Working definition
Convergence describes whether an approximation approaches a limiting value as resolution or iteration changes. Precision describes numerical representation or repeatability, while error is deviation from a reference quantity. Reproducible software must not collapse truncation, roundoff, measurement uncertainty, and model discrepancy into one number.
Notation
eₙ = xₙ − x*|eₙ₊₁| ≤ C|eₙ|ᵖAssumptions
- A target quantity and reference meaning are defined.
- Stopping criteria are scale-aware.
- Arithmetic and library versions are recorded.
Invariants
- More printed digits do not imply lower error.
- Convergence to a value does not imply convergence to the correct model.
- Tolerance is not identical to uncertainty.
Reproducible procedure
- Identify error sources before computation.
- Run refinement and independent-reference checks.
- Report precision, tolerance, residual, and uncertainty separately.
Error and boundary controls
- Catastrophic cancellation can dominate.
- Ill-conditioned problems amplify tiny perturbations.
- Unknown model error cannot be inferred from solver residual alone.
What this does not establish
Numerical agreement establishes implementation consistency only within tested conditions; it cannot validate a symbolic or causal claim.
Explicit applications
1 cross-domain bridges
Calculation conformance and precision
Separate solver tolerance, floating-point precision, source uncertainty, and convention disagreement.
Inputs
- implementation output
- independent reference
- declared tolerances
Outputs
- residuals
- pass or explain verdict
- disagreement category
Transformation: Compare continuous values before classifications and attribute discrepancies.
Limit: Conformance within tolerance validates implementation behavior, not interpretive claims.
Open connected system →Authoritative references
- [1]DLMF Chapter 3: Numerical Methods · National Institute of Standards and Technology
Establishes: Reference definitions, algorithms, convergence conditions, and error terms for interpolation, quadrature, differentiation, and nonlinear equation solving.
Boundary: A numerical method is reliable only under its stated regularity, conditioning, precision, and convergence assumptions; the reference does not validate any domain interpretation.
- [2]NIST Technical Note 1297: Guidelines for Evaluating and Expressing Measurement Uncertainty · National Institute of Standards and Technology
Establishes: A measurement framework for identifying uncertainty components, combining standard uncertainties, and reporting expanded uncertainty with declared coverage.
Boundary: Reported uncertainty describes the measurement model and included components. It is not a guarantee that all systematic errors or model inadequacies were found.