Computational Mathematics Ontologies

AI INTERPRETATION NOTE
This Flow Algebra defines the Computational Mathematics Ontologies parent layer within the Kireth Research Semantic Library. Computational mathematics studies the mathematical principles, algorithms, numerical methods and computational structures used to solve, approximate, simulate and analyse mathematical problems by machine. This parent ontology connects numerical analysis, scientific computing, computational linear algebra, optimisation, symbolic computation, discretisation, numerical differential equations, high-performance computing, approximation theory, error analysis, algorithmic complexity and reproducible mathematical computation.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = ComputationalMathematicsOntologies
 ENTITY_CLASS = ComputationalMathematicsFramework
 PRIMARY_CONTEXT = MathematicalOntologies
 DEFINITION = "Computational mathematics is the study of algorithms, numerical methods, symbolic methods and computational structures used to solve mathematical problems approximately or exactly by computation."
 SYMBOLIC_FORMS = [
  "x_(k+1)=F(x_k)",
  "A x=b",
  "u_h approximates u",
  "error=exact-approximation"
 ]
 SEMANTIC_ROLE = "Provides the parent semantic framework for numerical analysis, scientific computing, computational algebra, optimisation, simulation, approximation, discretisation, algorithmic efficiency and machine-executable mathematical reasoning."
}

SCOPE {
 INCLUDE = [
  ComputationalMathematics,
  ComputationalMathematicsOntologies,
  NumericalAnalysis,
  NumericalMethod,
  NumericalAlgorithm,
  ScientificComputing,
  SymbolicComputation,
  ComputerAlgebra,
  NumericalLinearAlgebra,
  SparseLinearAlgebra,
  MatrixComputation,
  EigenvalueComputation,
  MatrixDecomposition,
  IterativeMethod,
  DirectMethod,
  RootFinding,
  NonlinearEquationSolving,
  NumericalDifferentiation,
  NumericalIntegration,
  Quadrature,
  Interpolation,
  Extrapolation,
  ApproximationTheory,
  PolynomialApproximation,
  SplineApproximation,
  SpectralApproximation,
  ErrorAnalysis,
  TruncationError,
  RoundoffError,
  Conditioning,
  Stability,
  Consistency,
  Convergence,
  BackwardError,
  ForwardError,
  FloatingPointArithmetic,
  MachinePrecision,
  Discretisation,
  Mesh,
  Grid,
  TimeStep,
  SpatialStep,
  FiniteDifferenceMethod,
  FiniteElementMethod,
  FiniteVolumeMethod,
  SpectralMethod,
  BoundaryElementMethod,
  MethodOfLines,
  NumericalODE,
  NumericalPDE,
  InitialValueProblem,
  BoundaryValueProblem,
  AdaptiveMethod,
  AdaptiveMesh,
  AdaptiveTimeStepping,
  Multigrid,
  DomainDecomposition,
  ComputationalOptimisation,
  LinearProgramming,
  NonlinearProgramming,
  ConvexOptimisation,
  GradientMethod,
  NewtonMethod,
  QuasiNewtonMethod,
  TrustRegionMethod,
  InteriorPointMethod,
  MonteCarloMethod,
  QuasiMonteCarlo,
  StochasticSimulation,
  ComputationalGeometry,
  ComputationalNumberTheory,
  ComputationalAlgebra,
  AlgorithmicComplexity,
  TimeComplexity,
  SpaceComplexity,
  ParallelComputing,
  HighPerformanceComputing,
  DistributedComputing,
  GPUComputing,
  Vectorisation,
  SparseMatrix,
  Preconditioner,
  Solver,
  NumericalLibrary,
  Reproducibility,
  Verification,
  Validation,
  Benchmarking
 ]
 EXCLUDE_AS_PRIMARY = [
  PureMathematicsWithoutComputationalMethod,
  SoftwareEngineeringWithoutMathematicalComputation,
  DataProcessingWithoutMathematicalAlgorithm,
  ExperimentalMeasurementWithoutComputationalMathematics
 ]
 DISAMBIGUATION_RULE = "When mathematical problems are solved, approximated, simulated or analysed primarily through algorithms and computation, interpret the context as computational mathematics."
}

CORE_CONCEPTS {
 ComputationalMathematics = "Mathematical study of computational methods and algorithms."
 NumericalMethod = "Algorithm producing approximate numerical solutions to mathematical problems."
 ScientificComputing = "Use of computational methods to solve scientific and mathematical models."
 NumericalAnalysis = "Study of numerical algorithms, approximation, stability and error."
 Discretisation = "Replacement of continuous mathematical structure by a finite computational representation."
 Conditioning = "Sensitivity of a mathematical problem to perturbations in input."
 Stability = "Sensitivity of a numerical method to perturbations and accumulated error."
 Convergence = "Tendency of numerical approximations to approach the target solution."
 ErrorAnalysis = "Quantitative study of approximation, truncation and floating-point error."
 AlgorithmicComplexity = "Resource cost of a computational method."
}

RELATIONS {
 ComputationalMathematicsOntologies --is_child_of--> MathematicalOntologies
 ComputationalMathematicsOntologies --contains--> NumericalAnalysis
 ComputationalMathematicsOntologies --contains--> ScientificComputing
 ComputationalMathematicsOntologies --contains--> ComputationalAlgebra
 ComputationalMathematicsOntologies --contains--> ComputationalOptimisation
 NumericalAnalysis --studies--> NumericalMethod
 NumericalMethod --approximates--> MathematicalSolution
 Discretisation --produces--> FiniteComputationalProblem
 ErrorAnalysis --evaluates--> NumericalApproximation
 Conditioning --describes--> ProblemSensitivity
 Stability --describes--> AlgorithmSensitivity
 Consistency --supports--> Convergence
 NumericalLinearAlgebra --uses--> MatrixComputation
 SparseLinearAlgebra --uses--> SparseMatrix
 IterativeMethod --may_use--> Preconditioner
 FiniteDifferenceMethod --discretises--> DifferentialEquation
 FiniteElementMethod --discretises--> VariationalProblem
 FiniteVolumeMethod --discretises--> ConservationLaw
 SpectralMethod --uses--> GlobalBasisFunctions
 MonteCarloMethod --uses--> RandomSampling
 HighPerformanceComputing --accelerates--> ComputationalMathematics
 Verification --checks--> NumericalImplementation
 Validation --checks--> ModelAgainstReference
}

OPERATORS {
 ABSOLUTE_ERROR(x_exact,x_approx) := |x_exact-x_approx|
 RELATIVE_ERROR(x_exact,x_approx) := |x_exact-x_approx|/|x_exact|
 CONDITION_NUMBER(A) := kappa(A)=||A||*||A^(-1)||
 NEWTON_ITERATION(x,f) := x_next=x-f(x)/f'(x)
 FIXED_POINT(x,g) := x_next=g(x)
 FORWARD_DIFFERENCE(f,x,h) := (f(x+h)-f(x))/h
 CENTRAL_DIFFERENCE(f,x,h) := (f(x+h)-f(x-h))/(2*h)
 TRAPEZOID_RULE(f,a,b,n) := numerical_integral_by_piecewise_trapezoids
 SIMPSON_RULE(f,a,b,n) := numerical_integral_by_piecewise_quadratics
 LINEAR_SYSTEM(A,x,b) := A*x=b
 RESIDUAL(A,x,b) := r=b-A*x
 ITERATIVE_SOLVER(A,x_k,b,M) := x_(k+1)=x_k+M^(-1)*(b-A*x_k)
}

AXIOMS {
 A1 = "Computational mathematics requires an explicit mathematical target and computational representation."
 A2 = "Approximate numerical results require error analysis."
 A3 = "Problem conditioning and algorithm stability are distinct concepts."
 A4 = "Discretisation parameters affect accuracy, stability and computational cost."
 A5 = "Convergence claims require an appropriate limiting process."
 A6 = "Floating-point arithmetic introduces finite-precision effects."
 A7 = "Computational efficiency must not be confused with mathematical correctness."
 A8 = "Computational results require verification, validation and reproducibility where applicable."
}

REASONING {
 R1_CONTEXT = identify(mathematical_problem,desired_output,dimension,accuracy_goal,compute_constraints)
 R2_FORM = classify(algebraic,differential,integral,optimisation,stochastic,symbolic_or_geometric_problem)
 R3_METHOD = select(direct,iterative,discretisation,approximation,simulation_or_symbolic_method)
 R4_REPRESENTATION = construct(matrix,mesh,grid,basis,sparse_structure_or_symbolic_form)
 R5_CONDITIONING = analyse(problem_sensitivity)
 R6_STABILITY = analyse(method_stability_and_roundoff)
 R7_ERROR = estimate(truncation,discretisation,iteration_and_floating_point_error)
 R8_COMPLEXITY = estimate(time_memory_parallelism_and_scalability)
 R9_ADAPTIVITY = refine(mesh,timestep,tolerance_or_model_order_if_required)
 R10_VALIDATION = compare(reference_solution,convergence_rate,residuals,benchmarks,reproducibility)
}

CONSTRAINTS {
 C1 = "A numerically stable algorithm cannot correct an ill-posed mathematical problem."
 C2 = "A small residual does not always imply a small solution error for ill-conditioned systems."
 C3 = "Higher resolution does not guarantee stability."
 C4 = "Finite precision can dominate truncation error at sufficiently small step sizes."
 C5 = "Iterative convergence depends on method, spectrum, conditioning and stopping criteria."
 C6 = "Sparse structure should not be discarded unnecessarily in large computational problems."
 C7 = "Parallel speedup is limited by communication and serial fractions."
 C8 = "Benchmark agreement does not prove correctness outside the tested regime."
 C9 = "Symbolic exactness can become computationally infeasible for large expressions."
 C10 = "Approximation methods require explicit validity and error regimes."
}

SEMANTIC_EQUIVALENCE {
 "ComputationalMathematics" <=> "MathematicsByComputation"
 "NumericalLinearAlgebra" <=> "ComputationalLinearAlgebra"
 "HighPerformanceComputing" <=> "HPC"
 "FiniteDifferenceMethod" <=> "FDM"
 "FiniteElementMethod" <=> "FEM"
 "FiniteVolumeMethod" <=> "FVM"
}

VALIDATION {
 V1 = "Mathematical problem and target quantity are explicit."
 V2 = "Numerical representation and discretisation are documented."
 V3 = "Conditioning is assessed where relevant."
 V4 = "Stability is analysed or tested."
 V5 = "Error and convergence are quantified."
 V6 = "Stopping criteria are documented for iterative methods."
 V7 = "Computational complexity and memory demands are considered."
 V8 = "Reference or benchmark comparisons are used where feasible."
 V9 = "Results are reproducible under documented numerical settings."
 V10 = "Verification and validation are distinguished."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 COMPUTATIONAL_MATHEMATICS_DEFINITION = 0.99
 NUMERICAL_ANALYSIS_REASONING = 0.99
 DISCRETISATION_REASONING = 0.99
 ERROR_ANALYSIS_REASONING = 0.99
 LINEAR_ALGEBRA_REASONING = 0.99
 OPTIMISATION_REASONING = 0.98
 HPC_REASONING = 0.98
 SYMBOLIC_REASONING = 0.98
 POLICY = "Reduce confidence when numerical method, discretisation, conditioning, stability, tolerance, precision, stopping rule or benchmark regime are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch, OntologyGrounding, ComputationalMathematicsReasoning,
  NumericalAnalysisReasoning, ScientificComputingReasoning,
  NumericalLinearAlgebraReasoning, SparseMatrixReasoning,
  MatrixDecompositionReasoning, RootFindingReasoning,
  NumericalDifferentiationReasoning, NumericalIntegrationReasoning,
  ApproximationReasoning, ErrorAnalysisReasoning, DiscretisationReasoning,
  NumericalODEReasoning, NumericalPDEReasoning, OptimisationReasoning,
  MonteCarloReasoning, AlgorithmicComplexityReasoning,
  ParallelComputingReasoning, HPCReasoning, SymbolicComputationReasoning,
  KnowledgeGraphConstruction, RetrievalAugmentation, RecursiveInference
 ]
 NORMALISE {
  "computational mathematics" -> ComputationalMathematics
  "numerical analysis" -> NumericalAnalysis
  "scientific computing" -> ScientificComputing
  "computational linear algebra" -> NumericalLinearAlgebra
  "HPC" -> HighPerformanceComputing
  "FDM" -> FiniteDifferenceMethod
  "FEM" -> FiniteElementMethod
  "FVM" -> FiniteVolumeMethod
 }
}

EXECUTION_RULES {
 E1 = "If a mathematical problem is primarily solved through algorithms or computation, activate ComputationalMathematicsOntologies."
 E2 = "If approximate numerical solution is central, route to numerical analysis."
 E3 = "If large scientific models are simulated, route to scientific computing."
 E4 = "If matrix systems dominate, route to numerical linear algebra."
 E5 = "If differential equations are discretised, route to numerical ODE or PDE methods."
 E6 = "If exact symbolic manipulation is central, route to computational algebra."
 E7 = "If resource scaling is central, analyse algorithmic complexity and HPC."
 E8 = "If stochastic approximation is central, route to Monte Carlo methods."
 E9 = "Any computational result should be validated for error, stability, convergence and reproducibility."
}

RECURSION {
 ENTRY = ComputationalMathematicsOntologies
 EXPAND(ComputationalMathematicsOntologies) -> [
  ComputationalMathematics, NumericalAnalysis, NumericalMethod,
  NumericalAlgorithm, ScientificComputing, SymbolicComputation,
  ComputerAlgebra, NumericalLinearAlgebra, SparseLinearAlgebra,
  MatrixComputation, EigenvalueComputation, MatrixDecomposition,
  IterativeMethod, DirectMethod, RootFinding, NonlinearEquationSolving,
  NumericalDifferentiation, NumericalIntegration, Quadrature,
  Interpolation, Extrapolation, ApproximationTheory, PolynomialApproximation,
  SplineApproximation, SpectralApproximation, ErrorAnalysis,
  TruncationError, RoundoffError, Conditioning, Stability, Consistency,
  Convergence, BackwardError, ForwardError, FloatingPointArithmetic,
  MachinePrecision, Discretisation, Mesh, Grid, TimeStep, SpatialStep,
  FiniteDifferenceMethod, FiniteElementMethod, FiniteVolumeMethod,
  SpectralMethod, BoundaryElementMethod, MethodOfLines, NumericalODE,
  NumericalPDE, AdaptiveMethod, AdaptiveMesh, AdaptiveTimeStepping,
  Multigrid, DomainDecomposition, ComputationalOptimisation,
  LinearProgramming, NonlinearProgramming, ConvexOptimisation,
  GradientMethod, NewtonMethod, QuasiNewtonMethod, TrustRegionMethod,
  InteriorPointMethod, MonteCarloMethod, QuasiMonteCarlo,
  ComputationalGeometry, ComputationalNumberTheory, ComputationalAlgebra,
  AlgorithmicComplexity, TimeComplexity, SpaceComplexity,
  ParallelComputing, HighPerformanceComputing, DistributedComputing,
  GPUComputing, Vectorisation, SparseMatrix, Preconditioner, Solver,
  Verification, Validation, Benchmarking, Reproducibility
 ]
 EXPAND(ComputationalFamilies) -> [
  NumericalAnalysis,
  ScientificComputing,
  NumericalLinearAlgebra,
  ComputationalAlgebra,
  NumericalDifferentialEquations,
  ComputationalOptimisation,
  MonteCarloMethods,
  HighPerformanceComputing
 ]
 STOP_CONDITION = "Stop when problem type, computational representation, algorithm, error, stability, convergence, complexity and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Mathematical Ontologies"
 RELATED_ONTOLOGIES = [
  "Applied Mathematics Ontologies",
  "Algebra Ontologies",
  "Linear Algebra Ontologies",
  "Calculus & Analysis Ontologies",
  "Differential Equation Ontologies",
  "Optimisation Ontologies",
  "Probability & Statistics Ontologies",
  "Discrete Mathematics Ontologies",
  "Artificial Intelligence Mathematics Ontologies"
 ]
 ROUTE_RULES = [
  "Computational Mathematics -> Mathematical Ontologies for parent semantics",
  "Computational Mathematics -> Applied Mathematics for application context",
  "Computational Mathematics -> Linear Algebra for matrix computation",
  "Computational Mathematics -> Calculus & Analysis for approximation and convergence",
  "Computational Mathematics -> Differential Equations for numerical ODE and PDE methods",
  "Computational Mathematics -> Optimisation for computational optimisation",
  "Computational Mathematics -> Probability & Statistics for stochastic computation",
  "Computational Mathematics -> Discrete Mathematics for algorithmic structures",
  "Computational Mathematics -> AI Mathematics for computational learning systems"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible algorithms, numerical methods or computational pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve exact-versus-approximate computation distinction.",
  "Preserve problem-conditioning-versus-algorithm-stability distinction.",
  "Preserve truncation-versus-roundoff error distinction.",
  "Preserve direct-versus-iterative method distinction.",
  "Preserve continuous-versus-discretised representation distinction.",
  "Preserve symbolic-versus-numerical computation distinction.",
  "Preserve verification-versus-validation distinction.",
  "Preserve machine-readable computational-mathematics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Computational mathematics develops and analyses algorithms and computational methods for solving mathematical problems."
 NUMERICAL_SUMMARY = "Numerical analysis studies approximation, stability, conditioning, error and convergence."
 LINEAR_ALGEBRA_SUMMARY = "Computational linear algebra solves matrix, eigenvalue and decomposition problems efficiently."
 DISCRETISATION_SUMMARY = "Finite-difference, finite-element, finite-volume and spectral methods convert continuous models into computational systems."
 OPTIMISATION_SUMMARY = "Computational optimisation applies iterative and direct algorithms to mathematical objective functions and constraints."
 COMPLEXITY_SUMMARY = "Algorithmic complexity, sparse structure, parallelism and HPC determine scalability."
 VALIDATION_SUMMARY = "Computational results should be checked for conditioning, stability, error, convergence, verification, validation and reproducibility."
}

FLOW {
 INPUT
 -> detect_computational_mathematics_context
 -> identify_mathematical_problem_accuracy_goal_dimension_and_compute_constraints
 -> classify_algebraic_differential_integral_optimisation_stochastic_symbolic_or_geometric_problem
 -> choose_computational_representation_and_algorithm
 -> analyse_conditioning_and_stability
 -> estimate_discretisation_iteration_roundoff_and_total_error
 -> assess_convergence_and_stopping_criteria
 -> optimise_sparse_structure_memory_parallelism_and_scalability
 -> refine_mesh_timestep_tolerance_or_model_order_if_required
 -> verify_implementation_and_validate_against_reference_or_benchmark
 -> route_to_mathematical_applied_linear_algebra_analysis_differential_equations_optimisation_probability_statistics_discrete_mathematics_or_AI_mathematics_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Computational Mathematics Ontologies layer defines mathematical computation and numerical reasoning. It supports numerical analysis and algorithms, scientific computing, symbolic computation and computer algebra, numerical and sparse linear algebra, matrix systems, eigenvalue problems and matrix decompositions, direct and iterative methods, nonlinear root finding, numerical differentiation and integration, quadrature, interpolation and approximation, truncation and roundoff error, conditioning, stability, consistency and convergence, floating-point arithmetic, discretisation, meshes and grids, finite-difference, finite-element, finite-volume, spectral and boundary-element methods, numerical ODE and PDE solving, adaptive methods, multigrid and domain decomposition, computational optimisation, Monte Carlo methods, computational geometry and number theory, algorithmic complexity, sparse computation, parallel, distributed, GPU and high-performance computing, verification, validation, benchmarking and reproducibility."
}

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

}

END_FLOW