Newton's method for singular nonlinear equations using approximate left and right nullspaces of the Jacobian

DOI10.1016/j.apnum.2004.09.029zbMath1081.65046OpenAlexW2005164796MaRDI QIDQ557946

Yun-Qiu Shen, Tjalling J. Ypma

Publication date: 30 June 2005

Published in: Applied Numerical Mathematics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.apnum.2004.09.029

zbMATH Keywords

Newton's methodnumerical examplesSingular value decompositionNonlinear algebraic equationsQuadratic convergenceRank deficiencySingular Jacobian matrices

Mathematics Subject Classification ID

Numerical computation of solutions to systems of equations (65H10)

Related Items (12)

An efficient algorithm for the separable nonlinear least squares problem ⋮ Improved two-step Newton's method for computing simple multiple zeros of polynomial systems ⋮ Practical quasi-Newton algorithms for singular nonlinear systems ⋮ Solving rank-deficient separable nonlinear equations ⋮ Verified error bounds for isolated singular solutions of polynomial systems: case of breadth one ⋮ Finite-difference method for singular nonlinear systems ⋮ Two-step Newton's method for deflation-one singular zeros of analytic systems ⋮ Numerical approximation of multiple isolated roots of analytical systems ⋮ Verified error bounds for eigenvalues of geometric multiplicity \(q\) and corresponding invariant subspaces ⋮ Solving separable nonlinear least squares problems using the QR factorization ⋮ On isolation of simple multiple zeros and clusters of zeros of polynomial systems ⋮ Verified error bounds for singular solutions of nonlinear systems

Uses Software

• KELLEY

Cites Work

• Bifurcations via singular value decompositions
• Practical bifurcation and stability analysis: from equilibrium to chaos.
• Computation of a simple bifurcation point using one singular value decomposition nearby
• Characterization and Computation of Generalized Turning Points
• Local Convergence of Inexact Newton Methods
• On Solving Nonlinear Equations with Simple Singularities or Nearly Singular Solutions
• Characterization and Computation of Singular Points with Maximum Rank Deficiency
• Direct Methods for Solving Singular Nonlinear Equations
• Convergence of Newton's Method for Singular Smooth and Nonsmooth Equations Using Adaptive Outer Inverses
• Historical Development of the Newton–Raphson Method
• Numerical Methods for Bifurcations of Dynamical Equilibria
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