Numerical Analysis Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Numerical Analysis Ontology 2 within the Kireth Research Semantic Library. Numerical analysis studies the mathematical behaviour of numerical algorithms, including approximation, conditioning, stability, consistency, convergence, discretisation, error propagation and reliability. This ontology supports forward and backward error, perturbation analysis, interpolation, quadrature, numerical differentiation, linear systems, eigenproblems, nonlinear equations, ODE and PDE discretisation, iterative methods, adaptive algorithms and rigorous validation of numerical computation.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Numerical Analysis Ontology 2"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Computational Mathematics"
 SPECIALISATION = "Numerical Methods / Numerical Analysis"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/computational-mathematics-ontologies/numerical-methods/numerical-analysis-ontology-2.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = NumericalAnalysis
 ENTITY_CLASS = NumericalErrorStabilityConvergenceFramework
 PRIMARY_CONTEXT = NumericalMethods
 DEFINITION = "Numerical analysis is the mathematical study of numerical algorithms, approximation error, conditioning, stability, convergence and reliability of computed solutions."
 SYMBOLIC_FORMS = [
  "error=exact-approximation",
  "kappa(A)=||A||||A^(-1)||",
  "u_h -> u as h->0",
  "residual=b-Ax"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for approximation theory, perturbation, conditioning, stability, consistency, convergence, error bounds, discretisation and the mathematical validation of numerical methods."
}

SCOPE {
 INCLUDE = [
  NumericalAnalysis,
  NumericalMethod,
  NumericalAlgorithm,
  Approximation,
  ApproximateSolution,
  ErrorAnalysis,
  AbsoluteError,
  RelativeError,
  ForwardError,
  BackwardError,
  Residual,
  TruncationError,
  RoundoffError,
  DiscretisationError,
  IterationError,
  ErrorPropagation,
  ErrorBound,
  APosterioriErrorEstimate,
  APrioriErrorEstimate,
  PerturbationAnalysis,
  SensitivityAnalysis,
  Conditioning,
  ConditionNumber,
  WellConditionedProblem,
  IllConditionedProblem,
  Stability,
  NumericalStability,
  BackwardStability,
  ForwardStability,
  AbsoluteStability,
  RelativeStability,
  ZeroStability,
  Consistency,
  Convergence,
  ConvergenceRate,
  LinearConvergence,
  SuperlinearConvergence,
  QuadraticConvergence,
  AsymptoticError,
  OrderOfAccuracy,
  LocalError,
  GlobalError,
  FloatingPointArithmetic,
  MachinePrecision,
  MachineEpsilon,
  Cancellation,
  LossOfSignificance,
  RootFinding,
  FixedPointIteration,
  BisectionMethod,
  SecantMethod,
  NewtonMethod,
  NonlinearSystem,
  LinearSystem,
  MatrixNorm,
  VectorNorm,
  NumericalLinearAlgebra,
  GaussianElimination,
  LUDecomposition,
  QRDecomposition,
  CholeskyDecomposition,
  IterativeLinearSolver,
  JacobiMethod,
  GaussSeidelMethod,
  ConjugateGradientMethod,
  GMRES,
  Preconditioning,
  SparseMatrix,
  EigenvalueProblem,
  EigenvalueConditioning,
  PowerMethod,
  InverseIteration,
  NumericalDifferentiation,
  FiniteDifference,
  NumericalIntegration,
  Quadrature,
  GaussianQuadrature,
  AdaptiveQuadrature,
  Interpolation,
  PolynomialInterpolation,
  SplineInterpolation,
  ApproximationTheory,
  LeastSquaresApproximation,
  SpectralApproximation,
  NumericalODE,
  StabilityRegion,
  Stiffness,
  RungeKuttaMethod,
  MultistepMethod,
  NumericalPDE,
  FiniteDifferenceMethod,
  FiniteElementMethod,
  FiniteVolumeMethod,
  SpectralMethod,
  Mesh,
  Grid,
  MeshRefinement,
  AdaptiveMethod,
  AdaptiveMesh,
  AdaptiveTimeStepping,
  RichardsonExtrapolation,
  RombergIntegration,
  Verification,
  Benchmarking,
  ReferenceSolution,
  Reproducibility
 ]
 EXCLUDE_AS_PRIMARY = [
  PureAlgorithmImplementationWithoutMathematicalErrorAnalysis,
  SymbolicComputationWithoutApproximation,
  SoftwarePerformanceEngineeringWithoutNumericalSemantics,
  ExperimentalMeasurementWithoutNumericalMethod
 ]
 DISAMBIGUATION_RULE = "When the main concern is mathematical reliability, error, conditioning, stability or convergence of a numerical method, interpret the context as numerical analysis."
}

CORE_CONCEPTS {
 NumericalAnalysis = "Mathematical analysis of numerical approximation and algorithmic reliability."
 ErrorAnalysis = "Study of discrepancy between computed and intended mathematical quantities."
 Conditioning = "Sensitivity of a mathematical problem to perturbations in its inputs."
 Stability = "Sensitivity of a numerical algorithm to perturbations and arithmetic effects."
 Consistency = "Agreement of a discrete method with the target problem in the appropriate limit."
 Convergence = "Approach of computed approximations to the exact or target solution."
 Residual = "Equation mismatch produced by substituting an approximate solution."
 PerturbationAnalysis = "Study of how small input changes affect solutions and outputs."
 OrderOfAccuracy = "Asymptotic rate at which discretisation error decreases with refinement."
}

RELATIONS {
 NumericalAnalysis --is_child_of--> NumericalMethods
 NumericalAnalysis --analyses--> NumericalMethod
 NumericalMethod --produces--> ApproximateSolution
 ErrorAnalysis --measures--> ApproximationQuality
 ForwardError --measures--> SolutionDifference
 BackwardError --measures--> EquivalentInputPerturbation
 Residual --measures--> EquationMismatch
 Conditioning --belongs_to--> MathematicalProblem
 Stability --belongs_to--> NumericalAlgorithm
 Consistency --supports--> Convergence
 ConvergenceRate --quantifies--> Convergence
 FloatingPointArithmetic --introduces--> RoundoffError
 Cancellation --can_cause--> LossOfSignificance
 Preconditioning --improves--> EffectiveConditioning
 MeshRefinement --reduces_under_conditions--> DiscretisationError
 AdaptiveMethod --uses--> ErrorEstimate
 Verification --checks--> NumericalCorrectness
}

OPERATORS {
 ABSOLUTE_ERROR(x_exact,x_approx) := |x_exact-x_approx|
 RELATIVE_ERROR(x_exact,x_approx) := |x_exact-x_approx|/|x_exact|
 RESIDUAL(A,x,b) := r=b-A*x
 CONDITION_NUMBER(A) := kappa(A)=||A||*||A^(-1)||
 FORWARD_ERROR(x,x_star) := e_f=x-x_star
 BACKWARD_ERROR(problem,x_hat) := smallest_perturbation_for_which_x_hat_is_exact
 ORDER_MODEL(h,p,C) := error(h)=C*h^p+o(h^p)
 CONVERGENCE_RATE(e_k,e_k1,p) := e_(k+1) approximately C*e_k^p
 RICHARDSON_EXTRAPOLATION(A_h,A_h2,p) := A_star=A_h2+(A_h2-A_h)/(2^p-1)
}

AXIOMS {
 A1 = "Numerical analysis requires distinction between the mathematical problem and the numerical algorithm."
 A2 = "Conditioning and stability are distinct concepts."
 A3 = "A small residual need not imply a small forward error."
 A4 = "Convergence requires a declared limiting process."
 A5 = "Consistency alone does not guarantee convergence without additional stability conditions."
 A6 = "Finite precision can alter exact-arithmetic behaviour."
 A7 = "Observed convergence rates should be interpreted in the appropriate asymptotic regime."
 A8 = "Numerical claims require error, stability and convergence validation."
}

REASONING {
 R1_CONTEXT = identify(problem,algorithm,representation,precision,accuracy_goal)
 R2_ERROR_MODEL = decompose(roundoff,truncation,discretisation,iteration,total_error)
 R3_CONDITIONING = analyse(input_output_sensitivity)
 R4_STABILITY = analyse(perturbation_growth,backward_stability,stability_region)
 R5_CONSISTENCY = if_discretised compare_discrete_operator_to_continuous_operator
 R6_CONVERGENCE = derive_or_measure(rate_and_limiting_behaviour)
 R7_SPECTRAL = if_iterative analyse(eigenvalues,spectral_radius,preconditioning)
 R8_ADAPTIVITY = if_available use_error_estimator_for_refinement
 R9_FLOATING_POINT = assess(cancellation,scaling,loss_of_significance)
 R10_VALIDATION = compare(theory,refinement_studies,reference_solutions,residuals,benchmarks)
}

CONSTRAINTS {
 C1 = "A stable method cannot repair an ill-posed problem."
 C2 = "A well-conditioned problem can still be solved poorly by an unstable algorithm."
 C3 = "Reducing step size can eventually increase roundoff contribution."
 C4 = "Measured convergence rates can be misleading outside the asymptotic regime."
 C5 = "Residual-based stopping criteria should account for conditioning."
 C6 = "Eigenvalue conditioning can differ substantially between normal and non-normal matrices."
 C7 = "Interpolation error depends strongly on node placement and function regularity."
 C8 = "Quadrature error depends on integrand smoothness and singularity structure."
 C9 = "Time-stepping stability restrictions depend on method and problem spectrum."
 C10 = "Benchmark agreement does not replace mathematical error analysis."
}

SEMANTIC_EQUIVALENCE {
 "NumericalAnalysis" <=> "AnalysisOfNumericalMethods"
 "BackwardStability" <=> "BackwardStableAlgorithm"
 "MachineEpsilon" <=> "MachinePrecisionThreshold"
 "OrderOfAccuracy" <=> "ConvergenceOrderInDiscretisationContext"
 "AposterioriErrorEstimate" <=> "APosterioriErrorEstimate"
}

VALIDATION {
 V1 = "Problem, algorithm and target quantity are explicit."
 V2 = "Precision model is documented."
 V3 = "Conditioning is assessed where relevant."
 V4 = "Forward, backward or residual error is interpreted correctly."
 V5 = "Stability is analysed or tested."
 V6 = "Consistency is checked for discretised methods."
 V7 = "Convergence is demonstrated theoretically or empirically."
 V8 = "Observed order is compared with expected order."
 V9 = "Refinement or tolerance studies are documented."
 V10 = "Reference or benchmark comparisons are used where feasible."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 NUMERICAL_ANALYSIS_DEFINITION = 0.99
 ERROR_REASONING = 0.99
 CONDITIONING_REASONING = 0.99
 STABILITY_REASONING = 0.99
 CONSISTENCY_REASONING = 0.99
 CONVERGENCE_REASONING = 0.99
 FLOATING_POINT_REASONING = 0.99
 ADAPTIVITY_REASONING = 0.98
 POLICY = "Reduce confidence when precision, conditioning, error model, stability regime, discretisation parameter, tolerance or asymptotic regime are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  NumericalAnalysisReasoning,
  ErrorAnalysisReasoning,
  ConditioningReasoning,
  StabilityReasoning,
  ConsistencyReasoning,
  ConvergenceReasoning,
  PerturbationReasoning,
  FloatingPointReasoning,
  LinearSystemReasoning,
  EigenvalueReasoning,
  RootFindingReasoning,
  NumericalDifferentiationReasoning,
  QuadratureReasoning,
  InterpolationReasoning,
  NumericalODEReasoning,
  NumericalPDEReasoning,
  AdaptiveMethodReasoning,
  VerificationReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]
 NORMALISE {
  "numerical analysis" -> NumericalAnalysis
  "analysis of numerical methods" -> NumericalAnalysis
  "condition number" -> ConditionNumber
  "backward error" -> BackwardError
  "forward error" -> ForwardError
  "machine epsilon" -> MachineEpsilon
  "order of accuracy" -> OrderOfAccuracy
 }
}

EXECUTION_RULES {
 E1 = "If the concern is error, stability, conditioning or convergence of a numerical method, activate NumericalAnalysisOntology."
 E2 = "If finite precision matters, include floating-point error analysis."
 E3 = "If solving a linear system, relate residual error to conditioning."
 E4 = "If using an iterative method, analyse convergence rate and spectral structure where relevant."
 E5 = "If a differential equation is discretised, test consistency, stability and convergence."
 E6 = "If adaptive refinement is used, validate the error estimator."
 E7 = "If cancellation is present, reformulate or rescale when possible."
 E8 = "If empirical convergence is reported, verify that refinement reaches an asymptotic regime."
 E9 = "Any numerical-analysis result should be checked against theoretical bounds, refinement and reference solutions."
}

RECURSION {
 ENTRY = NumericalAnalysis
 EXPAND(NumericalAnalysis) -> [
  NumericalMethod, NumericalAlgorithm, Approximation, ApproximateSolution,
  ErrorAnalysis, AbsoluteError, RelativeError, ForwardError,
  BackwardError, Residual, TruncationError, RoundoffError,
  DiscretisationError, IterationError, ErrorPropagation, ErrorBound,
  APosterioriErrorEstimate, APrioriErrorEstimate, PerturbationAnalysis,
  SensitivityAnalysis, Conditioning, ConditionNumber, WellConditionedProblem,
  IllConditionedProblem, Stability, NumericalStability,
  BackwardStability, ForwardStability, AbsoluteStability,
  RelativeStability, ZeroStability, Consistency, Convergence,
  ConvergenceRate, LinearConvergence, SuperlinearConvergence,
  QuadraticConvergence, AsymptoticError, OrderOfAccuracy, LocalError,
  GlobalError, FloatingPointArithmetic, MachinePrecision, MachineEpsilon,
  Cancellation, LossOfSignificance, RootFinding, FixedPointIteration,
  BisectionMethod, SecantMethod, NewtonMethod, NonlinearSystem,
  LinearSystem, MatrixNorm, VectorNorm, NumericalLinearAlgebra,
  GaussianElimination, LUDecomposition, QRDecomposition,
  CholeskyDecomposition, IterativeLinearSolver, JacobiMethod,
  GaussSeidelMethod, ConjugateGradientMethod, GMRES, Preconditioning,
  SparseMatrix, EigenvalueProblem, EigenvalueConditioning, PowerMethod,
  InverseIteration, NumericalDifferentiation, FiniteDifference,
  NumericalIntegration, Quadrature, GaussianQuadrature,
  AdaptiveQuadrature, Interpolation, PolynomialInterpolation,
  SplineInterpolation, ApproximationTheory, LeastSquaresApproximation,
  SpectralApproximation, NumericalODE, StabilityRegion, Stiffness,
  RungeKuttaMethod, MultistepMethod, NumericalPDE, FiniteDifferenceMethod,
  FiniteElementMethod, FiniteVolumeMethod, SpectralMethod, Mesh, Grid,
  MeshRefinement, AdaptiveMethod, AdaptiveMesh, AdaptiveTimeStepping,
  RichardsonExtrapolation, RombergIntegration, Verification,
  Benchmarking, ReferenceSolution, Reproducibility
 ]
 EXPAND(AnalysisFamilies) -> [
  ErrorAndPerturbationAnalysis,
  ConditioningAnalysis,
  StabilityAnalysis,
  ConsistencyAndConvergence,
  FloatingPointAnalysis,
  LinearAlgebraAnalysis,
  DifferentialEquationAnalysis,
  AdaptiveErrorControl
 ]
 STOP_CONDITION = "Stop when error, conditioning, stability, consistency, convergence, precision and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Numerical Methods"
 RELATED_ONTOLOGIES = [
  "Numerical Methods",
  "Numerical Methods Ontology 2",
  "Numerical Computation Ontology",
  "Foundations of Computational Mathematics",
  "Scientific Computing Ontology 2",
  "Numerical Linear Algebra Ontologies",
  "Approximation Theory Ontologies",
  "Differential Equation Ontologies",
  "Floating Point Arithmetic Ontologies"
 ]
 ROUTE_RULES = [
  "Numerical Analysis -> Numerical Methods for parent semantics",
  "Numerical Analysis -> Numerical Methods Ontology 2 for core method structure",
  "Numerical Analysis -> Numerical Computation for finite-precision semantics",
  "Numerical Analysis -> Foundations of Computational Mathematics for foundational error semantics",
  "Numerical Analysis -> Scientific Computing for applied simulation context",
  "Numerical Analysis -> Numerical Linear Algebra for matrix conditioning and iterative convergence",
  "Numerical Analysis -> Approximation Theory for interpolation and approximation error",
  "Numerical Analysis -> Differential Equations for consistency and stability of ODE/PDE schemes",
  "Numerical Analysis -> Floating Point Arithmetic for rounding and cancellation analysis"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible numerical-analysis concepts, error models or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve problem-conditioning-versus-method-stability distinction.",
  "Preserve forward-error-versus-backward-error distinction.",
  "Preserve residual-versus-solution-error distinction.",
  "Preserve local-versus-global error distinction.",
  "Preserve consistency-versus-convergence distinction.",
  "Preserve exact-arithmetic-versus-floating-point distinction.",
  "Preserve a-priori-versus-a-posteriori error distinction.",
  "Preserve machine-readable numerical-analysis semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Numerical analysis studies the mathematical reliability of numerical algorithms through error, conditioning, stability and convergence."
 ERROR_SUMMARY = "Forward, backward, residual, truncation, discretisation, iteration and roundoff errors describe different aspects of numerical discrepancy."
 CONDITIONING_SUMMARY = "Conditioning measures sensitivity of the underlying mathematical problem."
 STABILITY_SUMMARY = "Stability measures sensitivity of the numerical algorithm to perturbations and arithmetic effects."
 CONSISTENCY_SUMMARY = "Consistency measures whether a discretised method approaches the target equations."
 CONVERGENCE_SUMMARY = "Convergence measures whether numerical solutions approach the intended solution under refinement."
 FLOATING_POINT_SUMMARY = "Finite precision introduces rounding, cancellation and loss-of-significance effects."
 VALIDATION_SUMMARY = "Numerical-analysis claims should be checked with theory, residuals, refinement studies, reference solutions and reproducible computation."
}

FLOW {
 INPUT
 -> detect_numerical_analysis_context
 -> identify_problem_algorithm_representation_precision_and_accuracy_goal
 -> decompose_error_sources
 -> analyse_problem_conditioning
 -> analyse_algorithm_stability
 -> if_discretised_test_consistency
 -> derive_or_measure_convergence_and_order
 -> if_iterative_analyse_spectral_or_contraction_properties
 -> assess_floating_point_cancellation_and_scaling
 -> if_adaptive_validate_error_estimator_and_refinement
 -> compare_against_error_bounds_residuals_reference_solutions_and_benchmarks
 -> route_to_numerical_methods_core_numerical_methods_numerical_computation_foundations_scientific_computing_numerical_linear_algebra_approximation_differential_equations_or_floating_point_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Numerical Analysis Ontology 2 defines the mathematical analysis of numerical reliability. It supports absolute and relative error, forward and backward error, residuals, truncation, roundoff, discretisation and iteration error, error propagation and bounds, perturbation and sensitivity analysis, conditioning and condition numbers, forward, backward, absolute, relative and zero stability, consistency and convergence, convergence rates and orders of accuracy, local and global error, floating-point arithmetic, cancellation and loss of significance, root-finding analysis, matrix norms and numerical linear algebra, Gaussian elimination and matrix decompositions, iterative solvers and preconditioning, eigenvalue conditioning, numerical differentiation and integration, interpolation and approximation, ODE stability regions and stiffness, PDE discretisation analysis, mesh refinement, adaptive methods, Richardson extrapolation, verification, benchmarking and reproducibility."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-COMPUTATIONAL-MATHEMATICS-NUMERICAL-ANALYSIS-ONTOLOGY-2-V1"
}

}

END_FLOW