#include <LOCA_Bifurcation_TPBord_NicDayModifiedBorderingGroup.H>
Inheritance diagram for LOCA::Bifurcation::TPBord::NicDayModifiedBorderingGroup:
The NicDayModifiedBorderingGroup 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 algorithm is similar to the ModifedBorderingGroup algorithm but uses 5 solves of the original Jacobian instead of bordered solves. 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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Performs a deflated solve. The current implemention solves each right hand side serially. |
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_340.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_341.png)
and 
. The five solves of the matrix ![\[ \begin{bmatrix} J & s \\ n^T & 0 \end{bmatrix} \]](form_344.png)