FeynmanKac.ForcingTerm (JMSL Numerical Library)

Enclosing class:

FeynmanKac
```
public static interface FeynmanKac.ForcingTerm
```
Public interface for non-zero forcing term in the Feynman-Kac equation.

Method Summary

Methods
Modifier and Type	Method and Description
`void`	`force(int interval, double[] y, double time, double width, double[] xlocal, double[] qw, double[][] u, double[] phi, double[][] dphi)` Computes approximations to the forcing term and its derivative .

- Method Detail
  - force
```
void force(int interval,
         double[] y,
         double time,
         double width,
         double[] xlocal,
         double[] qw,
         double[][] u,
         double[] phi,
         double[][] dphi)
```
    Computes approximations to the forcing term and its derivative .
    
    Parameters:
    interval - an int, the index related to the integration interval [xGrid[interval-1], xGrid[interval]].
    y - 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 $y_i=f_i, , y_{i+1}=f_i', , y_{i+2}=f_i'', ; i=1,4,7,ldots,3cdot mbox{xGrid.length}-2,$ are stored as successive triplets in y.
    time - a double, the time point.
    width - a double, the width of the integration interval, width=xGrid[interval]-xGrid[interval-1].
    xlocal - an input double array containing the Gauss-Legendre points translated and normalized to the interval [xGrid[interval-1], xGrid[interval]].
    qw - an input double array containing the Gauss-Legendre weights.
    u - an input double array of dimension 12 by xlocal.length containing the basis function values that define 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 - 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 - an output double array of dimension 6 by 6 containing a Gauss-Legendre approximation for the Jacobian of the local contributions 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.

Interface FeynmanKac.ForcingTerm

Method Summary

Method Detail

force