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);
}
}
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)));
}
}