#include <LOCA_Bifurcation_TPBord_ModifiedBorderingGroup.H>
Inheritance diagram for LOCA::Bifurcation::TPBord::ModifiedBorderingGroup:
The ModifiedBorderingGroup is derived from the LOCA::Bifurcation::TPBord::ExtendedGroup and reimplements the applyJacobianInverse() and applyJacobianInverseMulti() methods using a modified bordering algorithm that avoids nearly-singular matrix solves. The four solves in the original bordering algorithm are replaced by five solves of an 
system using the applyBorderedJacobianInverse() method of the underlying group. Therefore, to use this algorithm the underlying group must have an effective way of solving these bordered systems (such as using a matrix-free mode). See applyJacobianInverse() for a description of the algorithm.
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Applies the inverse of the extended Jacobian matrix using a modified bordering algorithm. Computes the solution to the turning point system
using the following modified bordering algorithm:
where
are computed via the applyBorderedJacobianInverse() method of the underlying group. Reimplemented from LOCA::Bifurcation::TPBord::ExtendedGroup. |
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Applies the inverse of the extended Jacobian matrix using a modified bordering algorithm for multiple right-hand sides. The current implemention solves each right hand side serially. Reimplemented from LOCA::Bifurcation::TPBord::ExtendedGroup. |
1.3.9.1
![\[ \begin{bmatrix} J & 0 & \frac{\partial F}{\partial p} \\ \frac{\partial Jn}{\partial x} & J & \frac{\partial Jn}{\partial p} \\ 0 & l^T & 0 \end{bmatrix} \begin{bmatrix} u \\ v \\ w \end{bmatrix} = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \]](form_327.png)
![\[ \begin{split} \begin{bmatrix} J & s \\ n^T & 0 \end{bmatrix} \begin{bmatrix} a \\ \bar{a} \end{bmatrix} &= \begin{bmatrix} x \\ 0 \end{bmatrix} \\ \begin{bmatrix} J & s \\ n^T & 0 \end{bmatrix} \begin{bmatrix} b \\ \bar{b} \end{bmatrix}&= \begin{bmatrix} \frac{\partial F}{\partial p} \\ 0 \end{bmatrix} \\ \begin{bmatrix} J & s \\ n^T & 0 \end{bmatrix} \begin{bmatrix} c \\ \bar{c} \end{bmatrix}&= \begin{bmatrix} y - \frac{\partial Jn}{\partial x}a \\ 0 \end{bmatrix} \\ \begin{bmatrix} J & s \\ n^T & 0 \end{bmatrix} \begin{bmatrix} d \\ \bar{d} \end{bmatrix}&= \begin{bmatrix} \frac{\partial Jn}{\partial p} - \frac{\partial Jn}{\partial x}a \\ 0 \end{bmatrix} \\ \begin{bmatrix} J & s \\ n^T & 0 \end{bmatrix} \begin{bmatrix} d \\ \bar{d} \end{bmatrix}&= \begin{bmatrix} \frac{\partial Jn}{\partial x}n \\ 0 \end{bmatrix} \\ \begin{bmatrix} \sigma & \bar{e} & -l^Te \\ 0 & \sigma & l^Tv \\ \bar{b} & \bar{d} & -l^Td \end{bmatrix} \begin{bmatrix} \alpha \\ \beta \\ w \end{bmatrix} &= \begin{bmatrix} \bar{a} \\ \bar{c} \\ z - l^T c \end{bmatrix} \\ u &= a - w b + \alpha v\\ v &= c - w d - \alpha e + \beta v \end{split} \]](form_328.png)
and 
. The five solves of the matrix ![\[ \begin{bmatrix} J & s \\ n^T & 0 \end{bmatrix} \]](form_331.png)