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.
Direct answer
- 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.
Mechanism and method
- Identify error sources before computation.
- Run refinement and independent-reference checks.
- Report precision, tolerance, residual, and uncertainty separately.
What is measured
- 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.
Interval addition, executed
- Method: interval-add, inclusive integer intervals with outward rounding on signed 64-bit integers. Units: nanometre, chosen only because the operation needs a unit; the arithmetic is unitless.
- Widths add: [1000,1010] + [500,505] → [1500,1515]. The inputs are 10 and 5 wide and the result is 15, so the bound on the sum is the sum of the bounds and nothing is hidden.
- Error accumulates: [1500,1515] + [500,505] → [2000,2020], now 20 wide. A second addition widens the interval again, which is the page's point that tolerance is not uncertainty.
- The loosest term dominates: [1000,1001] + [0,500] → [1000,1501], 501 wide. A tightly known quantity added to a poorly known one is poorly known, however many digits the first one had.
- Assumptions: Endpoints are exact integers in the stated unit, and the interval is inclusive of both. Rounding is outward, so a computed interval never claims to be narrower than the true one.
- Uncertainty: The interval is the uncertainty statement. Addition is exact on integers and aborts on overflow rather than wrapping, so no width is lost to the arithmetic itself.
- Executed by kernel sha256:9046cc9127b3992b962e1a1f2be7bcf62c9a8e355bd29a4d58f2fb2a597df7f3. Rebuild the kernel with npm run build:wasm-kernel, which is deterministic, confirm its digest matches kernelSha256, then re-execute each operation on the recorded inputs.
- What the receipt does not establish: That the kernel is correct. A receipt records what it computed, not whether the computation is right. That the operation models anything outside itself. That the page’s claims are true. These cases exhibit stated invariants on specific values, and exhibiting is not proving.
Limitations
- Catastrophic cancellation can dominate.
- Ill-conditioned problems amplify tiny perturbations.
- Unknown model error cannot be inferred from solver residual alone.
- A target quantity and reference meaning are defined.
- Stopping criteria are scale-aware.
- Arithmetic and library versions are recorded.
What this does not establish
- Numerical agreement establishes implementation consistency only within tested conditions; it cannot validate a symbolic or causal claim.
Bridge: Calculation conformance and precision
- Separate solver tolerance, floating-point precision, source uncertainty, and convention disagreement.
- Input: implementation output
- Input: independent reference
- Input: declared tolerances
- Output: residuals
- Output: pass or explain verdict
- Output: disagreement category
- Limit: Conformance within tolerance validates implementation behavior, not interpretive claims.