A Shamanskii-Like acceleration scheme for nonlinear equations at singular roots
Mathematics of Computation
Numerical continuation methods: an introduction
Numerical continuation methods: an introduction
On the computation of impasse points of quasi-linear differential-algebraic equations
Mathematics of Computation
Using dynamical systems methods to solve minimization problems
NUMDIFF-7 Selected papers of the seventh conference on Numerical treatment of differential equations
Algorithm 777: HOMPACK90: a suite of Fortran 90 codes for globally convergent homotopy algorithms
ACM Transactions on Mathematical Software (TOMS)
Iterative solution of nonlinear equations in several variables
Iterative solution of nonlinear equations in several variables
Adaptive cellular integration of linearly implicit differential equations
Proceedings of the on Numerical methods for differential equations
Practical Quasi-Newton algorithms for singular nonlinear systems
Numerical Algorithms
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This paper addresses the use of dynamical system theory to tackle singular root-finding problems. The use of continuous-time methods leads to implicit differential systems when applied to singular nonlinear equations. The analysis is based on a taxonomy of singularities and uses previous stability results proved in the context of quasilinear implicit ODEs. The proposed approach provides a framework for the systematic formulation of quadratically convergent iterations to singular roots. The scope of the work includes also the introduction of discrete-time analysis techniques for singular problems which are based on continuous-time stability and numerical stability. Some numerical experiments illustrate the applicability of the proposed techniques.