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: 6.5.0.0

Syntax

C#
void Force( int interval, double[] y, double time, double width, double[] xlocal, double[] qw, double[,] u, double[] phi, double[,] dphi )

Visual Basic (Declaration)
Sub Force ( _ interval As Integer, _ y As Double(), _ time As Double, _ width As Double, _ xlocal As Double(), _ qw As Double(), _ u As Double(,), _ phi As Double(), _ dphi As Double(,) _ )

Visual C++
void Force( int interval, array<double>^ y, double time, double width, array<double>^ xlocal, array<double>^ qw, array<double,2>^ u, array<double>^ phi, array<double,2>^ dphi )

interval: Type: System..::.Int32
An int, the index related to the integration interval [xGrid[interval-1], xGrid[interval]].

y: Type: array< System..::.Double >[]()[]
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.

width: Type: System..::.Double
A double scalar, the width of the integration interval, width=xGrid[interval]-xGrid[interval-1].

xlocal: Type: array< System..::.Double >[]()[]
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: array< System..::.Double >[]()[]
An input double array containing the Gauss-Legendre weights. The array length is equal to the degree of the Gauss-Legendre polynomial.

u: Type: array< System..::.Double ,2>[,](,)[,]
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: array< System..::.Double >[]()[]
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: array< System..::.Double ,2>[,](,)[,]
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