastRow(0.0, 1.0);
tmp[cH_.n_ - 1] = rightValue_;
break;
case CubicSplineInterpolation.BoundaryCondition.SecondDerivative:
L.setLastRow(1.0, 2.0);
tmp[cH_.n_ - 1] = 3.0 * S[cH_.n_ - 2] + rightValue_ * dx[cH_.n_ - 2] / 2.0;
break;
case CubicSplineInterpolation.BoundaryCondition.Periodic:
case CubicSplineInterpolation.BoundaryCondition.Lagrange:
// ignoring end condition value
throw new ApplicationException("this end condition is not implemented yet");
default:
throw new ApplicationException("unknown end condition");
}
// solve the system
tmp = L.solveFor(tmp);
for (int j = 0; i < cH_.monotonicityAdjustments_.Count; j++)
{
cH_.monotonicityAdjustments_[j] = false;
}
if (constrained_)
{
double correction;
double pm, pu, pd, M;
for (i = 0; i < cH_.n_; ++i)
{
if (i == 0)
{
if (tmp[i] * S[0] > 0.0)
{
correction = tmp[i] / Math.Abs(tmp[i]) *
Math.Min(Math.Abs(tmp[i]),
Math.Abs(3.0 * S[0]));
}
else
{
correction = 0.0;
}
if (correction != tmp[i])
{
tmp[i] = correction;
cH_.monotonicityAdjustments_[i] = true;
}
}
else if (i == cH_.n_ - 1)
{
if (tmp[i] * S[cH_.n_ - 2] > 0.0)
{
correction = tmp[i] / Math.Abs(tmp[i]) *
Math.Min(Math.Abs(tmp[i]),
Math.Abs(3.0 * S[cH_.n_ - 2]));
}
else
{
correction = 0.0;
}
if (correction != tmp[i])
{
tmp[i] = correction;
cH_.monotonicityAdjustments_[i] = true;
}
}
else
{
pm = (S[i - 1] * dx[i] + S[i] * dx[i - 1]) /
(dx[i - 1] + dx[i]);
M = 3.0 * Math.Min(Math.Min(Math.Abs(S[i - 1]),
Math.Abs(S[i])),
Math.Abs(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 = Math.Max(M, 1.5 * Math.Min(
Math.Abs(pm), Math.Abs(pd)));
}
}
}
if (i < cH_.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 = Math.Max(M, 1.5 * Math.Min(
Math.Abs(pm), Math.Abs(pu)));
}
}
}
if (tmp[i] * pm > 0.0)
{
correction = tmp[i] / Math.Abs(tmp[i]) *
Math.Min(Math.Abs(tmp[i]), M);
}
else
{
correction = 0.0;
}
if (correction != tmp[i])
{
tmp[i] = correction;
cH_.monotonicityAdjustments_[i] = true;
}
}
}
}
for (i = 0; i < cH_.n_ - 1; ++i)
{
cH_.a_[i] = tmp[i];
cH_.b_[i] = (3.0 * S[i] - tmp[i + 1] - 2.0 * tmp[i]) / dx[i];
cH_.c_[i] = (tmp[i + 1] + tmp[i] - 2.0 * S[i]) / (dx[i] * dx[i]);
}
cH_.primitiveConst_[0] = 0.0;
for (i = 1; i < cH_.n_ - 1; ++i)
{
cH_.primitiveConst_[i] = cH_.primitiveConst_[i - 1]
+ dx[i - 1] *
(this.yBegin_[i - 1] + dx[i - 1] *
(cH_.a_[i - 1] / 2.0 + dx[i - 1] *
(cH_.b_[i - 1] / 3.0 + dx[i - 1] * cH_.c_[i - 1] / 4.0)));
}
}
