The bordering algorithm and path following near singular points of higher nullity.

*(English)*Zbl 0536.65017The bordering algorithm is a block elimination method for solving systems with coefficient matrices of the form: \({\mathcal A} =\begin{pmatrix} A & B \\ C^ T & D \end{pmatrix}\),

where A is N by N, B is N by \(\nu\), \(C^ T\) is \(\nu\) by N, D is \(\nu\) by \(\nu\), and A is singular. The author previously considered the case \(\nu =1\), and treats general \(\nu\) here. Assuming \({\mathcal A}\) is nonsingular, it is shown how the system involving \({\mathcal A}\) can be solved by computing approximate null vectors for A from an LU decomposition of A. A heuristic test to determine the rank of A from the LU decomposition is given, and a perturbation analysis is done for the case when full pivoting is used. An argument is also given in conjunction with a two- point boundary value problem example, indicating that the bordering algorithm is successful in practice when merely block structure preserving forms of partial pivoting are used. Application of the algorithm (with \(\nu =1)\) to continuation methods is sketched. References are given to practical applications of the case \(\nu>1\). These include multiple limit points, Hopf bifurcation and period doubling, fold following, and critical boundary paths.

where A is N by N, B is N by \(\nu\), \(C^ T\) is \(\nu\) by N, D is \(\nu\) by \(\nu\), and A is singular. The author previously considered the case \(\nu =1\), and treats general \(\nu\) here. Assuming \({\mathcal A}\) is nonsingular, it is shown how the system involving \({\mathcal A}\) can be solved by computing approximate null vectors for A from an LU decomposition of A. A heuristic test to determine the rank of A from the LU decomposition is given, and a perturbation analysis is done for the case when full pivoting is used. An argument is also given in conjunction with a two- point boundary value problem example, indicating that the bordering algorithm is successful in practice when merely block structure preserving forms of partial pivoting are used. Application of the algorithm (with \(\nu =1)\) to continuation methods is sketched. References are given to practical applications of the case \(\nu>1\). These include multiple limit points, Hopf bifurcation and period doubling, fold following, and critical boundary paths.

Reviewer: B.Kearfott

##### MSC:

65F05 | Direct numerical methods for linear systems and matrix inversion |

65H10 | Numerical computation of solutions to systems of equations |

65L10 | Numerical solution of boundary value problems involving ordinary differential equations |