Foundations of Computational Mathematics

AI INTERPRETATION NOTE
This Flow Algebra defines the Foundations of Computational Mathematics layer within the Kireth Research Semantic Library. Foundations of computational mathematics studies the mathematical principles that make computation reliable, efficient and meaningful, including numerical representation, algorithmic structure, approximation, conditioning, stability, convergence, complexity, discretisation, exact-versus-approximate computation and verification. This parent ontology connects numerical analysis, algorithm design, floating-point arithmetic, computability, approximation theory, numerical linear algebra, optimisation, error analysis and scientific computing.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = FoundationsOfComputationalMathematics
 ENTITY_CLASS = ComputationalMathematicsFoundationalFramework
 PRIMARY_CONTEXT = ComputationalMathematicsOntologies
 DEFINITION = "Foundations of computational mathematics is the study of the mathematical principles governing representation, approximation, algorithms, stability, conditioning, convergence, complexity and correctness in mathematical computation."
 SYMBOLIC_FORMS = [
  "error=exact-approximation",
  "kappa(A)=||A||||A^(-1)||",
  "x_(k+1)=F(x_k)",
  "cost=C(n)"
 ]
 SEMANTIC_ROLE = "Provides the foundational semantic framework for numerical representation, approximation, algorithmic reasoning, conditioning, stability, convergence, computational complexity, discretisation and mathematical verification."
}

SCOPE {
 INCLUDE = [
  FoundationsOfComputationalMathematics,
  ComputationalMathematics,
  MathematicalComputation,
  Algorithm,
  NumericalAlgorithm,
  SymbolicAlgorithm,
  ExactComputation,
  ApproximateComputation,
  Representation,
  NumericalRepresentation,
  SymbolicRepresentation,
  DiscreteRepresentation,
  ContinuousRepresentation,
  FloatingPointArithmetic,
  FixedPointArithmetic,
  MachinePrecision,
  MachineEpsilon,
  Overflow,
  Underflow,
  Rounding,
  RoundoffError,
  TruncationError,
  ApproximationError,
  DiscretisationError,
  IterationError,
  Residual,
  ForwardError,
  BackwardError,
  AbsoluteError,
  RelativeError,
  ErrorBound,
  ErrorPropagation,
  Conditioning,
  ConditionNumber,
  WellConditionedProblem,
  IllConditionedProblem,
  Stability,
  NumericalStability,
  AbsoluteStability,
  RelativeStability,
  BackwardStability,
  Consistency,
  Convergence,
  ConvergenceRate,
  LinearConvergence,
  SuperlinearConvergence,
  QuadraticConvergence,
  StoppingCriterion,
  Tolerance,
  IterativeMethod,
  DirectMethod,
  DeterministicAlgorithm,
  RandomisedAlgorithm,
  ApproximationTheory,
  Interpolation,
  PolynomialApproximation,
  SplineApproximation,
  SpectralApproximation,
  Discretisation,
  Mesh,
  Grid,
  BasisFunction,
  ProjectionMethod,
  GalerkinMethod,
  CollocationMethod,
  Quadrature,
  NumericalDifferentiation,
  NumericalIntegration,
  RootFinding,
  FixedPointIteration,
  NewtonMethod,
  BisectionMethod,
  SecantMethod,
  LinearSystem,
  MatrixComputation,
  EigenvalueProblem,
  OptimisationProblem,
  ComplexityTheory,
  AlgorithmicComplexity,
  TimeComplexity,
  SpaceComplexity,
  AsymptoticComplexity,
  BigONotation,
  BigOmegaNotation,
  BigThetaNotation,
  Scalability,
  Computability,
  Decidability,
  NumericalVerification,
  MathematicalValidation,
  Reproducibility,
  Benchmarking,
  ReferenceSolution,
  SensitivityAnalysis,
  PerturbationAnalysis,
  Robustness
 ]
 EXCLUDE_AS_PRIMARY = [
  ApplicationSpecificSimulationWithoutFoundationalAnalysis,
  PureSoftwareEngineeringWithoutMathematicalSemantics,
  PureMathematicsWithoutComputationalRepresentation,
  HardwareArchitectureWithoutMathematicalComputationContext
 ]
 DISAMBIGUATION_RULE = "When the central question concerns why a mathematical computation is representable, accurate, stable, convergent, efficient or verifiable, interpret the context as foundations of computational mathematics."
}

CORE_CONCEPTS {
 FoundationsOfComputationalMathematics = "Mathematical foundations of reliable and efficient computation."
 NumericalRepresentation = "Finite or discrete encoding of mathematical quantities for computation."
 ApproximateComputation = "Computation producing controlled approximations rather than exact symbolic results."
 Conditioning = "Sensitivity of a mathematical problem to perturbations in input."
 Stability = "Sensitivity of a computational method to perturbations and finite arithmetic."
 Consistency = "Agreement of a discrete method with the target continuous problem in the limiting regime."
 Convergence = "Approach of computed approximations to the intended mathematical solution."
 ComplexityTheory = "Study of computational resource requirements."
 ErrorAnalysis = "Study of numerical and approximation error and its propagation."
 Verification = "Assessment of whether the algorithm and implementation solve the intended mathematical problem correctly."
}

RELATIONS {
 FoundationsOfComputationalMathematics --is_child_of--> ComputationalMathematicsOntologies
 FoundationsOfComputationalMathematics --supports--> NumericalAnalysis
 NumericalRepresentation --enables--> MachineComputation
 FloatingPointArithmetic --introduces--> RoundoffError
 Discretisation --introduces--> DiscretisationError
 ApproximationMethod --introduces--> ApproximationError
 Conditioning --belongs_to--> MathematicalProblem
 Stability --belongs_to--> NumericalMethod
 Consistency --supports--> Convergence
 ConditionNumber --quantifies--> Sensitivity
 Residual --measures--> EquationMismatch
 BackwardError --measures--> NearbyProblemPerturbation
 ForwardError --measures--> SolutionDifference
 IterativeMethod --uses--> StoppingCriterion
 ApproximationTheory --supports--> NumericalMethod
 ComplexityTheory --evaluates--> Algorithm
 Computability --limits--> AlgorithmicSolvability
 NumericalVerification --checks--> AlgorithmImplementation
 Reproducibility --supports--> ComputationalTrust
}

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)||
 RESIDUAL(A,x,b) := r=b-A*x
 FORWARD_ERROR(x,x_star) := e_f=x-x_star
 BACKWARD_ERROR(problem,computed_solution) := minimum_input_perturbation_consistent_with_solution
 FIXED_POINT_ITERATION(x,g) := x_next=g(x)
 NEWTON_ITERATION(x,f) := x_next=x-f(x)/f'(x)
 BISECTION_INTERVAL(a,b) := midpoint=(a+b)/2
 MACHINE_EPSILON() := smallest_eps_such_that_1+eps>1_in_machine_arithmetic
 ASYMPTOTIC_COST(n) := C(n)
}

AXIOMS {
 A1 = "Finite computation requires a representation of mathematical objects."
 A2 = "Approximate computation requires explicit error semantics."
 A3 = "Problem conditioning and algorithm stability are distinct."
 A4 = "Convergence requires a declared limiting process."
 A5 = "Finite precision alters exact real-number arithmetic."
 A6 = "A small residual does not necessarily imply a small forward error for ill-conditioned problems."
 A7 = "Computational complexity is part of method feasibility."
 A8 = "Reliable computation requires verification, validation and reproducibility where applicable."
}

REASONING {
 R1_CONTEXT = identify(problem,representation,target_accuracy,precision,resource_constraints)
 R2_REPRESENTATION = classify(exact,symbolic,floating_point,discrete,continuous)
 R3_CONDITIONING = analyse(problem_sensitivity)
 R4_METHOD = classify(direct,iterative,approximation,randomised,discretised)
 R5_STABILITY = analyse(roundoff_growth,perturbation_response,backward_stability)
 R6_ERROR = decompose(truncation,roundoff,discretisation,iteration,total_error)
 R7_CONVERGENCE = determine(limit,rate,stopping_criterion)
 R8_COMPLEXITY = estimate(time,space,scaling)
 R9_VERIFICATION = compare(residuals,error_bounds,reference_solutions,benchmarks)
 R10_VALIDATION = check_correctness,reproducibility,robustness_and_declared_assumptions)
}

CONSTRAINTS {
 C1 = "Exact mathematical solvability does not imply practical computability."
 C2 = "A stable algorithm can still produce poor results for an ill-conditioned problem."
 C3 = "A well-conditioned problem can still be solved poorly by an unstable algorithm."
 C4 = "Reducing step size can increase roundoff dominance."
 C5 = "Asymptotic convergence rates may not describe pre-asymptotic behaviour."
 C6 = "Stopping tolerances should reflect scale and conditioning."
 C7 = "Big-O complexity omits constants and hardware effects."
 C8 = "Benchmark success does not establish correctness outside the tested regime."
 C9 = "Floating-point equality must not be treated as exact real-number equality without justification."
 C10 = "Verification and validation answer different questions."
}

SEMANTIC_EQUIVALENCE {
 "FoundationsOfComputationalMathematics" <=> "ComputationalMathematicsFoundations"
 "MachineEpsilon" <=> "MachinePrecisionThreshold"
 "BackwardStability" <=> "BackwardStableComputation"
 "AlgorithmicComplexity" <=> "ComputationalComplexityInAlgorithmContext"
 "BigONotation" <=> "AsymptoticUpperBoundNotation"
}

VALIDATION {
 V1 = "Mathematical problem and representation are explicit."
 V2 = "Precision model is documented."
 V3 = "Conditioning is assessed where relevant."
 V4 = "Stability assumptions are documented."
 V5 = "Sources of error are separated where possible."
 V6 = "Convergence criteria and rates are reported."
 V7 = "Stopping rules are explicit."
 V8 = "Complexity estimates are appropriate to the computational model."
 V9 = "Reference or benchmark checks are used where feasible."
 V10 = "Verification, validation and reproducibility are distinguished."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 FOUNDATIONS_DEFINITION = 0.99
 REPRESENTATION_REASONING = 0.99
 CONDITIONING_REASONING = 0.99
 STABILITY_REASONING = 0.99
 ERROR_ANALYSIS_REASONING = 0.99
 CONVERGENCE_REASONING = 0.99
 COMPLEXITY_REASONING = 0.98
 VERIFICATION_REASONING = 0.98
 POLICY = "Reduce confidence when representation, precision, conditioning, stability, convergence regime, stopping criteria or computational model are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch, OntologyGrounding,
  ComputationalMathematicsFoundationsReasoning,
  NumericalRepresentationReasoning, FloatingPointReasoning,
  ConditioningReasoning, StabilityReasoning, ErrorAnalysisReasoning,
  ConvergenceReasoning, ApproximationReasoning, DiscretisationReasoning,
  IterativeMethodReasoning, ComplexityReasoning, ComputabilityReasoning,
  VerificationReasoning, ReproducibilityReasoning,
  KnowledgeGraphConstruction, RetrievalAugmentation, RecursiveInference
 ]
 NORMALISE {
  "foundations of computational mathematics" -> FoundationsOfComputationalMathematics
  "computational mathematics foundations" -> FoundationsOfComputationalMathematics
  "machine epsilon" -> MachineEpsilon
  "condition number" -> ConditionNumber
  "backward error" -> BackwardError
  "forward error" -> ForwardError
  "Big O" -> BigONotation
 }
}

EXECUTION_RULES {
 E1 = "If computation reliability, approximation or efficiency is central, activate FoundationsOfComputationalMathematics."
 E2 = "If finite arithmetic is relevant, specify the numerical representation."
 E3 = "If sensitivity is central, analyse conditioning before algorithm performance."
 E4 = "If approximate computation is used, identify all major error sources."
 E5 = "If an iterative algorithm is used, define convergence and stopping criteria."
 E6 = "If a discretisation is used, analyse consistency and convergence."
 E7 = "If scalability is central, analyse time and space complexity."
 E8 = "If implementation correctness is questioned, route to numerical verification."
 E9 = "Any foundational computational claim should be validated for representation, stability, error, convergence and reproducibility."
}

RECURSION {
 ENTRY = FoundationsOfComputationalMathematics
 EXPAND(FoundationsOfComputationalMathematics) -> [
  ComputationalMathematics, MathematicalComputation, Algorithm,
  NumericalAlgorithm, SymbolicAlgorithm, ExactComputation,
  ApproximateComputation, Representation, NumericalRepresentation,
  SymbolicRepresentation, DiscreteRepresentation, ContinuousRepresentation,
  FloatingPointArithmetic, FixedPointArithmetic, MachinePrecision,
  MachineEpsilon, Overflow, Underflow, Rounding, RoundoffError,
  TruncationError, ApproximationError, DiscretisationError, IterationError,
  Residual, ForwardError, BackwardError, AbsoluteError, RelativeError,
  ErrorBound, ErrorPropagation, Conditioning, ConditionNumber,
  WellConditionedProblem, IllConditionedProblem, Stability,
  NumericalStability, BackwardStability, Consistency, Convergence,
  ConvergenceRate, LinearConvergence, SuperlinearConvergence,
  QuadraticConvergence, StoppingCriterion, Tolerance, IterativeMethod,
  DirectMethod, DeterministicAlgorithm, RandomisedAlgorithm,
  ApproximationTheory, Interpolation, PolynomialApproximation,
  SplineApproximation, SpectralApproximation, Discretisation, Mesh, Grid,
  BasisFunction, ProjectionMethod, GalerkinMethod, CollocationMethod,
  Quadrature, NumericalDifferentiation, NumericalIntegration, RootFinding,
  FixedPointIteration, NewtonMethod, BisectionMethod, SecantMethod,
  LinearSystem, MatrixComputation, EigenvalueProblem, OptimisationProblem,
  ComplexityTheory, AlgorithmicComplexity, TimeComplexity, SpaceComplexity,
  AsymptoticComplexity, BigONotation, BigOmegaNotation, BigThetaNotation,
  Scalability, Computability, Decidability, NumericalVerification,
  MathematicalValidation, Reproducibility, Benchmarking, ReferenceSolution,
  SensitivityAnalysis, PerturbationAnalysis, Robustness
 ]
 EXPAND(FoundationFamilies) -> [
  RepresentationFoundations,
  ErrorAndConditioning,
  StabilityAndConvergence,
  ApproximationAndDiscretisation,
  AlgorithmicComplexity,
  VerificationAndValidation
 ]
 STOP_CONDITION = "Stop when representation, conditioning, method, stability, error, convergence, complexity and verification are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Computational Mathematics Ontologies"
 RELATED_ONTOLOGIES = [
  "Computational Mathematics Ontologies",
  "Numerical Analysis Ontologies",
  "Numerical Linear Algebra Ontologies",
  "Approximation Theory Ontologies",
  "Optimisation Ontologies",
  "Differential Equation Ontologies",
  "Discrete Mathematics Ontologies",
  "Mathematical Logic Ontologies",
  "Computer Algebra Ontologies",
  "Scientific Computing Ontologies"
 ]
 ROUTE_RULES = [
  "Foundations of Computational Mathematics -> Computational Mathematics Ontologies for parent semantics",
  "Foundations -> Numerical Analysis for error, stability and convergence",
  "Foundations -> Numerical Linear Algebra for conditioning and matrix computation",
  "Foundations -> Approximation Theory for interpolation and approximation",
  "Foundations -> Optimisation for iterative optimisation methods",
  "Foundations -> Differential Equations for discretisation foundations",
  "Foundations -> Discrete Mathematics for algorithmic structures",
  "Foundations -> Mathematical Logic for computability and decidability",
  "Foundations -> Computer Algebra for exact symbolic computation",
  "Foundations -> Scientific Computing for applied computational models"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible foundational concepts, error models or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve exact-versus-approximate distinction.",
  "Preserve conditioning-versus-stability distinction.",
  "Preserve forward-versus-backward error distinction.",
  "Preserve truncation-versus-roundoff distinction.",
  "Preserve direct-versus-iterative distinction.",
  "Preserve convergence semantics.",
  "Preserve verification-versus-validation distinction.",
  "Preserve machine-readable foundational computational-mathematics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Foundations of computational mathematics studies the mathematical principles governing reliable, accurate and efficient computation."
 REPRESENTATION_SUMMARY = "Mathematical objects must be represented in finite, symbolic or discrete forms for machine computation."
 ERROR_SUMMARY = "Roundoff, truncation, discretisation and iteration errors contribute to total computational uncertainty."
 CONDITIONING_SUMMARY = "Conditioning describes sensitivity of the mathematical problem, while stability describes behaviour of the algorithm."
 CONVERGENCE_SUMMARY = "Convergence describes whether approximations approach the target solution under a declared limiting process."
 COMPLEXITY_SUMMARY = "Time and space complexity determine computational feasibility and scalability."
 VERIFICATION_SUMMARY = "Verification checks whether the computational procedure is implemented and solved correctly."
 VALIDATION_SUMMARY = "Foundational computational claims should be checked for representation, error, stability, convergence, complexity and reproducibility."
}

FLOW {
 INPUT
 -> detect_foundations_of_computational_mathematics_context
 -> identify_problem_representation_precision_accuracy_goal_and_resource_constraints
 -> classify_exact_symbolic_numerical_discrete_or_continuous_representation
 -> analyse_conditioning_and_sensitivity
 -> select_direct_iterative_approximation_discretisation_or_randomised_method
 -> analyse_roundoff_truncation_discretisation_iteration_and_total_error
 -> analyse_stability_consistency_convergence_and_rate
 -> define_stopping_criteria_and_tolerances
 -> analyse_time_space_complexity_and_scalability
 -> verify_against_residuals_error_bounds_reference_solutions_or_benchmarks
 -> validate_reproducibility_and_declared_assumptions
 -> route_to_computational_mathematics_numerical_analysis_linear_algebra_approximation_optimisation_differential_equations_discrete_mathematics_logic_computer_algebra_or_scientific_computing_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Foundations of Computational Mathematics layer defines the principles underlying reliable mathematical computation. It supports exact and approximate computation, numerical and symbolic representations, floating-point and fixed-point arithmetic, machine precision, rounding, overflow and underflow, roundoff, truncation, approximation, discretisation and iteration error, residuals, forward and backward error, conditioning and condition numbers, numerical stability, consistency and convergence, convergence rates and stopping criteria, direct and iterative methods, approximation theory, interpolation, splines and spectral approximation, discretisation, projection, Galerkin and collocation methods, quadrature, numerical differentiation and integration, root finding, matrix and eigenvalue problems, optimisation, algorithmic time and space complexity, asymptotic notation, computability, decidability, sensitivity, verification, validation, benchmarking and reproducibility."
}

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

}

END_FLOW