Nonlinear systems and fixed points

Use this page when several coupled constraints must vanish together and you need to distinguish a locally accurate Newton step from a globally reliable solve.

The scientific question¶

A nonlinear system asks for a state whose coupled residuals all vanish. The scientific task is not only to obtain a small residual. It is to decide whether the variables, residual scales, constraints, and selected solution branch make that state meaningful.

Local convergence describes behavior once an iterate is sufficiently near a regular solution. Globalization describes how an algorithm tries to reach such a neighborhood from a less favorable initial state. A successful local model does not imply global convergence or uniqueness.

Mathematical objects¶

Let $\mathbf{F}:\mathbb{R}^n\rightarrow\mathbb{R}^n$ map a state $\mathbf{x}$ to residuals. Its Jacobian has entries $J_{ij}=\partial F_i/\partial x_j$. Residual units can differ by equation, so both state scaling and residual scaling are part of the numerical problem.

A termination policy normally combines a scaled residual norm, a scaled step norm, iteration limits, and finite-value checks. A singular or ill-conditioned Jacobian can make the Newton direction undefined or extremely sensitive even when the current residual is finite.

Core derivation¶

Linearize the residual around the current iterate and require the linear model to vanish at the next state. This gives the Newton system:

The linear solve in (1) is a local model, not a certificate that the full nonlinear residual decreases. Damping, line search, trust regions, or continuation can globalize a method, but each introduces its own branch and stopping behavior.

What the ecosystem already owns¶

Optimistix is the ecosystem owner for nonlinear solves and related fixed-point, least-squares, and minimization problems. Its solver selection, iteration mechanics, transforms, termination, and result diagnostics belong there. Linear subproblems should use the operator and solver abstractions chosen by that ecosystem rather than a second Jaxstro solver stack.

What Jaxstro may add¶

A later consumer-driven adapter could attach unit-aware variable and residual scales, record the solver configuration in provenance, and connect termination telemetry to a scientific evidence report. No such adapter is implemented or promised by this guide.

The adapter boundary would remain narrow: it could prepare a problem and interpret evidence, but it would not fork Newton, fixed-point, or globalization algorithms.

Evidence required before implementation¶

Before an adapter could be called ready, it would need:

• problems with analytic solutions and independent residual checks;

• scale-change tests showing that equivalent unit choices do not alter the physical answer;

• singular and ill-conditioned Jacobian cases that fail or warn explicitly;

• multiple-root cases that report dependence on initialization and branch;

• transform tests for the exact derivative contract the adapter claims; and

• provenance round trips for tolerances, solver choice, status, and telemetry.

Claim boundary¶

This page derives the local Newton model and states ownership. It does not benchmark Optimistix, certify any solver for a particular science problem, or claim a Jaxstro nonlinear-systems runtime.

Connected foundations and methods¶

Review Linear algebra as the language of change for Jacobians and linear maps, and Sensitivity, conditioning, and identifiability for conditioning and identifiability. Compare the scalar contract in Root-finding, objective-based formulations in Optimization helpers, and derivative mechanics in Autodiff products.