public double ADIPrice(string scheme, double thet, HParam param, double S0, double V0, double K, double r, double q, double[] S, double[] V, double[] T, string GridType)
{
// Alternating Direction Implicit (ADI) scheme for the Heston model.
// INPUTS
// scheme = 'DO' Douglas, 'CS' Craig-Sneyd,
// 'MSC' Modified CS, 'HV' Hundsdorfer-Verwer
// thet = weighing parameter 0 = Explicit Scheme
// 1 = Implicit Scheme, 0.5 = Crank-Nicolson
// param = Heston parameters
// S0 = Spot price on which to price
// V0 = Volatility on which to price
// K = Strike price
// r = Risk free rate
// q = Dividend yield
// S = Stock price grid - uniform
// V = Volatility grid - uniform
// T = Maturity grid - uniform
Interpolation IP = new Interpolation();
MatrixOps MO = new MatrixOps();
// Heston parameters
double kappa = param.kappa;
double theta = param.theta;
double sigma = param.sigma;
double v0 = param.v0;
double rho = param.rho;
double lambda = param.lambda;
// Length of grids;
int NS = S.Length;
int NV = V.Length;
int NT = T.Length;
double dt = T[1] - T[0];
// Build the matrices for the derivatives and other matrices
BuildDerivativesNU BN = new BuildDerivativesNU();
BuildDerivativesU BU = new BuildDerivativesU();
LMatrices LMat;
if (GridType == "Uniform")
{
LMat = BU.BuildDerivatives(S, V, T);
}
else
{
LMat = BN.BuildDerivativesNonUniform(S, V, T);
}
double[,] derS = LMat.derS;
double[,] derSS = LMat.derSS;
double[,] derV1 = LMat.derV1;
double[,] derV2 = LMat.derV2;
double[,] derVV = LMat.derVV;
double[,] derSV = LMat.derSV;
double[,] R = LMat.R;
// Decompose the derivatives matrices and create the identity matrix
int N = NS * NV;
double[,] A0 = new double[N, N];
double[,] A1 = new double[N, N];
double[,] A2 = new double[N, N];
double[,] I = new double[N, N];
for (int i = 0; i <= N - 1; i++)
{
I[i, i] = 1.0;
for (int j = 0; j <= N - 1; j++)
{
A0[i, j] = rho * sigma * derSV[i, j];
A1[i, j] = (r - q) * derS[i, j] + 0.5 * derSS[i, j] - 0.5 * r * R[i, j];
A2[i, j] = kappa * theta * derV1[i, j] - kappa * derV2[i, j] + 0.5 * sigma * sigma * derVV[i, j] - 0.5 * r * R[i, j];
}
}
// Initialize the u vector, create identity matrix
// u plays the role of U(t-1)
double[] U = new double[N];
double[] u = new double[N];
// U(0) vector - value of U(T) at maturity
double[] Si = new double[N];
int k = 0;
for (int v = 0; v <= NV - 1; v++)
{
for (int s = 0; s <= NS - 1; s++)
{
Si[k] = S[s];
U[k] = Math.Max(Si[k] - K, 0.0);
k += 1;
}
}
// Matrices for the ADI method
double[] Y0 = new double[N];
double[] Y1 = new double[N];
double[] Y2 = new double[N];
double[] Y0_ = new double[N];
double[] Y1_ = new double[N];
double[] Y2_ = new double[N];
double[] Y0h = new double[N];
//// Loop through the time increment
for (int t = 1; t <= NT - 1; t++)
{
u = U;
// Vectors common to all ADI schemes
double[,] SumA = MO.MAdd(MO.MAdd(A0, A1), A2);
double[,] M1 = MO.MMultS(SumA, dt);
double[,] IM1 = MO.MAdd(I, M1);
Y0 = MO.MVMult(IM1, u);
double[,] dtA1 = MO.MMultS(A1, dt * thet);
double[,] IminusA1 = MO.MSub(I, dtA1);
double[,] A1dt = MO.MMultS(A1, dt * thet);
double[] A1u = MO.MVMult(A1dt, u);
double[] Y0minusA1 = MO.VSub(Y0, A1u);
Y1 = MO.MVMult(MO.MInvLU(IminusA1), Y0minusA1);
double[,] dtA2 = MO.MMultS(A2, dt * thet);
double[,] IminusA2 = MO.MSub(I, dtA2);
double[,] A2dt = MO.MMultS(A2, dt * thet);
double[] A2u = MO.MVMult(A2dt, u);
double[] Y1minusA2 = MO.VSub(Y1, A2u);
Y2 = MO.MVMult(MO.MInvLU(IminusA2), Y1minusA2);
if (scheme == "DO")
{
// Douglas ADI scheme
U = Y2;
}
else if (scheme == "CS")
{
// Craig-Sneyd ADI scheme
double[] A0Y2 = MO.MVMult(A0, Y2);
double[] A0u = MO.MVMult(A0, u);
double[] A0A0 = MO.VSub(A0Y2, A0u);
double[] dtA = MO.VMultS(A0A0, dt * 0.5);
Y0_ = MO.VAdd(Y0, dtA);
double[] Y0_minusA1 = MO.VSub(Y0_, A1u);
Y1_ = MO.MVMult(MO.MInvLU(IminusA1), Y0_minusA1);
double[] Y1_minusA2 = MO.VSub(Y1_, A2u);
Y2_ = MO.MVMult(MO.MInvLU(IminusA2), Y1_minusA2);
U = Y2_;
}
else if (scheme == "MCS")
{
// Modified Craig-Sneyd ADI scheme
double[] A0Y2 = MO.MVMult(A0, Y2);
double[] A0u = MO.MVMult(A0, u);
double[] A0A0 = MO.VSub(A0Y2, A0u);
double[] dtA = MO.VMultS(A0A0, dt * thet);
Y0h = MO.VAdd(Y0, dtA);
double[] SumAY2 = MO.MVMult(SumA, Y2);
double[] SumAu = MO.MVMult(SumA, u);
double[] ThetDt = MO.VMultS(MO.VSub(SumAY2, SumAu), (0.5 - thet) * dt);
Y0_ = MO.VAdd(Y0h, ThetDt);
double[] Y0_minusA1 = MO.VSub(Y0_, A1u);
Y1_ = MO.MVMult(MO.MInvLU(IminusA1), Y0_minusA1);
double[] Y1_minusA2 = MO.VSub(Y1_, A2u);
Y2_ = MO.MVMult(MO.MInvLU(IminusA2), Y1_minusA2);
U = Y2_;
}
else if (scheme == "HV")
{
// Hundsdorfer-Verver ADI scheme
double[] A0Y2 = MO.MVMult(SumA, Y2);
double[] A0u = MO.MVMult(SumA, u);
double[] A0A0 = MO.VSub(A0Y2, A0u);
double[] dtA = MO.VMultS(A0A0, dt * 0.5);
Y0_ = MO.VAdd(Y0, dtA);
double[] A1Y2 = MO.VMultS(MO.MVMult(A1, Y2), thet * dt);
double[] Y0_minusA1Y2 = MO.VSub(Y0_, A1Y2);
Y1_ = MO.MVMult(MO.MInvLU(IminusA1), Y0_minusA1Y2);
double[] A2Y2 = MO.VMultS(MO.MVMult(A2, Y2), thet * dt);
double[] Y1_minusA2Y2 = MO.VSub(Y1_, A2Y2);
Y2_ = MO.MVMult(MO.MInvLU(IminusA2), Y1_minusA2Y2);
U = Y2_;
}
}
// Restack the U vector to output a matrix
double[,] UU = new double[NS, NV];
k = 0;
for (int v = 0; v <= NV - 1; v++)
{
for (int s = 0; s <= NS - 1; s++)
{
UU[s, v] = U[k];
k += 1;
}
}
// Interpolate to get the price at S0 and v0
return(IP.interp2(V, S, UU, V0, S0));
}
public LMatrices BuildDerivativesNonUniform(double[] S, double[] V, double[] T)
{
// Build the first- and second-order derivatives for the "L" operator matrix
// for the Weighted method
// Requires a uniform grid for S and V
// INPUTS
// S = vector for uniform stock price grid
// V = vector for uniform volatility grid
// T = vector for uniform maturity grid
// thet = parameter for the weighted scheme
// OUTPUTS
// Matrices of dimnension N x N (N=NS+NV)
// derS = Matrix for first-order derivative dU/dS
// derSS = Matrix for second-order derivative dU2/dS2
// derV1 = Matrix for first-order derivative dU/dV, kappa*theta portion
// derV2 = Matrix for first-order derivative dU/dV, -kappa*V portion
// derVV = Matrix for first-order derivative dU2/dV2
// derSV = Matrix for first-order derivative dU2/dSdV
// R = Matrix for r*S(v,t) portion of the PDE
BuildDerivativesU BU = new BuildDerivativesU();
// Length of stock price, volatility, and maturity
int NS = S.Length;
int NV = V.Length;
int NT = T.Length;
double Smin = S[0]; double Smax = S[NS - 1];
double Vmin = V[0]; double Vmax = V[NV - 1];
double Tmin = T[0]; double Tmax = T[NT - 1];
// Preliminary quantities
// Size of the U(t) vector and L matrix
int N = NS * NV;
// The vectors for S and V, stacked
double[] Si = new double[N];
double[] Vi = new double[N];
int k = 0;
for (int v = 0; v <= NV - 1; v++)
{
for (int s = 0; s <= NS - 1; s++)
{
Si[k] = S[s];
Vi[k] = V[v];
k += 1;
}
}
// Identification of the boundary points
int[] VminB = new int[N];
int[] VmaxB = new int[N];
int[] SminB = new int[N];
int[] SmaxB = new int[N];
// Vmin and Vmax
for (int v = 0; v <= NS - 2; v++)
{
VminB[v] = 1;
}
for (int v = N - NS + 1; v <= N - 2; v++)
{
VmaxB[v] = 1;
}
VmaxB[N - 1] = 1;
// Smin and Smax
k = 0;
for (int v = 0; v <= NV - 1; v++)
{
for (int s = 0; s <= NS - 1; s++)
{
if (s == 0)
{
SminB[k] = 1;
}
else if (s == NS - 1)
{
SmaxB[k] = 1;
}
k += 1;
}
}
SminB[0] = 0;
SmaxB[NS - 1] = 0;
SmaxB[N - 1] = 0;
// Identification of the non-boundary points
int[] NB = new int[N];
for (k = 0; k <= N - 1; k++)
{
if (SminB[k] == 0 & SmaxB[k] == 0 & VminB[k] == 0 & VmaxB[k] == 0)
{
NB[k] = 1;
}
}
NB[NS - 1] = 1;
// Forward, backward and central differences for S
int[] Cs = new int[N];
int[] Fs = new int[N];
int[] Bs = new int[N];
k = 0;
for (int v = 0; v <= NV - 2; v++)
{
for (int s = 0; s <= NS - 1; s++)
{
if (s == 1)
{
Fs[k] = 1;
}
else if (s == NS - 2)
{
Bs[k] = 1;
}
else if (s >= 2 & s <= NS - 3)
{
Cs[k] = 1;
}
k += 1;
}
}
Fs[1] = 0;
Bs[NS - 2] = 0;
for (k = 2; k <= NS - 3; k++)
{
Cs[k] = 0;
}
// Forward, backward and central differences for V
int[] Cv = new int[N];
int[] Fv = new int[N];
int[] Bv = new int[N];
for (k = NS + 1; k <= 2 * NS - 2; k++)
{
Fv[k] = 1;
}
for (k = (NV - 2) * NS + 1; k <= (NV - 1) * NS - 2; k++)
{
Bv[k] = 1;
}
k = 0;
for (int v = 0; v <= NV - 3; v++)
{
for (int s = 0; s <= NS - 1; s++)
{
if (s >= 1 & s <= NS - 2)
{
Cv[k] = 1;
}
k += 1;
}
}
for (k = 0; k <= 2 * NS - 1; k++)
{
Cv[k] = 0;
}
// Central difference for SV-derivatives
int[] Csv = new int[N];
k = 0;
for (int v = 0; v <= NV - 2; v++)
{
for (int s = 0; s <= NS - 1; s++)
{
if (s >= 1 & s <= NS - 2)
{
Csv[k] = 1;
}
k += 1;
}
}
for (k = 0; k <= NS - 1; k++)
{
Csv[k] = 0;
}
// for(k=0;k<=N-1;k++)
// Console.WriteLine("{0} {1} {2} {3} {4} {5} {6} {7} {8} {9} {10} {11}",SminB[k],SmaxB[k],VminB[k],VmaxB[k],NB[k],Fs[k],Bs[k],Cs[k],Fv[k],Bv[k],Cv[k],Csv[k]);
// Create the matrices for the derivatives
double[,] derS = new double[N, N];
double[,] derSS = new double[N, N];
double[,] derV1 = new double[N, N];
double[,] derV2 = new double[N, N];
double[,] derVV = new double[N, N];
double[,] derSV = new double[N, N];
double[,] R = new double[N, N];
for (int i = 0; i <= N - 1; i++)
{
for (int j = 0; j <= N - 1; j++)
{
if (i == j)
{
R[i, j] = 1.0;
}
}
}
int M;
int[] I;
double ds, dv;
// Create the matrices for the derivatives
// NON BOUNDARY POINTS ----------------------------------
I = BU.Find(Cs);
M = I.Length;
for (k = 0; k <= M - 1; k++)
{ // Central differences
ds = Si[I[k]] - Si[I[k] - 1];
derS[I[k], I[k] - 1] = -0.5 / ds * Si[I[k]]; // U(s-1,v)
derSS[I[k], I[k] - 1] = 1.0 / ds / ds * Vi[I[k]] * Si[I[k]] * Si[I[k]]; // U(s-1,v)
ds = Si[I[k] + 1] - Si[I[k]];
derS[I[k], I[k] + 1] = 0.5 / ds * Si[I[k]]; // U(s+1,v)
derSS[I[k], I[k] + 1] = 1.0 / ds / ds * Vi[I[k]] * Si[I[k]] * Si[I[k]]; // U(s+1,v)
ds = (Si[I[k] + 1] - Si[I[k] - 1]) / 2.0;
derS[I[k], I[k]] = 0.0; // U(s,v)
derSS[I[k], I[k]] = -2.0 / ds / ds * Vi[I[k]] * Si[I[k]] * Si[I[k]]; // U(s,v)
}
I = BU.Find(Fs);
M = I.Length;
for (k = 0; k <= M - 1; k++)
{ // Forward differences
ds = (Si[I[k] + 1] - Si[I[k] - 1]) / 2.0;
derS[I[k], I[k]] = -1.5 / ds * Si[I[k]]; // U(s,v)
derSS[I[k], I[k]] = 1.0 / ds / ds * Vi[I[k]] * Si[I[k]] * Si[I[k]]; // U(s,v)
ds = Si[I[k] + 1] - Si[I[k]];
derS[I[k], I[k] + 1] = 2.0 / ds * Si[I[k]]; // U(s+1,v)
derSS[I[k], I[k] + 1] = -2.0 / ds / ds * Vi[I[k]] * Si[I[k]] * Si[I[k]]; // U(s+1,v)
ds = (Si[I[k] + 2] - Si[I[k]]) / 2.0;
derS[I[k], I[k] + 2] = -1.0 / ds * Si[I[k]]; // U(s+2,v)
derSS[I[k], I[k] + 2] = 1.0 / ds / ds * Vi[I[k]] * Si[I[k]] * Si[I[k]]; // U(s+2,v)
}
I = BU.Find(Bs);
M = I.Length;
for (k = 0; k <= M - 1; k++)
{ // Backward differences
ds = (Si[I[k]] - Si[I[k] - 2]) / 2.0;
derS[I[k], I[k] - 2] = -0.5 / ds * Si[I[k]]; // U(s-2,v)
derSS[I[k], I[k] - 2] = 1.0 / ds / ds * Vi[I[k]] * Si[I[k]] * Si[I[k]]; // U(s-2,v)
ds = (Si[I[k]] - Si[I[k] - 1]);
derS[I[k], I[k] - 1] = -2.0 / ds * Si[I[k]]; // U(s-1,v)
derSS[I[k], I[k] - 1] = -2.0 / ds / ds * Vi[I[k]] * Si[I[k]] * Si[I[k]]; // U(s-1,v)
ds = (Si[I[k] + 1] - Si[I[k] - 1]) / 2.0;
derSS[I[k], I[k]] = 1.0 / ds / ds * Vi[I[k]] * Si[I[k]] * Si[I[k]]; // U(s,v)
derS[I[k], I[k]] = 1.5 / ds * Si[I[k]]; // U(s,v)
}
// Create the matrix for V-derivatives
I = BU.Find(Cv);
M = I.Length;
for (k = 0; k <= M - 1; k++)
{ // Central differences
dv = Vi[I[k]] - Vi[I[k] - NS];
derV1[I[k], I[k] - NS] = -0.5 / dv; // U(s,v-1)
derV2[I[k], I[k] - NS] = -0.5 / dv * Vi[I[k]]; // U(s,v-1)
derVV[I[k], I[k] - NS] = 1.0 / dv / dv * Vi[I[k]]; // U(s,v-1)
dv = Vi[I[k] + NS] - Vi[I[k]];
derV1[I[k], I[k] + NS] = 0.5 / dv; // U(s,v+1)
derV2[I[k], I[k] + NS] = 0.5 / dv * Vi[I[k]]; // U(s,v+1)
derVV[I[k], I[k] + NS] = 1.0 / dv / dv * Vi[I[k]]; // U(s,v+1)
dv = (Vi[I[k] + NS] - Vi[I[k] - NS]) / 2.0;
derVV[I[k], I[k]] = -2.0 / dv / dv * Vi[I[k]]; // U(s,v)
derV1[I[k], I[k]] = 0.0; // U(s,v)
derV2[I[k], I[k]] = 0.0; // U(s,v)
}
I = BU.Find(Fv);
M = I.Length;
for (k = 0; k <= M - 1; k++)
{ // Forward differences
dv = (Vi[I[k] + NS] - Vi[I[k] - NS]) / 2.0;
derV1[I[k], I[k]] = -1.5 / dv; // U(s,v)
derV2[I[k], I[k]] = -1.5 / dv * Vi[I[k]]; // U(s,v)
derVV[I[k], I[k]] = 1.0 / dv / dv * Vi[I[k]]; // U(s,v)
dv = Vi[I[k] + NS] - Vi[I[k]];
derV1[I[k], I[k] + NS] = 2.0 / dv; // U(s,v+1)
derV2[I[k], I[k] + NS] = 2.0 / dv * Vi[I[k]]; // U(s,v+1)
derVV[I[k], I[k] + NS] = -2.0 / dv / dv * Vi[I[k]]; // U(s,v+1)
dv = (Vi[I[k] + 2 * NS] - Vi[I[k]]) / 2.0;
derV1[I[k], I[k] + 2 * NS] = -1.0 / dv; // U(s,v+2)
derV2[I[k], I[k] + 2 * NS] = -1.0 / dv * Vi[I[k]]; // U(s,v+2)
derVV[I[k], I[k] + 2 * NS] = 1.0 / dv / dv * Vi[I[k]]; // U(s,v+2)
}
I = BU.Find(Bv);
M = I.Length;
for (k = 0; k <= M - 1; k++)
{ // Backward differences
dv = (Vi[I[k]] - Vi[I[k] - 2 * NS]) / 2.0;
derV1[I[k], I[k] - 2 * NS] = -0.5 / dv; // U(s,v-2)
derV2[I[k], I[k] - 2 * NS] = -0.5 / dv * Vi[I[k]]; // U(s,v-2)
derVV[I[k], I[k] - 2 * NS] = 1.0 / dv / dv * Vi[I[k]]; // U(s,v-2)
dv = Vi[I[k]] - Vi[I[k] - NS];
derV1[I[k], I[k] - NS] = -2.0 / dv; // U(s,v-1)
derV2[I[k], I[k] - NS] = -2.0 / dv * Vi[I[k]]; // U(s,v-1)
derVV[I[k], I[k] - NS] = -2.0 / dv / dv * Vi[I[k]]; // U(s,v-1)
dv = (Vi[I[k] + NS] - Vi[I[k] - NS]) / 2.0;
derV1[I[k], I[k]] = 1.5 / dv; // U(s,v)
derV2[I[k], I[k]] = 1.5 / dv * Vi[I[k]]; // U(s,v)
derVV[I[k], I[k]] = 1.0 / dv / dv * Vi[I[k]]; // U(s,v)
}
// Create the matrix for SV-derivatives - simplified version
/* I = Find(Csv);
* M = I.Length;
* for(k=0;k<=M-1;k++)
* {
* dv = (Vi[I[k]+NS] - Vi[I[k]-NS])/2.0;
* ds = (Si[I[k]+1] - Si[I[k]-1])/2.0;
* derSV[I[k],I[k]+NS+1] = 1.0/(4.0*ds*dv) * Vi[I[k]]*Si[I[k]]; // U(s+1,v+1)
* derSV[I[k],I[k]+NS-1] = -1.0/(4.0*ds*dv) * Vi[I[k]]*Si[I[k]]; // U(s-1,v+1)
* derSV[I[k],I[k]-NS-1] = 1.0/(4.0*ds*dv) * Vi[I[k]]*Si[I[k]]; // U(s-1,v-1)
* derSV[I[k],I[k]-NS+1] = -1.0/(4.0*ds*dv) * Vi[I[k]]*Si[I[k]]; // U(s+1,v-1)
* }
*/
// Create the matrix for SV-derivatives
I = BU.Find(Csv);
M = I.Length;
double a1, a2, a3, a4, a5, a6, a7, a8, a9, ds1, dv1;
for (k = 0; k <= M - 1; k++)
{
ds = Si[I[k]] - Si[I[k] - 1];
if (I[k] + 1 <= NS)
{
ds1 = Si[I[k + 1]] - S[I[k]]; // Correct for off-grid point
}
else
{
ds1 = ds;
}
dv = Vi[I[k]] - Vi[I[k] - NS];
dv1 = Vi[I[k] + NS] - Vi[I[k]];
a1 = ds1 / ds / (ds + ds1) * dv1 / dv / (dv + dv1);
a2 = -ds1 / ds / (ds + ds1) * (dv1 - dv) / dv / dv1;
a3 = -ds1 / ds / (ds + ds1) * dv / dv1 / (dv + dv1);
a4 = (ds1 - ds) / ds / ds1 * (-dv1) / dv / (dv + dv1);
a5 = (ds1 - ds) / ds / ds1 * (dv1 - dv) / dv1 / dv;
a6 = (ds1 - ds) / ds / ds1 * dv / dv1 / (dv + dv1);
a7 = ds / ds / (ds + ds1) * (-dv1) / dv / (dv + dv1);
a8 = ds / ds1 / (ds + ds1) * (dv1 - dv) / dv / dv1;
a9 = ds / ds1 / (ds1 + ds) * dv / dv1 / (dv + dv1);
derSV[I[k], I[k] - NS - 1] = a1 * Vi[I[k]] * Si[I[k]]; // U(s-1,v-1)
derSV[I[k], I[k] - 1] = a2 * Vi[I[k]] * Si[I[k]]; // U(s-1,v)
derSV[I[k], I[k] + NS - 1] = a3 * Vi[I[k]] * Si[I[k]]; // U(s-1,v+1)
derSV[I[k], I[k] - NS] = a4 * Vi[I[k]] * Si[I[k]]; // U(s,v-1)
derSV[I[k], I[k]] = a5 * Vi[I[k]] * Si[I[k]]; // U(s,v)
derSV[I[k], I[k] + NS] = a6 * Vi[I[k]] * Si[I[k]]; // U(s,v+1)
derSV[I[k], I[k] - NS + 1] = a7 * Vi[I[k]] * Si[I[k]]; // U(s+1,v-1)
derSV[I[k], I[k] + 1] = a8 * Vi[I[k]] * Si[I[k]]; // U(s+1,v)
derSV[I[k], I[k] + NS + 1] = a9 * Vi[I[k]] * Si[I[k]]; // U(s+1,v+1)
}
// BOUNDARY POINTS ----------------------------------
// Boundary for Smin
I = BU.Find(SminB);
M = I.Length;
for (k = 0; k <= M - 1; k++)
{
derS[I[k], I[k]] = 0.0;
derSS[I[k], I[k]] = 0.0;
derV1[I[k], I[k]] = 0.0;
derV2[I[k], I[k]] = 0.0;
derVV[I[k], I[k]] = 0.0;
derSV[I[k], I[k]] = 0.0;
R[I[k], I[k]] = 0.0;
}
// Boundary condition for Smax
I = BU.Find(SmaxB);
M = I.Length;
for (k = 0; k <= M - 1; k++)
{
derS[I[k], I[k]] = Si[I[k]];
derSS[I[k], I[k]] = 0.0;
derSV[I[k], I[k]] = 0.0; // Central difference
derV1[I[k], I[k]] = 0; // U(s,v)
derV2[I[k], I[k]] = 0.0; // U(s,v)
dv = Vi[I[k]] - Vi[I[k] - NS];
derV1[I[k], I[k] - NS] = -0.5 / dv; // U(s,v-1)
derV2[I[k], I[k] - NS] = -0.5 / dv * Vi[I[k]]; // U(s,v-1)
derVV[I[k], I[k] - NS] = 1.0 / dv / dv * Vi[I[k]]; // U(s,v-1)
dv = Vi[I[k] + NS] - Vi[I[k]];
derV1[I[k], I[k] + NS] = 0.5 / dv; // U(s,v+1)
derV2[I[k], I[k] + NS] = 0.5 / dv * Vi[I[k]]; // U(s,v+1)
derVV[I[k], I[k] + NS] = 1.0 / dv / dv * Vi[I[k]]; // U(s,v+1)
dv = (Vi[I[k] + NS] - Vi[I[k] - NS]) / 2.0;
derVV[I[k], I[k]] = -2.0 / dv / dv * Vi[I[k]]; // U(s,v)
}
// Boundary condition for Vmax. Only the LS submatrix is non-zero
I = BU.Find(VmaxB);
M = I.Length;
for (k = 0; k <= M - 1; k++)
{
derS[I[k], I[k]] = Si[I[k]];
}
// Boundary condition for Vmin
// First row
ds = Si[1] - Si[0];
derS[0, 0] = -1.5 / ds * Si[0]; // U(s,v)
derS[0, 1] = 2.0 / ds * Si[0]; // U(s+1,v)
derS[0, 2] = -1.0 / ds * Si[0]; // U(s+2,v)
dv = Vi[2 * NS] - Vi[NS];
derV1[0, 0] = -1.5 / dv; // U(s,v)
derV1[0, NS] = 2.0 / dv; // U(s,v+1)
derV1[0, 2 * NS] = -1.0 / dv; // U(s,v+2)
// Last row
ds = Si[N - 2] - Si[N - 3];
derS[N - 2, N - 4] = -0.5 / ds * Si[N - 2]; // U(s-2,v)
derS[N - 2, N - 3] = -2.0 / ds * Si[N - 2]; // U(s-1,v)
derS[N - 2, N - 2] = 1.5 / ds * Si[N - 2]; // U(s,v)
dv = Vi[2 * NS] - Vi[NS];
derV1[NS - 2, NS - 2 + 2 * NS] = -0.5 / dv * Vi[NS - 2]; // U(s,v+2)
derV1[NS - 2, NS - 2 + NS] = 2.0 / dv * Vi[NS - 2]; // U(s,v+1)
derV1[NS - 2, NS - 2] = -1.5 / dv * Vi[NS - 2]; // U(s,v)
// Other rows
for (int i = 1; i <= NS - 3; i++)
{
ds = Si[i] - Si[i - 1];
derS[i, i - 1] = -0.5 / ds * Si[i]; // U(s-1,v)
derS[i, i] = 0.0; // U(s,v)
derS[i, i + 1] = 0.5 / ds * Si[i]; // U(s+1,v)
dv = Vi[2 * NS] - Vi[NS];
derV1[i, i] = -1.5 / dv; // U(s,v)
derV1[i, i + NS] = 2.0 / dv; // U(s,v+1)
derV1[i, i + 2 * NS] = -1.0 / dv; // U(s,v+2)
}
// Output the sub-matrices
LMatrices LMat;
LMat.derS = derS;
LMat.derSS = derSS;
LMat.derV1 = derV1;
LMat.derV2 = derV2;
LMat.derVV = derVV;
LMat.derSV = derSV;
LMat.R = R;
return(LMat);
}
