Local convergence

A numerical method's guarantee that every sufficiently nearby initial approximation converges to a specified solution.

Version

v1 · 2026-10-25 · History

Domain-specific #

14410

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomain

Numerical Analysis → Mathematics

Core Idea

An iterative numerical method is locally convergent to a particular solution if there is some neighborhood of that solution such that every permitted starting approximation in it produces iterates approaching that solution.^[1] The claim concerns the update rule, the target, and a set of starts. A single successful run cannot establish it.

“Local” identifies where a guarantee holds. It does not say that more distant starts must fail, nor does it state how quickly the iterates approach the target. A separate theorem can give a rate such as quadratic convergence.^[1]

Scope of Application

The property applies to a specified iterative update and specified target under conditions that keep the update defined. NIST discusses it for Newton's scalar root method and Bairstow's iteration on polynomial-factor coefficients.^[1] Theorems for those methods have different conditions; a result for one does not prove a result for the other.^[1]

Clarity

To evaluate a claim, ask: What is the update? Which solution is the target? What starts are permitted? Which assumptions give a neighborhood in which all of those starts converge? A table of iterates answers what happened from its displayed start, while a local-convergence theorem covers a neighborhood. NIST gives both a worked Newton table and a separate simple-zero local result.^[1]

Manages Complexity

There may be many possible starting approximations. A local theorem summarizes an entire neighborhood of them with one conditional guarantee, allowing an analyst to check whether a proposed start is covered. It need not reveal the full set of starts that succeed. Reported failures farther away do not locate the boundary of every convergent start.

Abstract Reasoning

1. Specify the update, target, admissible state space, and theorem assumptions.
2. Identify the neighborhood for which convergence is established, or state that a particular source proves only its existence.
3. Check whether the proposed initial approximation satisfies the proved conditions.
4. Conclude convergence to that target only for starts the theorem covers. If a start is outside the proved region, the theorem is silent; it does not prove failure.

Keep a rate conclusion separate. Local quadratic convergence describes error reduction under its conditions; it does not make an arbitrary initial approximation safe.^[1]

Knowledge Transfer

The same four questions—update, target, starting neighborhood, and all-starts guarantee—apply to unlike numerical methods. Newton updates one root estimate, while Bairstow updates a pair of factor coefficients.^[1] What transfers is the question, not a theorem's method-specific assumptions.

Example

For f(x) = x − tan(x), NIST's Newton table starts at x₀ = 4.65 and approaches the first positive nonzero root, approximately 4.4934094579.^[1] The one run illustrates the method. The local guarantee comes from the separate simple-zero result, provided its conditions hold.^[1] NIST also reports starting values that do not reach this required root, but its reported failure thresholds do not prove that every start between them succeeds.^[1]

Neighborhood in Abstraction Space

Local convergence sits in a sparse region of the domain-specific corpus (63^rd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Dynamical Systems & Differential Structures (38 abstractions)

Nearest neighbors

• Numerov's method — 0.85
• Ridders' Method — 0.85
• Floyd's Cycle-Finding Algorithm — 0.84
• Residence Time (Statistics) — 0.84
• Equicontinuity — 0.84

Computed from structural-signature embeddings · 2026-10-25

Not to Be Confused With

• Convergence of one sequence: that concerns one initial approximation; local convergence quantifies over every permitted start in some neighborhood.
• Convergence order: quadratic or another rate describes speed near the target, not the covered set of starts.
• Global convergence: a claim over a broader declared starting domain requires its own proof. A local result neither gives nor denies it.
• Γ-convergence: the separate domain-specific entry named convergence concerns variational functionals, not this numerical-method property.

References

[1] National Institute of Standards and Technology. “Nonlinear Equations”. Digital Library of Mathematical Functions, §3.8, especially §§3.8(i), 3.8(ii), and 3.8(iv), with the cited equations and worked tables. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j