A neural network modeled by an adaptive Lotka-Volterra system
SIAM Journal on Applied Mathematics
Solving eigenvalue problems of real nonsymmetric matrices with real homotopies
SIAM Journal on Numerical Analysis
Note on the end game in homotopy zero curve tracking
ACM Transactions on Mathematical Software (TOMS)
Computing Hopf Bifurcations II: Three Examples from Neurophysiology
SIAM Journal on Scientific Computing
Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation
ACM Transactions on Mathematical Software (TOMS)
Numerical methods for bifurcations of dynamical equilibria
Numerical methods for bifurcations of dynamical equilibria
Numerical Polynomial Algebra
A Rank-Revealing Method with Updating, Downdating, and Applications
SIAM Journal on Matrix Analysis and Applications
Computing the multiplicity structure in solving polynomial systems
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Newton's method with deflation for isolated singularities of polynomial systems
Theoretical Computer Science
Solving parametric polynomial systems
Journal of Symbolic Computation
Adaptive Multiprecision Path Tracking
SIAM Journal on Numerical Analysis
| Hi-index | 7.29 |
Many problems give rise to polynomial systems. These systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters. Using predictor-corrector methods we track the solution paths. A point along a solution path is critical when the Jacobian matrix is rank deficient. The simplest case of quadratic turning points is well understood, but these methods no longer work for general types of singularities. In order not to miss any singular solutions along a path we propose to monitor the determinant of the Jacobian matrix. We examine the operation range of deflation and relate the effectiveness of deflation to the winding number. Computational experiments on systems coming from different application fields are presented.