Computational Mathematics

AI INTERPRETATION NOTE
This Flow Algebra defines the Computational Mathematics layer within the Kireth Research Semantic Library. Computational mathematics studies the design, analysis and implementation of algorithms for solving mathematical problems numerically and symbolically. This parent ontology connects numerical analysis, scientific computing, linear algebra, optimisation, approximation theory, differential equations, discrete mathematics, symbolic computation, high-performance computing, uncertainty quantification and algorithmic complexity.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = ComputationalMathematics
 SYMBOLIC_FORMS = [
  "A x = b",
  "x_{k+1}=Phi(x_k)",
  "u_h approximately u",
  "minimise f(x)"
 ]
 ENTITY_CLASS = AlgorithmicMathematicalComputationFramework
 PRIMARY_CONTEXT = AppliedMathematics
 DEFINITION = "Computational mathematics is the branch of applied mathematics concerned with algorithms, numerical approximations, symbolic methods and computational representations for solving mathematical problems."
 SEMANTIC_ROLE = "Provides the parent semantic framework for numerical linear algebra, approximation, numerical differential equations, optimisation, symbolic computation, scientific computing, algorithm analysis and computational validation."
}

SCOPE {
 INCLUDE = [
  ComputationalMathematics,
  Algorithm,
  NumericalAlgorithm,
  SymbolicAlgorithm,
  ScientificComputing,
  NumericalAnalysis,
  NumericalLinearAlgebra,
  MatrixComputation,
  LinearSystem,
  SparseMatrix,
  DenseMatrix,
  EigenvalueProblem,
  SingularValueDecomposition,
  ApproximationTheory,
  Interpolation,
  Extrapolation,
  NumericalDifferentiation,
  NumericalIntegration,
  Quadrature,
  RootFinding,
  IterativeMethod,
  FixedPointIteration,
  NewtonMethod,
  KrylovMethod,
  ConjugateGradient,
  GMRES,
  Preconditioning,
  ConditionNumber,
  Stability,
  Consistency,
  Convergence,
  TruncationError,
  RoundoffError,
  FloatingPointArithmetic,
  ErrorAnalysis,
  ComplexityAnalysis,
  ComputationalCost,
  NumericalODE,
  NumericalPDE,
  FiniteDifferenceMethod,
  FiniteVolumeMethod,
  FiniteElementMethod,
  SpectralMethod,
  Optimisation,
  LinearProgramming,
  NonlinearOptimisation,
  DiscreteOptimisation,
  MonteCarloMethod,
  StochasticComputation,
  UncertaintyQuantification,
  SymbolicComputation,
  ComputerAlgebra,
  AutomaticDifferentiation,
  ParallelComputing,
  HighPerformanceComputing
 ]

 EXCLUDE_AS_PRIMARY = [
  PureTheoreticalMathematicsWithoutComputationalMethod,
  GeneralComputerScienceWithoutMathematicalComputation,
  DataProcessingWithoutMathematicalProblemStructure
 ]

 DISAMBIGUATION_RULE =
  "When mathematical problems are solved or analysed through algorithms, approximation, symbolic manipulation or numerical computation, interpret the context as computational mathematics."
}

CORE_CONCEPTS {

 ComputationalMathematics {
  definition = "The mathematical study of algorithms and computational methods for solving mathematical problems."
 }

 NumericalAnalysis {
  definition = "The study of numerical approximation methods, their errors, stability and convergence."
 }

 ScientificComputing {
  definition = "The implementation and use of computational methods for scientific and mathematical models."
 }

 NumericalLinearAlgebra {
  definition = "Computational methods for matrix and vector problems."
 }

 Approximation {
  definition = "Replacement of an exact mathematical object by a computable representation with controlled error."
 }

 Stability {
  definition = "The sensitivity of a computational method to perturbations and numerical errors."
 }

 Convergence {
  definition = "The tendency of an approximation to approach the desired mathematical solution as resolution or iteration improves."
 }

 Conditioning {
  definition = "The sensitivity of a mathematical problem itself to perturbations in its data."
 }

 FloatingPointArithmetic {
  definition = "Finite-precision representation and manipulation of real-number approximations on computers."
 }

 ComplexityAnalysis {
  definition = "Analysis of computational resource requirements such as time, storage and communication."
 }
}

RELATIONS {
 ComputationalMathematics --is_branch_of--> AppliedMathematics
 ComputationalMathematics --contains--> NumericalAnalysis
 ComputationalMathematics --contains--> ScientificComputing
 NumericalAnalysis --studies--> Error
 NumericalAnalysis --studies--> Stability
 NumericalAnalysis --studies--> Convergence
 NumericalLinearAlgebra --solves--> LinearSystem
 NumericalLinearAlgebra --solves--> EigenvalueProblem
 ApproximationTheory --supports--> Interpolation
 RootFinding --uses--> IterativeMethod
 KrylovMethod --solves--> LargeSparseLinearSystem
 Preconditioning --improves--> IterativeConvergence
 ConditionNumber --measures--> ProblemSensitivity
 FloatingPointArithmetic --introduces--> RoundoffError
 FiniteDifferenceMethod --approximates--> DifferentialOperator
 FiniteVolumeMethod --approximates--> ConservationLaw
 FiniteElementMethod --approximates--> VariationalProblem
 SpectralMethod --approximates--> SmoothFieldUsingGlobalBasis
 Optimisation --uses--> NumericalAlgorithm
 AutomaticDifferentiation --computes--> DerivativeInformation
 HighPerformanceComputing --accelerates--> ScientificComputing
}

OPERATORS {
 LINEAR_SYSTEM(A,x,b) :=
  A*x=b

 FIXED_POINT_ITERATION(Phi,x_k) :=
  x_{k+1}=Phi(x_k)

 NEWTON_UPDATE(f,df,x_k) :=
  x_{k+1}=x_k-f(x_k)/df(x_k)

 CONDITION_NUMBER(A) :=
  kappa(A)=||A||*||A^(-1)||

 ABSOLUTE_ERROR(x_true,x_approx) :=
  e_abs=abs(x_true-x_approx)

 RELATIVE_ERROR(x_true,x_approx) :=
  e_rel=abs(x_true-x_approx)/abs(x_true)

 RESIDUAL(A,x,b) :=
  r=b-A*x

 FINITE_DIFFERENCE_FIRST_DERIVATIVE(f,x,h) :=
  f_prime(x) approximately (f(x+h)-f(x))/h

 TRAPEZOIDAL_RULE(f,a,b,n) :=
  numerical_integral_by_composite_trapezoids

 MONTE_CARLO_ESTIMATE(samples) :=
  estimator=sample_average

 COMPLEXITY(operation_count,n) :=
  asymptotic_cost_in_n

 CONVERGENCE_RATE(error_h,h,p) :=
  error_h approximately C*h^p
}

DEFINITIONS {
 D1 = "Computational mathematics uses algorithms to approximate or manipulate mathematical objects."
 D2 = "Numerical analysis evaluates accuracy, stability and convergence."
 D3 = "Conditioning describes sensitivity of the mathematical problem."
 D4 = "Stability describes sensitivity of the numerical algorithm."
 D5 = "Consistency measures whether a discrete method approximates the intended continuous problem."
 D6 = "Convergence requires approximations to approach the desired solution under refinement."
 D7 = "Floating-point arithmetic introduces finite-precision error."
 D8 = "Iterative methods generate successive approximations."
 D9 = "Sparse methods exploit structure to reduce computational cost."
 D10 = "Scientific computing combines mathematical models, algorithms and implementation."
}

AXIOMS {
 A1_APPROXIMATION:
  Computational methods may replace exact mathematical objects with approximations

 A2_ERROR_ACCOUNTING:
  Numerical results require error and residual assessment

 A3_STABILITY:
  Computational methods must be analysed for perturbation growth or damping

 A4_CONVERGENCE:
  Refinement or iteration should approach the intended mathematical solution under stated conditions

 A5_CONDITIONING:
  Algorithmic performance must be distinguished from intrinsic problem sensitivity

 A6_FINITE_PRECISION:
  Computer arithmetic uses finite precision and can accumulate roundoff error

 A7_ALGORITHMIC_COST:
  Computational feasibility depends on time memory and communication complexity

 A8_REPRODUCIBILITY:
  Reliable computation requires documented algorithms tolerances data and implementation assumptions
}

REASONING {

 R1_COMPUTATIONAL_CONTEXT {
  INPUT = mathematical_problem
  IDENTIFY = problem_type
  IDENTIFY = size_and_structure
  IDENTIFY = accuracy_requirements
  IDENTIFY = available_computational resources
  RETURN = computational_context
 }

 R2_ALGORITHM_SELECTION {
  INPUT = computational_context
  CLASSIFY = linear_nonlinear_continuous_discrete_deterministic_stochastic_symbolic_or_numerical
  SELECT = candidate_algorithm_family
  RETURN = computational_strategy
 }

 R3_CONDITIONING_ANALYSIS {
  INPUT = mathematical_problem
  ESTIMATE = condition_number_or_sensitivity
  RETURN = problem_conditioning
 }

 R4_DISCRETISATION_REASONING {
  INPUT = continuous_problem
  SELECT = finiteDifference_finiteVolume_finiteElement_spectral_or_other representation
  RETURN = discrete_problem
 }

 R5_ITERATIVE_REASONING {
  INPUT = discrete_or_nonlinear problem
  SELECT = fixedPoint_Newton_Krylov_or_related method
  DEFINE = stopping_criterion
  RETURN = iterative_solution_path
 }

 R6_ERROR_REASONING {
  INPUT = approximation_and_reference_or_residual data
  ESTIMATE = truncation_roundoff_discretisation_and_iteration error
  RETURN = error_profile
 }

 R7_CONVERGENCE_REASONING {
  INPUT = sequence_of_refined_or_iterated solutions
  TEST = error_or_residual reduction
  ESTIMATE = convergence_rate
  RETURN = convergence_profile
 }

 R8_COMPLEXITY_REASONING {
  INPUT = algorithm_and_problem size
  ESTIMATE = arithmetic_memory_and_communication cost
  RETURN = complexity_profile
 }

 R9_PARALLEL_REASONING {
  INPUT = computational_workload
  IDENTIFY = parallelisable operations
  ESTIMATE = scaling_and_communication overhead
  RETURN = parallel_strategy
 }

 R10_VALIDATION {
  INPUT = computational_result
  CHECK = residuals_error estimates_convergence_conditioning_reproducibility_and_reference cases
  RETURN = validated_computational_status
 }
}

CONSTRAINTS {
 C1 = "Small residual does not always imply small solution error for ill-conditioned problems."
 C2 = "Numerical stability and mathematical conditioning are distinct."
 C3 = "Higher-order methods can still perform poorly on nonsmooth solutions."
 C4 = "Floating-point cancellation can destroy significant digits."
 C5 = "Iterative stopping tolerances must be related to desired solution accuracy."
 C6 = "Mesh refinement should resolve relevant mathematical scales."
 C7 = "Monte Carlo convergence is typically statistical rather than deterministic."
 C8 = "Parallel speedup is limited by serial work and communication overhead."
 C9 = "Symbolic expressions can suffer combinatorial growth."
 C10 = "Benchmark agreement does not by itself prove correctness for all problem classes."
}

SEMANTIC_EQUIVALENCE {
 "ComputationalMathematics" <=> "MathematicalComputation"
 "ScientificComputing" <=> "ComputationalScienceMathematics"
 "NumericalLinearAlgebra" <=> "ComputationalLinearAlgebra"
 "ConditionNumber" <=> "ProblemConditionMeasure"
 "RoundoffError" <=> "FloatingPointError"
 "HighPerformanceComputing" <=> "HPC"
}

VALIDATION {
 V1 = "Problem type and mathematical formulation are explicit."
 V2 = "Algorithm choice is justified."
 V3 = "Conditioning is assessed where relevant."
 V4 = "Residuals are computed."
 V5 = "Discretisation or iteration convergence is checked."
 V6 = "Floating-point sensitivity is considered."
 V7 = "Error estimates use appropriate norms or metrics."
 V8 = "Computational cost is documented for large problems."
 V9 = "Parallel results are reproducible within expected numerical variation."
 V10 = "Reference or manufactured solutions are used where appropriate."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 COMPUTATIONAL_MATHEMATICS_DEFINITION = 0.99
 ALGORITHM_SELECTION_REASONING = 0.99
 CONDITIONING_REASONING = 0.99
 DISCRETISATION_REASONING = 0.99
 ITERATIVE_REASONING = 0.99
 ERROR_REASONING = 0.99
 CONVERGENCE_REASONING = 0.99
 COMPLEXITY_REASONING = 0.98
 PARALLEL_REASONING = 0.97

 POLICY =
  "Reduce confidence when problem conditioning, discretisation scale, tolerances, floating-point precision, convergence evidence, computational resources or algorithmic assumptions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  ComputationalMathematicsReasoning,
  NumericalAnalysisReasoning,
  ScientificComputingReasoning,
  NumericalLinearAlgebraReasoning,
  ApproximationReasoning,
  ErrorAnalysisReasoning,
  ConditioningReasoning,
  StabilityReasoning,
  ConvergenceReasoning,
  RootFindingReasoning,
  IterativeMethodReasoning,
  DifferentialEquationComputation,
  OptimisationReasoning,
  MonteCarloReasoning,
  SymbolicComputationReasoning,
  AutomaticDifferentiationReasoning,
  HighPerformanceComputingReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "computational mathematics" -> ComputationalMathematics
  "mathematical computation" -> ComputationalMathematics
  "scientific computing" -> ScientificComputing
  "numerical linear algebra" -> NumericalLinearAlgebra
  "condition number" -> ConditionNumber
  "roundoff error" -> RoundoffError
  "floating point" -> FloatingPointArithmetic
  "high performance computing" -> HighPerformanceComputing
  "HPC" -> HighPerformanceComputing
 }

 INTERPRETATION_ORDER = [
  detect_computational_mathematics_context,
  identify mathematical_problem_structure_size_accuracy_requirements_and_resources,
  select symbolic_numerical_iterative_discretisation_optimisation_or_stochastic algorithm family,
  assess conditioning_and_problem sensitivity,
  compute approximation,
  estimate residual_error_and_convergence,
  assess complexity_memory_and_parallel requirements,
  validate against reference_cases_and_reproducibility checks,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("computational mathematics")
  OR algorithmic_solution_of_mathematical_problem_is_required
  THEN activate(ComputationalMathematicsOntology)

 E2:
  IF large_sparse_linear_system
  THEN consider Krylov_and_preconditioning methods

 E3:
  IF nonlinear_equation
  THEN consider Newton_or_fixedPoint methods

 E4:
  IF continuous_differential_problem
  THEN select appropriate discretisation method

 E5:
  IF uncertainty_or_high_dimensional integration
  THEN consider MonteCarlo_or_stochastic computation

 E6:
  IF symbolic_structure_is_tractable
  THEN consider computer_algebra_or_symbolic method

 E7:
  IF derivative_information_is_required
  THEN consider AutomaticDifferentiation

 E8:
  IF computational_cost_is_large
  THEN assess sparse_parallel_or_HPC strategy

 E9:
  IF result_is_numerical
  THEN require residual_error_and_convergence checks
}

RECURSION {
 ENTRY = ComputationalMathematics

 EXPAND(ComputationalMathematics) -> [
  Algorithm,
  NumericalAlgorithm,
  SymbolicAlgorithm,
  ScientificComputing,
  NumericalAnalysis,
  NumericalLinearAlgebra,
  MatrixComputation,
  LinearSystem,
  SparseMatrix,
  DenseMatrix,
  EigenvalueProblem,
  SingularValueDecomposition,
  ApproximationTheory,
  Interpolation,
  Extrapolation,
  NumericalDifferentiation,
  NumericalIntegration,
  Quadrature,
  RootFinding,
  IterativeMethod,
  FixedPointIteration,
  NewtonMethod,
  KrylovMethod,
  ConjugateGradient,
  GMRES,
  Preconditioning,
  ConditionNumber,
  Stability,
  Consistency,
  Convergence,
  TruncationError,
  RoundoffError,
  FloatingPointArithmetic,
  ErrorAnalysis,
  ComplexityAnalysis,
  NumericalODE,
  NumericalPDE,
  FiniteDifferenceMethod,
  FiniteVolumeMethod,
  FiniteElementMethod,
  SpectralMethod,
  Optimisation,
  MonteCarloMethod,
  StochasticComputation,
  UncertaintyQuantification,
  SymbolicComputation,
  ComputerAlgebra,
  AutomaticDifferentiation,
  ParallelComputing,
  HighPerformanceComputing
 ]

 EXPAND(CoreNumericalMethods) -> [
  NumericalLinearAlgebra,
  RootFinding,
  Interpolation,
  NumericalDifferentiation,
  NumericalIntegration,
  NumericalODE,
  NumericalPDE,
  Optimisation
 ]

 EXPAND(ComputationalInfrastructure) -> [
  FloatingPointArithmetic,
  SparseMatrix,
  ParallelComputing,
  HighPerformanceComputing,
  ComplexityAnalysis
 ]

 STOP_CONDITION =
  "Stop when mathematical problem structure, algorithm, conditioning, approximation, error, convergence, complexity and reproducibility are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Applied Mathematics Ontologies"

 IMMEDIATE_CONTEXTS = [
  "Applied Mathematics Ontologies",
  "Numerical Analysis Ontologies",
  "Scientific Computing Ontologies",
  "Linear Algebra Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Applied Mathematics Ontologies",
  "Numerical Analysis Ontologies",
  "Scientific Computing Ontologies",
  "Linear Algebra Ontologies",
  "Optimisation Ontologies",
  "Differential Equation Ontologies",
  "Discrete Mathematics Ontologies",
  "Probability & Statistics Ontologies",
  "Symbolic Computation Ontologies",
  "Algorithmic Complexity Ontologies",
  "High Performance Computing Ontologies",
  "Uncertainty Quantification Ontologies"
 ]

 ROUTE_RULES = [
  "Computational Mathematics -> Applied Mathematics Ontologies for parent semantics",
  "Computational Mathematics -> Numerical Analysis for approximation error and convergence",
  "Computational Mathematics -> Scientific Computing for implementation",
  "Computational Mathematics -> Linear Algebra for matrix computation",
  "Computational Mathematics -> Optimisation for numerical search",
  "Computational Mathematics -> Differential Equations for ODE and PDE computation",
  "Computational Mathematics -> Discrete Mathematics for combinatorial computation",
  "Computational Mathematics -> Probability & Statistics for stochastic methods",
  "Computational Mathematics -> Symbolic Computation for exact manipulation",
  "Computational Mathematics -> HPC for large-scale computation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible algorithms, computational models or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve algorithmic-computation semantics.",
  "Preserve conditioning-versus-stability distinction.",
  "Preserve consistency-and-convergence semantics.",
  "Preserve finite-precision semantics.",
  "Preserve residual-and-error distinction.",
  "Preserve symbolic-versus-numerical distinction.",
  "Preserve complexity and reproducibility semantics.",
  "Preserve machine-readable computational-mathematics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Computational mathematics studies algorithms and computational methods for solving mathematical problems numerically and symbolically."

 NUMERICAL_SUMMARY =
  "Numerical methods approximate mathematical solutions and require analysis of error, stability and convergence."

 LINEAR_ALGEBRA_SUMMARY =
  "Numerical linear algebra provides algorithms for matrix systems, eigenproblems and decompositions."

 DISCRETISATION_SUMMARY =
  "Finite-difference, finite-volume, finite-element and spectral methods convert continuous mathematical models into computable discrete systems."

 ERROR_SUMMARY =
  "Reliable computation distinguishes conditioning, truncation error, roundoff error, residual and convergence."

 COMPLEXITY_SUMMARY =
  "Algorithm choice depends on arithmetic cost, memory, sparsity, communication and scalability."

 SYMBOLIC_SUMMARY =
  "Symbolic computation manipulates exact mathematical expressions and can complement numerical computation."

 VALIDATION_SUMMARY =
  "Computational results should be checked using residuals, refinement, error estimates, reference problems and reproducibility."
}

FLOW {
 INPUT
 -> detect_computational_mathematics_context
 -> identify mathematical_problem_type_structure_size_accuracy_requirements_and_resources
 -> classify symbolic_numerical_continuous_discrete_linear_nonlinear_deterministic_or_stochastic structure
 -> assess conditioning_and_problem sensitivity
 -> select algorithm_and_data representation
 -> if_continuous choose discretisation_method
 -> compute numerical_or_symbolic solution
 -> estimate residual_truncation_roundoff_iteration_and_discretisation error
 -> test convergence_and_stability
 -> assess computational complexity_memory_and_parallel scalability
 -> validate against analytical_reference_manufactured_or_benchmark solution where available
 -> route to applied_mathematics numerical_analysis scientific_computing linear_algebra optimisation differential_equations discrete_mathematics probability_statistics symbolic_computation algorithmic_complexity HPC_or_uncertainty_quantification ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Computational Mathematics layer defines the algorithmic and numerical foundations of mathematical computation. It supports numerical and symbolic algorithms, scientific computing, numerical linear algebra, dense and sparse matrices, eigenvalue and singular-value problems, approximation, interpolation and extrapolation, numerical differentiation and integration, root finding, fixed-point and Newton methods, Krylov methods, preconditioning, conditioning, stability, consistency and convergence, truncation and roundoff error, floating-point arithmetic, numerical ODE and PDE methods, finite-difference, finite-volume, finite-element and spectral methods, optimisation, Monte Carlo computation, uncertainty quantification, computer algebra, automatic differentiation, parallel computing and high-performance computing."
}

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

}

END_FLOW