Gaussian Quadrature Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Gaussian Quadrature Ontology within the Kireth Research Semantic Library. Gaussian quadrature approximates weighted definite integrals using optimally chosen nodes and weights associated with orthogonal polynomials. This ontology supports Gauss-Legendre, Gauss-Hermite, Gauss-Laguerre, Gauss-Chebyshev and Gauss-Jacobi rules, polynomial exactness, node and weight construction, affine interval mapping, composite and adaptive use, error estimation, high-order accuracy, multidimensional extensions, sparse grids, finite precision and verification.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = GaussianQuadrature
 ENTITY_CLASS = OrthogonalPolynomialQuadratureFramework
 PRIMARY_CONTEXT = Quadrature
 DEFINITION = "Gaussian quadrature is a family of numerical integration rules whose nodes and weights are chosen so that an n-point rule integrates polynomials of degree up to 2n-1 exactly under the associated weight-function setting."
 SYMBOLIC_FORMS = [
  "integral_a^b w(x)f(x) dx approximately sum_(i=1)^n W_i*f(x_i)",
  "P_n(x_i)=0",
  "degree_of_exactness=2*n-1",
  "Q_n(f)=sum_i W_i*f(x_i)"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for Gaussian nodes, weights, orthogonal polynomials, weighted integration, polynomial exactness, interval transformations, error behaviour and scalable high-order quadrature."
}

SCOPE {
 INCLUDE = [
  GaussianQuadrature, GaussianRule, GaussianIntegration,
  NPointGaussianQuadrature, Quadrature, NumericalIntegration,
  WeightedIntegral, WeightFunction, OrthogonalPolynomial,
  PolynomialFamily, QuadratureNode, GaussianNode, QuadratureWeight,
  GaussianWeight, DegreeOfExactness, PolynomialExactness,
  GaussLegendreQuadrature, LegendrePolynomial,
  GaussHermiteQuadrature, HermitePolynomial,
  GaussLaguerreQuadrature, LaguerrePolynomial,
  GaussChebyshevQuadrature, ChebyshevPolynomial,
  GaussJacobiQuadrature, JacobiPolynomial,
  GegenbauerPolynomial, UltrasphericalQuadrature,
  GaussLobattoQuadrature, GaussRadauQuadrature,
  EndpointIncludingQuadrature, EndpointFixedQuadrature,
  OrthogonalityRelation, ThreeTermRecurrence,
  RecurrenceCoefficient, JacobiMatrix, SymmetricTridiagonalMatrix,
  GolubWelschAlgorithm, EigenvalueComputation, EigenvectorComputation,
  ChristoffelNumber, Moment, MomentMatching, GaussianWeightConstruction,
  RootOfOrthogonalPolynomial, NodeOrdering,
  StandardInterval, IntervalMapping, AffineTransformation,
  FiniteInterval, InfiniteInterval, SemiInfiniteInterval,
  CompositeGaussianQuadrature, AdaptiveGaussianQuadrature,
  GaussKronrodQuadrature, KronrodExtension, EmbeddedQuadrature,
  LocalErrorEstimate, GlobalErrorEstimate, AdaptiveTolerance,
  HighOrderQuadrature, SmoothIntegrand, AnalyticIntegrand,
  ErrorTerm, QuadratureError, TruncationError,
  Convergence, ConvergenceRate, ExponentialConvergenceContext,
  SpectralAccuracyContext, Conditioning, Sensitivity,
  FloatingPointArithmetic, MachinePrecision, RoundOffError,
  NodeWeightAccuracy, Underflow, Overflow, Cancellation,
  TensorProductGaussianQuadrature, MultidimensionalQuadrature,
  Cubature, SparseGridQuadrature, SmolyakQuadrature,
  FunctionEvaluationCost, NodeReuse, WeightReuse,
  Verification, ReferenceIntegral, Benchmarking, Reproducibility
 ]
 EXCLUDE_AS_PRIMARY = [
  NewtonCotesQuadratureWithEquallySpacedNodes,
  MonteCarloIntegrationWithoutGaussianNodesAndWeights,
  SymbolicIntegrationWithoutNumericalQuadrature,
  ProbabilityGaussianDistributionWithoutQuadratureContext
 ]
 DISAMBIGUATION_RULE = "When an integral is approximated using nodes given by roots of orthogonal polynomials and corresponding optimal quadrature weights, interpret the context as Gaussian quadrature."
}

CORE_CONCEPTS {
 GaussianQuadrature = "High-order quadrature using optimally chosen nodes and weights."
 GaussianNode = "Quadrature node typically equal to a root of an orthogonal polynomial."
 GaussianWeight = "Quadrature coefficient associated with a Gaussian node."
 OrthogonalPolynomial = "Polynomial belonging to a family orthogonal under a specified weight function."
 DegreeOfExactness = "Highest polynomial degree integrated exactly by a quadrature rule."
 GolubWelschAlgorithm = "Algorithm computing Gaussian nodes and weights from a symmetric tridiagonal Jacobi matrix."
 GaussKronrodQuadrature = "Embedded Gaussian-related rule adding nodes to estimate integration error."
 IntervalMapping = "Transformation carrying a standard Gaussian rule to a target interval."
}

RELATIONS {
 GaussianQuadrature --is_child_of--> Quadrature
 GaussianQuadrature --uses--> OrthogonalPolynomial
 OrthogonalPolynomial --is_defined_relative_to--> WeightFunction
 GaussianNode --is_root_of--> OrthogonalPolynomial
 GaussianQuadrature --uses--> GaussianNode
 GaussianQuadrature --uses--> GaussianWeight
 NPointGaussianQuadrature --has_degree_of_exactness--> Degree2NMinus1
 GaussLegendreQuadrature --uses--> LegendrePolynomial
 GaussHermiteQuadrature --uses--> HermitePolynomial
 GaussLaguerreQuadrature --uses--> LaguerrePolynomial
 GaussChebyshevQuadrature --uses--> ChebyshevPolynomial
 GaussJacobiQuadrature --uses--> JacobiPolynomial
 GolubWelschAlgorithm --uses--> JacobiMatrix
 JacobiMatrix --is_type_of--> SymmetricTridiagonalMatrix
 EigenvalueComputation --produces--> GaussianNodesInGolubWelsch
 EigenvectorComputation --supports--> GaussianWeightsInGolubWelsch
 IntervalMapping --transforms--> GaussianNodesAndWeights
 GaussKronrodQuadrature --provides--> EmbeddedErrorEstimate
 TensorProductGaussianQuadrature --extends--> GaussianQuadrature
 SparseGridQuadrature --reduces--> TensorProductGrowth
 Verification --checks--> GaussianQuadratureResult
}

OPERATORS {
 GAUSSIAN_QUADRATURE(f,nodes,weights) := sum_i weights_i*f(nodes_i)
 WEIGHTED_GAUSSIAN_QUADRATURE(f,w,nodes,weights) := sum_i weights_i*f(nodes_i)
 AFFINE_MAP_NODE(t,a,b) := x=(a+b)/2+(b-a)*t/2
 AFFINE_MAP_WEIGHT(W,a,b) := W_mapped=(b-a)*W/2
 GAUSS_LEGENDRE(f,a,b,n) := mapped_sum_of_n_Legendre_nodes_and_weights
 DEGREE_OF_EXACTNESS(n) := 2*n-1
 GOLUB_WELSCH(alpha,beta) := eigendecompose_Jacobi_matrix_to_obtain_nodes_and_weights
 EMBEDDED_ERROR(Q_G,Q_K) := |Q_K-Q_G|
}

AXIOMS {
 A1 = "An n-point Gaussian quadrature rule integrates polynomials up to degree 2n-1 exactly under its associated weight-function setting."
 A2 = "Gaussian nodes lie at roots of the corresponding degree-n orthogonal polynomial."
 A3 = "Gaussian weights are positive for standard positive weight functions under usual assumptions."
 A4 = "Gauss-Legendre quadrature corresponds to unit weight on the standard finite interval."
 A5 = "Gauss-Hermite and Gauss-Laguerre rules encode nonuniform weight functions and infinite-domain structure."
 A6 = "Affine mapping preserves the order and exactness structure of Gauss-Legendre integration on finite intervals."
 A7 = "Error behaviour depends on integrand smoothness beyond polynomial exactness."
 A8 = "Gaussian quadrature results require node, weight, mapping, error and reproducibility validation."
}

REASONING {
 R1_CONTEXT = identify(integrand,domain,weight_function,dimension,smoothness,accuracy_goal)
 R2_FAMILY = select(Legendre,Hermite,Laguerre,Chebyshev,Jacobi,Lobatto,Radau_or_other_Gaussian_family)
 R3_ORDER = choose(number_of_nodes_from_accuracy_cost_and_integrand_structure)
 R4_CONSTRUCTION = obtain(nodes_and_weights_from_table_recurrence_GolubWelsch_or_stable_algorithm)
 R5_MAPPING = if_needed transform_standard_nodes_and_weights_to_target_domain
 R6_COMPUTE = evaluate_integrand_at_Gaussian_nodes_and_accumulate_weighted_sum
 R7_ERROR = estimate_by_order_refinement_embedded_GaussKronrod_or_reference_rule
 R8_ADAPT = if_needed subdivide_domain_and_reapply_local_Gaussian_rules
 R9_STABILITY = assess(node_weight_accuracy,roundoff,overflow,underflow_and_cancellation)
 R10_VALIDATE = check_polynomial_exactness_reference_integral_or_independent_quadrature)
}

CONSTRAINTS {
 C1 = "The Gaussian family must match the integral's weight function and domain structure."
 C2 = "Gauss-Legendre rules assume the standard unit-weight finite-interval setting before mapping."
 C3 = "Using incorrect node or weight tables invalidates polynomial exactness."
 C4 = "Very high-order rules can require careful node and weight computation for numerical stability."
 C5 = "Endpoint singularities can reduce expected efficiency unless transformations or specialised weighted rules are used."
 C6 = "Highly oscillatory integrals can require specialised oscillatory quadrature despite high Gaussian order."
 C7 = "Tensor-product Gaussian quadrature suffers exponential growth with dimension."
 C8 = "Adaptive Gaussian schemes require reliable local error estimation."
 C9 = "Infinite-interval rules require correct interpretation of their built-in weight functions."
 C10 = "Finite-precision node and weight errors can limit attainable high-order accuracy."
}

SEMANTIC_EQUIVALENCE {
 "GaussianQuadrature" <=> "GaussianIntegration"
 "GaussLegendreQuadrature" <=> "LegendreGaussianQuadrature"
 "GaussHermiteQuadrature" <=> "HermiteGaussianQuadrature"
 "GaussLaguerreQuadrature" <=> "LaguerreGaussianQuadrature"
 "GaussChebyshevQuadrature" <=> "ChebyshevGaussianQuadrature"
 "GaussJacobiQuadrature" <=> "JacobiGaussianQuadrature"
 "GolubWelschAlgorithm" <=> "JacobiMatrixGaussianQuadratureConstruction"
}

VALIDATION {
 V1 = "Integration domain and weight function are explicit."
 V2 = "Gaussian quadrature family is documented."
 V3 = "Node count is documented."
 V4 = "Nodes and weights are documented or reproducibly generated."
 V5 = "Interval mapping is documented where used."
 V6 = "Polynomial exactness is checked for test polynomials where feasible."
 V7 = "Error estimates or order-refinement studies are reported."
 V8 = "Node and weight accuracy are considered at high order."
 V9 = "Reference or analytic integrals are used where feasible."
 V10 = "Results are reproducible under documented arithmetic and quadrature settings."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 GAUSSIAN_QUADRATURE_DEFINITION = 0.99
 LEGENDRE_REASONING = 0.99
 HERMITE_REASONING = 0.99
 LAGUERRE_REASONING = 0.99
 JACOBI_REASONING = 0.99
 GOLUB_WELSCH_REASONING = 0.99
 ADAPTIVE_GAUSSIAN_REASONING = 0.98
 MULTIDIMENSIONAL_REASONING = 0.98
 POLICY = "Reduce confidence when weight function, Gaussian family, node count, node-weight construction, interval mapping, integrand regularity or arithmetic precision are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch, OntologyGrounding, GaussianQuadratureReasoning,
  OrthogonalPolynomialReasoning, GaussianNodeReasoning,
  GaussianWeightReasoning, PolynomialExactnessReasoning,
  GaussLegendreReasoning, GaussHermiteReasoning,
  GaussLaguerreReasoning, GaussChebyshevReasoning,
  GaussJacobiReasoning, GolubWelschReasoning,
  IntervalMappingReasoning, GaussKronrodReasoning,
  AdaptiveGaussianReasoning, MultidimensionalQuadratureReasoning,
  ErrorEstimationReasoning, FloatingPointReasoning,
  KnowledgeGraphConstruction, RetrievalAugmentation, RecursiveInference
 ]
 NORMALISE {
  "Gaussian quadrature" -> GaussianQuadrature
  "Gaussian integration" -> GaussianQuadrature
  "Gauss-Legendre" -> GaussLegendreQuadrature
  "Gauss-Hermite" -> GaussHermiteQuadrature
  "Gauss-Laguerre" -> GaussLaguerreQuadrature
  "Gauss-Chebyshev" -> GaussChebyshevQuadrature
  "Gauss-Jacobi" -> GaussJacobiQuadrature
  "Golub-Welsch" -> GolubWelschAlgorithm
 }
}

EXECUTION_RULES {
 E1 = "If a weighted integral is approximated using orthogonal-polynomial nodes and associated weights, activate GaussianQuadratureOntology."
 E2 = "Identify the integration domain and weight function before selecting the Gaussian family."
 E3 = "Use Gauss-Legendre for unit-weight finite intervals unless another weight structure is required."
 E4 = "Map standard nodes and weights consistently when changing intervals."
 E5 = "Generate high-order nodes and weights using stable recurrence or eigenvalue-based methods such as Golub-Welsch."
 E6 = "If error control is required, compare orders or use an embedded Gauss-Kronrod strategy."
 E7 = "If local difficulty varies, use adaptive subdivision with Gaussian rules."
 E8 = "For high-dimensional problems, consider sparse grids rather than unrestricted tensor products."
 E9 = "Validate through polynomial exactness, error studies, reference integrals and reproducibility."
}

RECURSION {
 ENTRY = GaussianQuadrature
 EXPAND(GaussianQuadrature) -> [
  GaussianRule,GaussianIntegration,NPointGaussianQuadrature,
  Quadrature,NumericalIntegration,WeightedIntegral,WeightFunction,
  OrthogonalPolynomial,PolynomialFamily,QuadratureNode,GaussianNode,
  QuadratureWeight,GaussianWeight,DegreeOfExactness,
  PolynomialExactness,GaussLegendreQuadrature,LegendrePolynomial,
  GaussHermiteQuadrature,HermitePolynomial,GaussLaguerreQuadrature,
  LaguerrePolynomial,GaussChebyshevQuadrature,ChebyshevPolynomial,
  GaussJacobiQuadrature,JacobiPolynomial,GegenbauerPolynomial,
  GaussLobattoQuadrature,GaussRadauQuadrature,OrthogonalityRelation,
  ThreeTermRecurrence,RecurrenceCoefficient,JacobiMatrix,
  SymmetricTridiagonalMatrix,GolubWelschAlgorithm,EigenvalueComputation,
  EigenvectorComputation,ChristoffelNumber,Moment,MomentMatching,
  RootOfOrthogonalPolynomial,NodeOrdering,StandardInterval,
  IntervalMapping,AffineTransformation,FiniteInterval,InfiniteInterval,
  SemiInfiniteInterval,CompositeGaussianQuadrature,
  AdaptiveGaussianQuadrature,GaussKronrodQuadrature,KronrodExtension,
  EmbeddedQuadrature,LocalErrorEstimate,GlobalErrorEstimate,
  AdaptiveTolerance,HighOrderQuadrature,SmoothIntegrand,
  AnalyticIntegrand,ErrorTerm,QuadratureError,Convergence,
  ConvergenceRate,SpectralAccuracyContext,Conditioning,Sensitivity,
  FloatingPointArithmetic,MachinePrecision,RoundOffError,
  NodeWeightAccuracy,TensorProductGaussianQuadrature,
  MultidimensionalQuadrature,Cubature,SparseGridQuadrature,
  SmolyakQuadrature,FunctionEvaluationCost,Verification,
  ReferenceIntegral,Benchmarking,Reproducibility
 ]
 EXPAND(GaussianFamilies) -> [
  GaussLegendreQuadrature,
  GaussHermiteQuadrature,
  GaussLaguerreQuadrature,
  GaussChebyshevQuadrature,
  GaussJacobiQuadrature,
  GaussLobattoAndRadauQuadrature,
  AdaptiveGaussKronrodQuadrature,
  MultidimensionalGaussianQuadrature
 ]
 STOP_CONDITION = "Stop when weight function, Gaussian family, nodes, weights, mapping, order, error, stability and verification are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Quadrature Ontology"
 RELATED_ONTOLOGIES = [
  "Quadrature Ontology",
  "Numerical Calculus",
  "Trapezoidal Rule Ontology",
  "Simpson's Rule Ontology",
  "Eigenvalue Computation Ontology",
  "Eigenvector Computation Ontology",
  "Numerical Methods",
  "Numerical Analysis Ontology 2",
  "Numerical Approximation Ontology",
  "Numerical Error Ontology",
  "Numerical Stability Ontology 2",
  "Floating Point Arithmetic Ontology",
  "Orthogonal Polynomial Ontologies"
 ]
 ROUTE_RULES = [
  "Gaussian Quadrature -> Quadrature for parent numerical-integration semantics",
  "Gaussian Quadrature -> Numerical Calculus for broader calculus context",
  "Gaussian Quadrature -> Trapezoidal and Simpson rules for Newton-Cotes comparison",
  "Gaussian Quadrature -> Eigenvalue Computation for Golub-Welsch node construction",
  "Gaussian Quadrature -> Eigenvector Computation for Golub-Welsch weight construction",
  "Gaussian Quadrature -> Numerical Analysis for exactness, convergence and error theory",
  "Gaussian Quadrature -> Numerical Approximation for orthogonal-polynomial approximation semantics",
  "Gaussian Quadrature -> Numerical Error for truncation and node-weight error",
  "Gaussian Quadrature -> Numerical Stability for high-order finite-precision behaviour",
  "Gaussian Quadrature -> Floating Point Arithmetic for node and weight precision limits"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible Gaussian families, construction algorithms or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve Gaussian-versus-Newton-Cotes distinction.",
  "Preserve node-versus-weight semantics.",
  "Preserve orthogonal-polynomial-versus-weight-function relationship.",
  "Preserve standard-domain-versus-mapped-domain distinction.",
  "Preserve polynomial-exactness-versus-general-integrand-error distinction.",
  "Preserve one-dimensional-versus-multidimensional distinction.",
  "Preserve deterministic-Gaussian-versus-stochastic-integration distinction.",
  "Preserve machine-readable Gaussian-quadrature semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Gaussian quadrature approximates weighted integrals using optimally chosen nodes and weights derived from orthogonal polynomials."
 EXACTNESS_SUMMARY = "An n-point Gaussian rule integrates polynomials of degree up to 2n-1 exactly under its associated weight-function setting."
 FAMILY_SUMMARY = "Legendre, Hermite, Laguerre, Chebyshev and Jacobi rules correspond to different domains and weight functions."
 CONSTRUCTION_SUMMARY = "Nodes are roots of orthogonal polynomials; stable algorithms such as Golub-Welsch compute nodes and weights from recurrence data."
 MAPPING_SUMMARY = "Affine transformations map standard finite-interval Gaussian rules to arbitrary finite intervals."
 ADAPTIVE_SUMMARY = "Gauss-Kronrod and related embedded rules provide efficient adaptive error estimation."
 HIGH_DIMENSION_SUMMARY = "Tensor-product and sparse-grid constructions extend Gaussian quadrature to multiple dimensions."
 VALIDATION_SUMMARY = "Gaussian quadrature results should be checked for family selection, node and weight accuracy, polynomial exactness, error estimates and reproducibility."
}

FLOW {
 INPUT
 -> detect_Gaussian_quadrature_context
 -> identify_integrand_domain_weight_function_dimension_smoothness_and_accuracy_goal
 -> select_Legendre_Hermite_Laguerre_Chebyshev_Jacobi_Lobatto_Radau_or_other_Gaussian_family
 -> choose_node_count_from_accuracy_cost_and_structure
 -> obtain_nodes_and_weights_from_stable_table_recurrence_GolubWelsch_or_equivalent_algorithm
 -> if_needed_map_standard_nodes_and_weights_to_target_domain
 -> evaluate_integrand_at_Gaussian_nodes_and_accumulate_weighted_sum
 -> estimate_error_by_order_refinement_embedded_GaussKronrod_or_reference_rule
 -> if_needed_apply_adaptive_subdivision
 -> assess_node_weight_accuracy_conditioning_roundoff_overflow_underflow_and_cancellation
 -> validate_polynomial_exactness_reference_integral_or_independent_quadrature
 -> route_to_related_quadrature_numerical_calculus_eigenvalue_eigenvector_analysis_approximation_error_stability_floating_point_and_orthogonal_polynomial_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Gaussian Quadrature Ontology defines high-order numerical integration through orthogonal-polynomial nodes and weights. It supports n-point Gaussian rules, weighted integrals, polynomial exactness through degree 2n-1, Gauss-Legendre, Hermite, Laguerre, Chebyshev and Jacobi families, Lobatto and Radau variants, orthogonality relations and three-term recurrences, Golub-Welsch construction through symmetric tridiagonal eigenproblems, moments and Christoffel numbers, interval mapping, composite and adaptive Gaussian quadrature, Gauss-Kronrod embedded rules, smooth and analytic integrands, error estimation, multidimensional tensor-product and sparse-grid Gaussian rules, finite-precision node and weight accuracy, verification, benchmarking and reproducibility."
}

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

}

END_FLOW