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A Newton's Iteration Converges Quadratically to Nonisolated Solutions Too

Zhonggang Zeng

TL;DR

The paper addresses solving nonlinear systems with nonisolated solutions where Jacobians may be rank-deficient, by introducing a rank-$r$ Newton's iteration $\mathbf{x}_{k+1}=\mathbf{x}_k- J_{\text{rank-$r$}}(\mathbf{x}_k)^{\dagger}\mathbf{f}(\mathbf{x}_k)$. It proves local quadratic convergence to semiregular zeros for exact data and linear convergence with explicit error bounds under perturbations, implying regularization of ill-posed zeros and a geometric interpretation as projection toward the nearest point on the solution manifold. The work also provides practical algorithms for computing the rank-$r$ pseudoinverse, and demonstrates versatility across numerical algebraic geometry, GCD computations, and defective eigenvalue problems. It further discusses how to handle ultrasingular zeros via deflation techniques, enabling robust convergence to semiregular zeros and preserving accuracy commensurate with data precision. Altogether, the rank-$r$ Newton's method broadens Newton's paradigm to nonisolated solutions and offers a unified framework for regularization, modeling, and computation in nonlinear systems.

Abstract

The textbook Newton's iteration is practically inapplicable on solutions of nonlinear systems with singular Jacobians. By a simple modification, a novel extension of Newton's iteration regains its local quadratic convergence toward nonisolated solutions that are semiregular as properly defined regardless of whether the system is square, underdetermined or overdetermined while Jacobians can be rank-deficient. Furthermore, the iteration serves as a regularization mechanism for computing singular solutions from empirical data. When a system is perturbed, its nonisolated solutions can be altered substantially or even disappear. The iteration still locally converges to a stationary point that approximates a singular solution of the underlying system with an error bound in the same order of the data accuracy. Geometrically, the iteration approximately approaches the nearest point on the solution manifold. The method simplifies the modeling of nonlinear systems by permitting nonisolated solutions and enables a wide range of applications in algebraic computation.

A Newton's Iteration Converges Quadratically to Nonisolated Solutions Too

TL;DR

The paper addresses solving nonlinear systems with nonisolated solutions where Jacobians may be rank-deficient, by introducing a rank- Newton's iteration r. It proves local quadratic convergence to semiregular zeros for exact data and linear convergence with explicit error bounds under perturbations, implying regularization of ill-posed zeros and a geometric interpretation as projection toward the nearest point on the solution manifold. The work also provides practical algorithms for computing the rank- pseudoinverse, and demonstrates versatility across numerical algebraic geometry, GCD computations, and defective eigenvalue problems. It further discusses how to handle ultrasingular zeros via deflation techniques, enabling robust convergence to semiregular zeros and preserving accuracy commensurate with data precision. Altogether, the rank- Newton's method broadens Newton's paradigm to nonisolated solutions and offers a unified framework for regularization, modeling, and computation in nonlinear systems.

Abstract

The textbook Newton's iteration is practically inapplicable on solutions of nonlinear systems with singular Jacobians. By a simple modification, a novel extension of Newton's iteration regains its local quadratic convergence toward nonisolated solutions that are semiregular as properly defined regardless of whether the system is square, underdetermined or overdetermined while Jacobians can be rank-deficient. Furthermore, the iteration serves as a regularization mechanism for computing singular solutions from empirical data. When a system is perturbed, its nonisolated solutions can be altered substantially or even disappear. The iteration still locally converges to a stationary point that approximates a singular solution of the underlying system with an error bound in the same order of the data accuracy. Geometrically, the iteration approximately approaches the nearest point on the solution manifold. The method simplifies the modeling of nonlinear systems by permitting nonisolated solutions and enables a wide range of applications in algebraic computation.

Paper Structure

This paper contains 13 sections, 8 theorems, 103 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Semiregular zeros of smooth mappings
  4. Convergence theorem on exact equations
  5. Convergence theorem on perturbed equations
  6. A geometric interpretation
  7. Algorithms for computing $J_{\hbox{\scriptsize rank-$r$}}(\mathbf{x}_k)^\dagger\,\mathbf{f}(\mathbf{x}_k)$
  8. Modeling with nonisolated solutions
  9. Numerical Algebraic Geometry
  10. Numerical greatest common divisor
  11. Accurate computation of defective eigenvalues
  12. A note on computing ultrasingular zeros
  13. Conclusion

Key Result

Lemma 2.1

Let $J(\mathbf{x})$ be the Jacobian of a smooth mapping $\mathbf{f}$ at any $\mathbf{x}$ in its open domain $\Omega$ in $\mathbbm{C}^n$ or $\mathbbm{R}^n$. Assume $\mathpzc{rank}\left(\,J(\mathbf{x}_*)\,\right)=r$ at $\mathbf{x}_*\in\Omega$. Then there is an open bounded convex subset

Figures (2)

  • Figure 1: Each sequence of iteration (\ref{['ni']}) asymptotically follows a normal line toward the solution set. Illustration is plotted using actual data in Example \ref{['e:dir']} from two initial iterates.
  • Figure 2: Solution sets of the system in (\ref{['ilu']})

Theorems & Definitions (26)

  • Lemma 2.1
  • Lemma 2.2
  • Definition 3.1: Dimension of a Zero
  • Definition 3.2: Semiregular Zero
  • Lemma 3.3: Local Invariance of Semiregularity
  • Lemma 3.4: Stationary Point Property
  • Remark 3.5: A note on terminology
  • Theorem 4.1: Convergence Theorem
  • Remark 4.2: Convergence near an ultrasingular zero
  • Remark 4.3: On the projection rank $r$
  • ...and 16 more