Quadrature Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Quadrature Ontology within the Kireth Research Semantic Library. Quadrature is the numerical approximation of definite integrals by weighted sums of function values. This ontology supports Newton-Cotes formulas, Gaussian quadrature, composite and adaptive rules, weighted integrals, singular and oscillatory integration, multidimensional cubature, sparse grids, error estimation, convergence, stability, finite precision and verification.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = Quadrature
 ENTITY_CLASS = NumericalIntegralApproximationFramework
 PRIMARY_CONTEXT = NumericalCalculus
 DEFINITION = "Quadrature is the numerical approximation of a definite integral using weighted sums of function evaluations at selected nodes."
 SYMBOLIC_FORMS = [
  "integral_a^b f(x) dx approximately sum_i w_i*f(x_i)",
  "Q_n(f)=sum_i w_i*f(x_i)",
  "I-Q_n(f)=E_n",
  "Q_h -> I"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for numerical integration rules, quadrature nodes and weights, composite and adaptive strategies, Gaussian formulas, error estimation, convergence and finite-precision validation."
}

SCOPE {
 INCLUDE = [
  Quadrature, NumericalIntegration, NumericalIntegral,
  QuadratureRule, QuadratureFormula, QuadratureNode, QuadratureWeight,
  WeightedSum, IntegrationInterval, Integrand, DefiniteIntegral,
  RiemannSum, LeftRiemannSum, RightRiemannSum, MidpointRule,
  TrapezoidalRule, CompositeTrapezoidalRule, SimpsonsRule,
  CompositeSimpsonsRule, NewtonCotesFormula, ClosedNewtonCotes,
  OpenNewtonCotes, BooleRule, WeddleRule,
  GaussianQuadrature, GaussLegendreQuadrature,
  GaussHermiteQuadrature, GaussLaguerreQuadrature,
  GaussChebyshevQuadrature, GaussJacobiQuadrature,
  OrthogonalPolynomial, WeightFunction, GaussianNode,
  GaussianWeight, DegreeOfExactness, PolynomialExactness,
  CompositeQuadrature, Subinterval, Mesh, GridSpacing,
  AdaptiveQuadrature, AdaptiveSimpson, LocalErrorEstimate,
  GlobalErrorEstimate, ErrorEstimator, RecursiveSubdivision,
  RombergIntegration, RichardsonExtrapolation,
  ClenshawCurtisQuadrature, FejerQuadrature,
  TanhSinhQuadrature, DoubleExponentialQuadrature,
  ImproperIntegral, InfiniteInterval, EndpointSingularity,
  InteriorSingularity, SingularIntegral, PrincipalValueIntegral,
  OscillatoryIntegral, FilonMethod, LevinMethod,
  PeriodicIntegral, EulerMaclaurinFormula,
  MultidimensionalQuadrature, Cubature, ProductRule,
  TensorProductQuadrature, SparseGridQuadrature,
  SmolyakQuadrature, MonteCarloIntegration,
  QuasiMonteCarloIntegration, ImportanceSampling,
  StratifiedSampling, LowDiscrepancySequence,
  ErrorBound, TruncationError, QuadratureError,
  DiscretisationError, RoundOffError, Cancellation,
  FunctionEvaluationCost, AdaptiveTolerance,
  AbsoluteTolerance, RelativeTolerance, Convergence,
  ConvergenceRate, OrderOfAccuracy, Stability,
  NumericalStability, Conditioning, Sensitivity,
  FloatingPointArithmetic, MachinePrecision,
  Overflow, Underflow, Verification, ReferenceIntegral,
  Benchmarking, Reproducibility
 ]
 EXCLUDE_AS_PRIMARY = [
  SymbolicIntegrationWithoutNumericalApproximation,
  NumericalDifferentiationWithoutIntegralApproximation,
  ProbabilityExpectationWithoutNumericalIntegrationContext,
  DifferentialEquationTimeSteppingWithoutQuadratureSemantics
 ]
 DISAMBIGUATION_RULE = "When a definite integral is approximated numerically by weighted function evaluations, interpret the context as quadrature."
}

CORE_CONCEPTS {
 Quadrature = "Numerical approximation of a definite integral."
 QuadratureNode = "Point at which the integrand is evaluated."
 QuadratureWeight = "Coefficient multiplying an integrand value in a quadrature formula."
 DegreeOfExactness = "Highest polynomial degree integrated exactly by a quadrature rule."
 CompositeQuadrature = "Application of a base quadrature rule across multiple subintervals."
 AdaptiveQuadrature = "Quadrature that refines subintervals according to estimated local error."
 GaussianQuadrature = "Quadrature with optimally chosen nodes and weights relative to a weight function."
 QuadratureError = "Difference between the exact integral and numerical quadrature approximation."
}

RELATIONS {
 Quadrature --is_child_of--> NumericalCalculus
 QuadratureRule --approximates--> DefiniteIntegral
 QuadratureRule --uses--> QuadratureNode
 QuadratureRule --uses--> QuadratureWeight
 NewtonCotesFormula --uses--> EquallySpacedNodes
 TrapezoidalRule --is_type_of--> NewtonCotesFormula
 SimpsonsRule --is_type_of--> NewtonCotesFormula
 GaussianQuadrature --uses--> OrthogonalPolynomial
 GaussianQuadrature --has--> DegreeOfExactness
 CompositeQuadrature --partitions--> IntegrationInterval
 AdaptiveQuadrature --uses--> ErrorEstimator
 AdaptiveQuadrature --refines--> Subinterval
 RombergIntegration --uses--> RichardsonExtrapolation
 ClenshawCurtisQuadrature --uses--> ChebyshevRelatedNodes
 TanhSinhQuadrature --handles_well--> EndpointSingularity
 SparseGridQuadrature --reduces--> TensorProductGrowth
 MonteCarloIntegration --uses--> RandomSampling
 QuasiMonteCarloIntegration --uses--> LowDiscrepancySequence
 Verification --checks--> NumericalIntegral
}

OPERATORS {
 QUADRATURE(f,nodes,weights) := sum_i weights_i*f(nodes_i)
 TRAPEZOID(f,a,b) := (b-a)*(f(a)+f(b))/2
 MIDPOINT(f,a,b) := (b-a)*f((a+b)/2)
 SIMPSON(f,a,b) := (b-a)*(f(a)+4*f((a+b)/2)+f(b))/6
 COMPOSITE_RULE(f,partition,Q) := sum_j Q(f,[x_j,x_(j+1)])
 GAUSSIAN_QUADRATURE(f,w,nodes,weights) := sum_i weights_i*f(nodes_i)
 ERROR(I,Q) := E=I-Q
 ADAPTIVE_SPLIT([a,b]) := [a,m] union [m,b], m=(a+b)/2
}

AXIOMS {
 A1 = "Quadrature approximates definite integrals through finite weighted sums."
 A2 = "Quadrature accuracy depends on integrand regularity, node placement and rule order."
 A3 = "Composite rules improve resolution by subdividing the integration interval."
 A4 = "Adaptive quadrature should refine where local error estimates indicate insufficient accuracy."
 A5 = "Gaussian quadrature maximises polynomial exactness for a given number of nodes under its weight-function setting."
 A6 = "Quadrature error and floating-point round-off error are distinct."
 A7 = "Improper, singular and oscillatory integrals can require specialised transformations or rules."
 A8 = "Quadrature results require explicit error, convergence and verification semantics."
}

REASONING {
 R1_CONTEXT = identify(integrand,interval,dimension,regularity,singularities,oscillation,accuracy_goal)
 R2_CLASS = classify(finite,improper,singular,oscillatory,weighted_or_multidimensional_integral)
 R3_METHOD = select(Newton_Cotes,Gaussian,adaptive,Romberg,Clenshaw_Curtis,tanh_sinh,sparse_grid,MonteCarlo_or_other_rule)
 R4_NODES = choose(nodes,weights,subintervals_or_sampling_budget)
 R5_COMPUTE = evaluate_integrand_and_accumulate_weighted_sum)
 R6_ERROR = estimate(local_global_quadrature_error_and_roundoff)
 R7_ADAPT = if_needed refine_subintervals_nodes_or_samples)
 R8_CONVERGENCE = compare_successive_resolutions_or_orders)
 R9_STABILITY = assess(cancellation,overflow,underflow,conditioning_and_function_evaluation_noise)
 R10_VALIDATE = compare(reference_integral,analytic_case,independent_rule_or_convergence_study)
}

CONSTRAINTS {
 C1 = "High-order closed Newton-Cotes formulas on equally spaced nodes can become numerically poor."
 C2 = "Composite Simpson rules require compatible subdivision structure."
 C3 = "Gaussian quadrature nodes and weights depend on the chosen weight function and interval mapping."
 C4 = "Adaptive error estimators can fail when unsampled local features dominate."
 C5 = "Improper integrals can require interval transformations or tail treatment."
 C6 = "Endpoint singularities can defeat standard polynomial-based quadrature without transformation."
 C7 = "Highly oscillatory integrals can require oscillation-aware methods."
 C8 = "Tensor-product quadrature suffers exponential growth with dimension."
 C9 = "Monte Carlo methods have stochastic sampling error."
 C10 = "Very fine subdivision can increase round-off and accumulation error."
}

SEMANTIC_EQUIVALENCE {
 "Quadrature" <=> "NumericalIntegrationInWeightedSumContext"
 "SimpsonsRule" <=> "SimpsonRule"
 "GaussLegendreQuadrature" <=> "LegendreGaussianQuadrature"
 "DoubleExponentialQuadrature" <=> "TanhSinhQuadrature"
 "Cubature" <=> "MultidimensionalQuadrature"
}

VALIDATION {
 V1 = "Integration interval and integrand are explicit."
 V2 = "Quadrature rule is documented."
 V3 = "Nodes, weights, subdivision or sampling configuration are documented."
 V4 = "Rule order or degree of exactness is documented where relevant."
 V5 = "Error estimates are reported."
 V6 = "Convergence under refinement is checked where feasible."
 V7 = "Singular, improper or oscillatory behaviour is treated explicitly."
 V8 = "Floating-point accumulation effects are considered."
 V9 = "Reference or analytic integrals are used where feasible."
 V10 = "Results are reproducible under documented arithmetic and method settings."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 QUADRATURE_DEFINITION = 0.99
 NEWTON_COTES_REASONING = 0.99
 GAUSSIAN_QUADRATURE_REASONING = 0.99
 ADAPTIVE_QUADRATURE_REASONING = 0.99
 SINGULAR_INTEGRAL_REASONING = 0.98
 MULTIDIMENSIONAL_QUADRATURE_REASONING = 0.98
 MONTE_CARLO_REASONING = 0.98
 POLICY = "Reduce confidence when integrand regularity, singularities, node and weight definitions, adaptive error estimates, dimension or arithmetic precision are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch, OntologyGrounding, QuadratureReasoning,
  NumericalIntegrationReasoning, NewtonCotesReasoning,
  TrapezoidalReasoning, SimpsonReasoning, GaussianQuadratureReasoning,
  AdaptiveQuadratureReasoning, RombergReasoning,
  ClenshawCurtisReasoning, TanhSinhReasoning,
  SingularIntegralReasoning, OscillatoryIntegralReasoning,
  CubatureReasoning, SparseGridReasoning,
  MonteCarloIntegrationReasoning, ErrorEstimationReasoning,
  ConvergenceReasoning, FloatingPointReasoning,
  KnowledgeGraphConstruction, RetrievalAugmentation, RecursiveInference
 ]
 NORMALISE {
  "quadrature" -> Quadrature
  "numerical integration" -> Quadrature
  "Simpson rule" -> SimpsonsRule
  "Gauss-Legendre" -> GaussLegendreQuadrature
  "double exponential quadrature" -> TanhSinhQuadrature
  "cubature" -> MultidimensionalQuadrature
 }
}

EXECUTION_RULES {
 E1 = "If a definite integral is approximated by weighted function evaluations, activate QuadratureOntology."
 E2 = "Classify interval structure, integrand regularity, singularities, oscillation and dimension before selecting a rule."
 E3 = "Use composite rules when fixed local formulas need greater resolution."
 E4 = "Use adaptive rules when integrand difficulty varies across the interval."
 E5 = "Use Gaussian rules when high polynomial exactness is useful and the weight-function setting is appropriate."
 E6 = "Use specialised transformations or rules for singular, improper or highly oscillatory integrals."
 E7 = "For higher dimensions, consider sparse grids, Monte Carlo or quasi-Monte Carlo methods."
 E8 = "Monitor both discretisation and floating-point error."
 E9 = "Validate quadrature through error estimates, refinement, reference integrals and reproducibility."
}

RECURSION {
 ENTRY = Quadrature
 EXPAND(Quadrature) -> [
  NumericalIntegration,NumericalIntegral,QuadratureRule,QuadratureFormula,
  QuadratureNode,QuadratureWeight,WeightedSum,IntegrationInterval,
  Integrand,DefiniteIntegral,RiemannSum,LeftRiemannSum,RightRiemannSum,
  MidpointRule,TrapezoidalRule,CompositeTrapezoidalRule,SimpsonsRule,
  CompositeSimpsonsRule,NewtonCotesFormula,ClosedNewtonCotes,
  OpenNewtonCotes,BooleRule,GaussianQuadrature,
  GaussLegendreQuadrature,GaussHermiteQuadrature,
  GaussLaguerreQuadrature,GaussChebyshevQuadrature,
  GaussJacobiQuadrature,OrthogonalPolynomial,WeightFunction,
  DegreeOfExactness,CompositeQuadrature,Subinterval,GridSpacing,
  AdaptiveQuadrature,AdaptiveSimpson,LocalErrorEstimate,
  GlobalErrorEstimate,RecursiveSubdivision,RombergIntegration,
  RichardsonExtrapolation,ClenshawCurtisQuadrature,FejerQuadrature,
  TanhSinhQuadrature,ImproperIntegral,EndpointSingularity,
  InteriorSingularity,SingularIntegral,PrincipalValueIntegral,
  OscillatoryIntegral,FilonMethod,LevinMethod,PeriodicIntegral,
  EulerMaclaurinFormula,MultidimensionalQuadrature,Cubature,
  ProductRule,TensorProductQuadrature,SparseGridQuadrature,
  SmolyakQuadrature,MonteCarloIntegration,QuasiMonteCarloIntegration,
  ImportanceSampling,StratifiedSampling,LowDiscrepancySequence,
  ErrorBound,QuadratureError,TruncationError,RoundOffError,
  FunctionEvaluationCost,AdaptiveTolerance,Convergence,
  OrderOfAccuracy,NumericalStability,Conditioning,
  FloatingPointArithmetic,MachinePrecision,Verification,
  ReferenceIntegral,Benchmarking,Reproducibility
 ]
 EXPAND(QuadratureFamilies) -> [
  NewtonCotesQuadrature,
  GaussianQuadrature,
  CompositeQuadrature,
  AdaptiveQuadrature,
  SingularAndImproperQuadrature,
  OscillatoryQuadrature,
  MultidimensionalQuadrature,
  StochasticIntegration
 ]
 STOP_CONDITION = "Stop when integral class, quadrature family, nodes, weights, resolution, error, convergence, stability and verification are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Numerical Calculus"
 RELATED_ONTOLOGIES = [
  "Numerical Calculus",
  "Numerical Methods",
  "Numerical Analysis Ontology 2",
  "Numerical Approximation Ontology",
  "Numerical Error Ontology",
  "Numerical Stability Ontology 2",
  "Numerical Convergence Ontology",
  "Floating Point Arithmetic Ontology",
  "Interpolation Ontologies",
  "Probability and Statistics Ontologies"
 ]
 ROUTE_RULES = [
  "Quadrature -> Numerical Calculus for parent semantics",
  "Quadrature -> Numerical Methods for integration algorithms",
  "Quadrature -> Numerical Analysis for convergence and error theory",
  "Quadrature -> Numerical Approximation for polynomial exactness and approximation quality",
  "Quadrature -> Numerical Error for discretisation and round-off semantics",
  "Quadrature -> Numerical Stability for finite-precision accumulation",
  "Quadrature -> Numerical Convergence for refinement behaviour",
  "Quadrature -> Floating Point Arithmetic for accumulation and cancellation",
  "Quadrature -> Interpolation for Newton-Cotes and polynomial-based formulas",
  "Quadrature -> Probability and Statistics for Monte Carlo and quasi-Monte Carlo integration"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible quadrature families, adaptive estimators or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve quadrature-versus-symbolic-integration distinction.",
  "Preserve deterministic-versus-stochastic integration distinction.",
  "Preserve fixed-rule-versus-adaptive-rule distinction.",
  "Preserve one-dimensional-versus-multidimensional quadrature distinction.",
  "Preserve quadrature-error-versus-round-off-error distinction.",
  "Preserve node-versus-weight semantics.",
  "Preserve finite-interval-versus-improper-integral distinction.",
  "Preserve machine-readable quadrature semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Quadrature approximates definite integrals by weighted sums of integrand evaluations."
 NEWTON_COTES_SUMMARY = "Midpoint, trapezoidal, Simpson and related Newton-Cotes rules use polynomial-based formulas on structured nodes."
 GAUSSIAN_SUMMARY = "Gaussian quadrature chooses nodes and weights to maximise polynomial exactness for a given number of evaluations."
 ADAPTIVE_SUMMARY = "Adaptive quadrature recursively refines regions with larger estimated local error."
 SPECIAL_CASE_SUMMARY = "Improper, singular and oscillatory integrals often require specialised transformations or quadrature rules."
 HIGH_DIMENSION_SUMMARY = "Cubature, sparse grids, Monte Carlo and quasi-Monte Carlo methods extend numerical integration beyond one dimension."
 VALIDATION_SUMMARY = "Quadrature results should be checked through error estimates, convergence studies, reference integrals and reproducibility."
}

FLOW {
 INPUT
 -> detect_quadrature_context
 -> identify_integrand_interval_dimension_regularity_singularities_oscillation_and_accuracy_goal
 -> classify_finite_improper_singular_oscillatory_weighted_or_multidimensional_integral
 -> select_Newton_Cotes_Gaussian_composite_adaptive_Romberg_Clenshaw_Curtis_tanh_sinh_sparse_grid_MonteCarlo_or_other_rule
 -> choose_nodes_weights_subintervals_or_sampling_budget
 -> evaluate_integrand_and_accumulate_weighted_sum
 -> estimate_local_global_quadrature_error_and_roundoff
 -> if_needed_refine_subintervals_nodes_or_samples
 -> assess_convergence_order_conditioning_cancellation_overflow_underflow_and_precision_limits
 -> validate_against_reference_integral_analytic_case_or_independent_rule
 -> route_to_related_numerical_calculus_methods_analysis_approximation_error_stability_convergence_floating_point_interpolation_or_probability_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Quadrature Ontology defines numerical approximation of definite integrals by weighted sums. It supports Riemann sums, midpoint, trapezoidal and Simpson rules, composite and Newton-Cotes formulas, Gaussian quadrature families including Gauss-Legendre, Hermite, Laguerre, Chebyshev and Jacobi rules, degree of exactness, adaptive quadrature and adaptive Simpson methods, Romberg and Richardson extrapolation, Clenshaw-Curtis, Fejer and tanh-sinh quadrature, improper, singular and principal-value integrals, oscillatory integration, multidimensional cubature, tensor-product and sparse-grid rules, Monte Carlo and quasi-Monte Carlo integration, error estimation, convergence, conditioning, finite precision, verification, benchmarking and reproducibility."
}

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

}

END_FLOW