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FeynmanKacIForcingTermForce Method
Computes approximations to the forcing term \phi(f,x,t) and its derivative \partial \phi/\partial y.

Namespace: Imsl.Math
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0
Syntax
Copy
void Force(
	int interval,
	double[] y,
	double time,
	double width,
	double[] xlocal,
	double[] qw,
	double[,] u,
	double[] phi,
	double[,] dphi
)

Parameters

interval
Type: SystemInt32
An int, the index related to the integration interval [xGrid[interval-1], xGrid[interval]].
y
Type: SystemDouble
An input double array of length 3*xGrid.Length containing the coefficients of the Hermite quintic spline representing the solution of the Feynman-Kac equation at time point time. For each
x \in [x_i,x_{i+1}], \, h_i=x_{i+1}-x_i, z_i=(x-x_i)/h_i,\,i=1,\ldots, \text{xGrid.Length}-1
the approximate solution is locally defined by
f(x,t) = f_ib_0(z)+f_{i+1}b_0(1-z)+h_i f_i' b_1(z)-h_i f_{i+1}' b_1(1-z)+ h_i^2f_i''b_2(z)+h_i^2f_{i+1}''b_2(1-z).
The values y_{3i-2}=f_i, \, y_{3i-1}=f_i', \, y_{3i}=f_i'', \; i=1,\ldots,\mbox{xGrid.Length}, are stored as successive triplets in y.
time
Type: SystemDouble
A double scalar, the time point.
width
Type: SystemDouble
A double scalar, the width of the integration interval, width=xGrid[interval]-xGrid[interval-1].
xlocal
Type: SystemDouble
An input double array containing the Gauss-Legendre points translated and normalized to the interval [xGrid[interval-1], xGrid[interval]]. The array length is equal to the degree of the Gauss-Legendre polynomial.
qw
Type: SystemDouble
An input double array containing the Gauss-Legendre weights. The array length is equal to the degree of the Gauss-Legendre polynomial.
u
Type: SystemDouble
An input double matrix of dimension 12 by xlocal.Length containing the basis function values that define \tilde{\beta}(x) at the Gauss-Legendre points xlocal. Setting
u_{k,i}:=\text{u[k,i]} \quad \mbox{and} \quad x_i:=\text{xlocal[i]} \,,
vector \tilde{\beta}(x_i) is defined as
\tilde{\beta}(x_i):=(\beta_{3*(\text{interval}-1)}(x_i),\ldots, \beta_{3*\text{interval}+2}(x_i))^T = (u_{0,i},u_{1,i},u_{2,i},u_{6,i},u_{7,i},u_{8,i})^T \,.
phi
Type: SystemDouble
An output double array of length 6 containing Gauss-Legendre approximations for the local contributions
\phi_t := \int_{\text{xgrid[interval-1]}}^{\text{xgrid[interval]}}\phi(f,x,t) \tilde{\beta}(x) dx,
where t=time and \tilde{\beta}(x):=(\beta_{3*(\text{interval}-1)}(x),\ldots,\beta_{3*\text{interval}+2}(x))^T. Denoting by degree the number of Gauss-Legendre points (xlocal.Length) and setting x_j:=\text{xlocal[j]}, vector phi contains elements
\text{phi[i]} = \text{width}*\sum_{j=0}^{\text{degree-1}}\text{qw}[j]\,\tilde{\beta}_i(x_j)\,\phi(f,x_j,t)
for i=0,...,5.
dphi
Type: SystemDouble
An output double matrix of dimension 6 by 6 containing a Gauss-Legendre approximation for the Jacobian of the local contributions \phi_t at time point t=time,
\frac{\partial \phi_t}{\partial y}=\int_{\text{xgrid[interval-1]}}^{\text{xgrid[interval]}} \frac{\partial \phi (f,x,t)}{\partial f} \,\tilde{\beta}(x)\,\tilde{\beta}^T(x)dx\,.
The approximation to this symmetric matrix is stored row-wise, i.e.
\text{dphi[i,j]} = \text{width} * \sum_{k=0}^{\text{degree-1}}\text{qw}[k]\,\tilde{\beta}_i(x_k)\,\tilde{\beta}_j(x_k) \left. \frac{\partial{\phi}}{\partial f}\right|_{x=\text{xlocal}[k],\,t=\text{time}}
for i,j=0,...,5.
See Also