com.imsl.math

Interface FeynmanKac.ForcingTerm

Enclosing class:

FeynmanKac

public static interface FeynmanKac.ForcingTerm

Public interface for non-zero forcing term in the Feynman-Kac equation.

Method Summary

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 \(\phi(f,x,t)\) and its derivative \(\partial \phi/\partial y\).

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 \(\phi(f,x,t)\) and its derivative \(\partial \phi/\partial y\).

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 \( y_i=f_i, \, y_{i+1}=f_i', \, y_{i+2}=f_i'', \; i=1,4,7,\ldots,3\cdot \mbox{xGrid.length}-2, \) 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 \(\tilde{\beta}(x)\) 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 - 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 \(\phi_t\) 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.