Numerical Error Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Numerical Error Ontology within the Kireth Research Semantic Library. Numerical error describes discrepancies introduced when mathematical quantities and solutions are represented or computed approximately.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Numerical Error Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Computational Mathematics"
 SPECIALISATION = "Numerical Methods / Numerical Error"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/computational-mathematics-ontologies/numerical-methods/numerical-error-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = NumericalError
 ENTITY_CLASS = ComputationalApproximationErrorFramework
 PRIMARY_CONTEXT = NumericalMethods
 DEFINITION = "Numerical error is the discrepancy between an exact or reference mathematical quantity and a value produced by finite representation, approximation, discretisation or numerical computation."
 SYMBOLIC_FORMS = ["e=x_exact-x_approx","e_abs=|x_exact-x_approx|","e_rel=|x_exact-x_approx|/|x_exact|","r=b-A*x_hat"]
 SEMANTIC_ROLE = "Provides machine-readable semantics for numerical discrepancy, error sources, propagation, bounds, conditioning relationships, residual interpretation and finite precision."
}

SCOPE {
 INCLUDE = [NumericalError,ApproximationError,AbsoluteError,RelativeError,PercentageError,ForwardError,BackwardError,Residual,ResidualNorm,TruncationError,RoundoffError,DiscretisationError,IterationError,InterpolationError,QuadratureError,DifferentiationError,IntegrationError,LinearSolveError,EigenvalueError,RootFindingError,ODEError,PDEError,LocalError,GlobalError,LocalTruncationError,GlobalTruncationError,FloatingPointError,RepresentationError,QuantisationError,RoundingError,ChoppingError,Cancellation,CatastrophicCancellation,LossOfSignificance,Overflow,Underflow,MachinePrecision,MachineEpsilon,SignificantDigits,ErrorPropagation,ErrorAmplification,ErrorAccumulation,Perturbation,Sensitivity,Conditioning,ConditionNumber,Stability,NumericalStability,BackwardStability,ForwardStability,ErrorBound,APrioriErrorEstimate,APosterioriErrorEstimate,ErrorEstimator,ErrorIndicator,AsymptoticError,ErrorConstant,OrderOfAccuracy,ConvergenceRate,RichardsonExtrapolation,GridConvergenceIndex,GCI,Norm,L1Norm,L2Norm,InfinityNorm,EnergyNorm,SobolevNorm,AbsoluteTolerance,RelativeTolerance,ResidualTolerance,StoppingCriterion,ErrorBudget,ErrorFloor,RoundoffFloor,NumericalUncertainty,Verification,RefinementStudy,PrecisionStudy,Benchmark,ReferenceSolution,Reproducibility]
 EXCLUDE_AS_PRIMARY = [MeasurementErrorWithoutNumericalComputation,StatisticalSamplingErrorWithoutNumericalApproximation,ModelFormErrorWithoutNumericalErrorContext,SoftwareBugWithoutMathematicalErrorSemantics]
 DISAMBIGUATION_RULE = "When discrepancy arises from numerical representation, approximation, discretisation, iteration or finite arithmetic, interpret the context as numerical error."
}

CORE_CONCEPTS {
 NumericalError = "Discrepancy introduced or observed in numerical representation and computation."
 AbsoluteError = "Magnitude of the difference between reference and approximate values."
 RelativeError = "Absolute error normalised by a reference magnitude."
 ForwardError = "Difference between computed and exact solutions of the original problem."
 BackwardError = "Smallest perturbation for which the computed result is exact."
 Residual = "Mismatch obtained by substituting an approximate solution into the governing equation."
 Truncation-Error = "Error caused by replacing an infinite or limiting process with a finite approximation."
 RoundoffError = "Error caused by finite-precision representation and arithmetic."
 Discretisation-Error = "Error introduced by replacing a continuous problem with a discrete representation."
 Iteration-Error = "Error remaining because an iterative process is terminated before its limiting solution."
}

RELATIONS {
 NumericalError --is_child_of--> NumericalMethods
 NumericalError --is_analysed_by--> NumericalAnalysis
 AbsoluteError --measures--> MagnitudeDifference
 RelativeError --normalises--> AbsoluteError
 ForwardError --compares--> ComputedAndExactSolution
 BackwardError --measures--> EquivalentProblemPerturbation
 Residual --measures--> EquationMismatch
 FloatingPointArithmetic --introduces--> RoundoffError
 Discretisation --introduces--> DiscretisationError
 IterativeMethod --leaves--> IterationError
 Cancellation --can_cause--> LossOfSignificance
 Conditioning --influences--> ErrorAmplification
 NumericalStability --controls--> ErrorAmplification
 ErrorEstimator --estimates--> NumericalError
 RefinementStudy --quantifies--> DiscretisationError
}

OPERATORS {
 ABSOLUTE_ERROR(x_ref,x_hat) := |x_ref-x_hat|
 RELATIVE_ERROR(x_ref,x_hat) := |x_ref-x_hat|/|x_ref|
 PERCENTAGE_ERROR(x_ref,x_hat) := 100*RELATIVE_ERROR(x_ref,x_hat)
 RESIDUAL(A,x_hat,b) := r=b-A*x_hat
 FORWARD_ERROR(x_hat,x_star) := e_f=x_hat-x_star
 BACKWARD_ERROR(problem,x_hat) := minimum_perturbation_making_x_hat_exact
 CONDITION_RELATION(kappa,eta) := relative_forward_error approximately kappa*relative_backward_error
 ERROR_PROPAGATION(J,delta_x) := delta_y approximately J*delta_x
 ASYMPTOTIC_ERROR(h,C,p) := E(h)=C*h^p+o(h^p)
}

AXIOMS {
 A1 = "Numerical error requires a reference quantity, governing relation or declared error estimator."
 A2 = "Absolute and relative error answer different scale-dependent questions."
 A3 = "Residual and forward error are distinct quantities."
 A4 = "Problem conditioning influences how backward error maps to forward error."
 A5 = "Roundoff and truncation error can trade off as numerical resolution changes."
 A6 = "Error estimates are meaningful only under their stated assumptions and norms."
 A7 = "Finite precision creates a practical error floor for many computations."
 A8 = "Numerical error claims require explicit source, metric, scale and validation."
}

REASONING {
 R1_CONTEXT = identify(reference_quantity,computed_quantity,algorithm,precision,error_metric)
 R2_CLASS = classify(roundoff,truncation,discretisation,iteration,interpolation,quadrature_or_other_error)
 R3_MEASURE = compute(absolute,relative,forward,backward_or_residual_error)
 R4_CONDITIONING = assess(problem_sensitivity_and_error_amplification)
 R5_PROPAGATION = propagate(input_and_intermediate_errors_through_computation)
 R6_BOUND = derive_or_estimate(a_priori,a_posteriori_or_asymptotic_error_bound)
 R7_BALANCE = assess(truncation_roundoff_iteration_and_discretisation_tradeoffs)
 R8_REFINEMENT = perform(mesh,timestep,iteration,order_or_precision_study_if_required)
 R9_TOLERANCE = compare(error_against_absolute_relative_or_residual_tolerance)
 R10_VALIDATION = compare(reference_solution,error_bound,observed_order,precision,reproducibility)
}

CONSTRAINTS {
 C1 = "Relative error can be undefined or misleading when the reference value is zero or near zero."
 C2 = "A small residual does not guarantee a small forward error for ill-conditioned problems."
 C3 = "Reducing step size can reduce truncation error while increasing roundoff influence."
 C4 = "Increasing iteration count cannot overcome an inaccurate discretisation indefinitely."
 C5 = "Cancellation can magnify relative error even when operations are correctly rounded."
 C6 = "Error components are not always independent."
 C7 = "A-priori bounds can be conservative."
 C8 = "A-posteriori estimators require validation for the problem class."
 C9 = "Reference solutions can themselves contain numerical error."
 C10 = "Model-form uncertainty must not be silently reported as numerical error."
}

SEMANTIC_EQUIVALENCE {
 "NumericalError" <=> "ComputationalNumericalError"
 "RoundingError" <=> "RoundoffError"
 "CatastrophicCancellation" <=> "SevereCancellation"
 "GridConvergenceIndex" <=> "GCI"
}

VALIDATION {
 V1 = "Reference quantity or governing relation is explicit."
 V2 = "Error metric and norm are documented."
 V3 = "Absolute and relative scales are interpreted correctly."
 V4 = "Residuals are distinguished from solution errors."
 V5 = "Conditioning is assessed where relevant."
 V6 = "Roundoff, truncation, discretisation and iteration errors are separated where feasible."
 V7 = "Refinement studies are used to assess discretisation error."
 V8 = "Precision studies are used where roundoff may dominate."
 V9 = "Error estimators are checked against reference solutions where feasible."
 V10 = "Numerical error results are reproducible under documented settings."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 NUMERICAL_ERROR_DEFINITION = 0.99
 FORWARD_BACKWARD_ERROR_REASONING = 0.99
 RESIDUAL_REASONING = 0.99
 ROUNDOFF_REASONING = 0.99
 TRUNCATION_REASONING = 0.99
 DISCRETISATION_ERROR_REASONING = 0.99
 POLICY = "Reduce confidence when reference value, error norm, conditioning, precision, discretisation, stopping tolerance or estimator assumptions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [SemanticSearch,OntologyGrounding,NumericalErrorReasoning,AbsoluteErrorReasoning,RelativeErrorReasoning,ForwardErrorReasoning,BackwardErrorReasoning,ResidualReasoning,RoundoffErrorReasoning,TruncationErrorReasoning,DiscretisationErrorReasoning,IterationErrorReasoning,ErrorPropagationReasoning,ConditioningReasoning,CancellationReasoning,ErrorBoundReasoning,ErrorEstimatorReasoning,RefinementStudyReasoning,PrecisionStudyReasoning,VerificationReasoning,KnowledgeGraphConstruction,RetrievalAugmentation,RecursiveInference]
 NORMALISE {
  "numerical error" -> NumericalError
  "rounding error" -> RoundoffError
  "forward error" -> ForwardError
  "backward error" -> BackwardError
  "GCI" -> GridConvergenceIndex
 }
}

EXECUTION_RULES {
 E1 = "If discrepancy arises from numerical approximation or finite computation, activate NumericalErrorOntology."
 E2 = "If a reference solution is available, compute absolute and relative error where meaningful."
 E3 = "If solving equations, distinguish residual, backward error and forward error."
 E4 = "If finite precision matters, analyse roundoff, cancellation, overflow and underflow."
 E5 = "If a continuous problem is discretised, quantify discretisation and truncation error."
 E6 = "If an iterative solver is stopped early, quantify iteration error."
 E7 = "If error changes under refinement, estimate asymptotic order where appropriate."
 E8 = "If a tolerance is used, scale it to the quantity and conditioning."
 E9 = "Validate numerical error through bounds, refinement, precision studies or reference solutions."
}

RECURSION {
 ENTRY = NumericalError
 EXPAND(NumericalError) -> [AbsoluteError,RelativeError,ForwardError,BackwardError,Residual,TruncationError,RoundoffError,DiscretisationError,IterationError,InterpolationError,QuadratureError,LocalError,GlobalError,FloatingPointError,Cancellation,LossOfSignificance,MachinePrecision,ErrorPropagation,ErrorAmplification,Perturbation,Conditioning,ConditionNumber,NumericalStability,ErrorBound,APrioriErrorEstimate,APosterioriErrorEstimate,ErrorEstimator,AsymptoticError,OrderOfAccuracy,ConvergenceRate,RichardsonExtrapolation,GridConvergenceIndex,Norm,AbsoluteTolerance,RelativeTolerance,ResidualTolerance,ErrorBudget,ErrorFloor,NumericalUncertainty,Verification,RefinementStudy,PrecisionStudy,ReferenceSolution,Reproducibility]
 STOP_CONDITION = "Stop when reference, error type, metric, propagation, conditioning, bounds, tolerances and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Numerical Methods"
 RELATED_ONTOLOGIES = ["Numerical Methods","Numerical Methods Ontology 2","Numerical Analysis Ontology 2","Numerical Approximation Ontology","Numerical Stability Ontology 2","Numerical Convergence Ontology","Numerical Computation Ontology","Floating Point Arithmetic Ontologies","Scientific Computing Ontology 2"]
 ROUTE_RULES = [
  "Numerical Error -> Numerical Analysis for formal error analysis",
  "Numerical Error -> Numerical Approximation for approximation error",
  "Numerical Error -> Numerical Stability for error amplification",
  "Numerical Error -> Numerical Convergence for refinement and asymptotic error",
  "Numerical Error -> Floating Point Arithmetic for roundoff and cancellation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible numerical error classes, estimators or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = ["Preserve absolute-versus-relative error distinction.","Preserve forward-versus-backward error distinction.","Preserve residual-versus-solution-error distinction.","Preserve truncation-versus-roundoff error distinction.","Preserve discretisation-versus-iteration error distinction.","Preserve numerical-error-versus-model-error distinction.","Preserve machine-readable numerical-error semantics."]
}

OUTPUT {
 CANONICAL_DEFINITION = "Numerical error is discrepancy associated with finite representation, approximation, discretisation, iteration or arithmetic in numerical computation."
 SCALE_SUMMARY = "Absolute error measures magnitude while relative error measures discrepancy relative to a reference scale."
 FORWARD_BACKWARD_SUMMARY = "Forward error measures solution discrepancy; backward error measures the perturbation required to make a computed result exact."
 RESIDUAL_SUMMARY = "Residual measures equation mismatch and must be interpreted with conditioning."
 SOURCE_SUMMARY = "Major sources include roundoff, truncation, discretisation, interpolation and iteration error."
 VALIDATION_SUMMARY = "Numerical error claims should be checked using bounds, refinement, precision studies, reference solutions and reproducible settings."
}

FLOW {
 INPUT
 -> detect_numerical_error_context
 -> identify_reference_quantity_computed_quantity_algorithm_precision_and_error_metric
 -> classify_error_source
 -> compute_absolute_relative_forward_backward_or_residual_error
 -> analyse_conditioning_and_error_amplification
 -> propagate_input_intermediate_and_roundoff_errors
 -> derive_or_estimate_error_bounds
 -> assess_truncation_roundoff_discretisation_and_iteration_tradeoffs
 -> perform_refinement_or_precision_study_if_required
 -> compare_against_tolerances_reference_solutions_and_expected_order
 -> route_to_related_numerical_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Numerical Error Ontology defines discrepancy in numerical computation, including absolute and relative error, forward and backward error, residuals, truncation, roundoff, discretisation and iteration error, local and global errors, floating-point representation, cancellation, machine precision, error propagation, conditioning and stability relationships, error bounds, a-priori and a-posteriori estimates, asymptotic error, convergence order, refinement studies, tolerances, numerical uncertainty, verification and reproducibility."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-COMPUTATIONAL-MATHEMATICS-NUMERICAL-ERROR-ONTOLOGY-V1"
}

}

END_FLOW