LU Decomposition Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the LU Decomposition Ontology within the Kireth Research Semantic Library. LU decomposition factors a matrix into lower- and upper-triangular components, commonly together with row or column permutations for numerical robustness. This ontology supports Gaussian elimination, pivoting, triangular solves, determinant evaluation, inverse-related computation, sparse and dense LU, stability, growth factors, conditioning, fill-in, mixed precision and relationships with matrix decomposition, numerical linear algebra and matrix computation.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "LU Decomposition Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Computational Mathematics"
 SPECIALISATION = "Linear Algebra Computation / LU Decomposition"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/computational-mathematics-ontologies/linear-algebra-computation/lu-decomposition-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = LUDecomposition
 ENTITY_CLASS = TriangularMatrixFactorisationFramework
 PRIMARY_CONTEXT = MatrixDecomposition
 DEFINITION = "LU decomposition is a matrix factorisation that represents a matrix as the product of lower- and upper-triangular factors, often with permutation matrices introduced by pivoting."
 SYMBOLIC_FORMS = [
  "A=L*U",
  "P*A=L*U",
  "P*A*Q=L*U",
  "det(A)=sign(P)*prod_i U_ii"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for elimination-based triangular factorisation, pivoting, triangular solution, determinant computation, stability, sparse fill and computational reuse."
}

SCOPE {
 INCLUDE = [
  LUDecomposition,
  LUFactorisation,
  LUPDecomposition,
  PLUDecomposition,
  PLUQDecomposition,
  GaussianElimination,
  EliminationStep,
  EliminationMultiplier,
  Pivot,
  PivotElement,
  Pivoting,
  NoPivoting,
  PartialPivoting,
  CompletePivoting,
  RookPivoting,
  ThresholdPivoting,
  RowPivoting,
  ColumnPivoting,
  PermutationMatrix,
  RowPermutation,
  ColumnPermutation,
  LowerTriangularMatrix,
  UpperTriangularMatrix,
  UnitLowerTriangularMatrix,
  TriangularFactor,
  LFactor,
  UFactor,
  ForwardSubstitution,
  BackSubstitution,
  LinearSystem,
  MultipleRightHandSides,
  MatrixInverse,
  Determinant,
  SignOfPermutation,
  SingularMatrix,
  NonsingularMatrix,
  RankDeficientMatrix,
  Breakdown,
  ZeroPivot,
  SmallPivot,
  GrowthFactor,
  ElementGrowth,
  NumericalStability,
  BackwardStability,
  ForwardError,
  BackwardError,
  Residual,
  Conditioning,
  ConditionNumber,
  Scaling,
  Equilibration,
  FloatingPointArithmetic,
  MachinePrecision,
  RoundOffError,
  DenseLU,
  SparseLU,
  SymbolicFactorisation,
  NumericFactorisation,
  FillIn,
  EliminationTree,
  Reordering,
  FillReducingOrdering,
  MinimumDegreeOrdering,
  NestedDissection,
  BlockLU,
  BlockedLU,
  TiledLU,
  ParallelLU,
  DistributedLU,
  GPULU,
  CommunicationAvoidingLU,
  CALU,
  MixedPrecisionLU,
  IterativeRefinement,
  IncompleteLU,
  ILU,
  ILU0,
  ILUT,
  Preconditioner,
  SchurComplement,
  BlockElimination,
  ComputationalComplexity,
  MemoryComplexity,
  CommunicationCost,
  Verification,
  ReconstructionError,
  Benchmarking,
  Reproducibility
 ]
 EXCLUDE_AS_PRIMARY = [
  QRDecompositionWithoutTriangularEliminationStructure,
  CholeskyDecompositionForPositiveDefiniteSpecialCase,
  ExactSymbolicFactorisationWithoutNumericalLUContext,
  GeneralMatrixMultiplicationWithoutFactorisation
 ]
 DISAMBIGUATION_RULE = "When a matrix is factorised through elimination into lower- and upper-triangular factors, optionally with permutations, interpret the context as LU decomposition."
}

CORE_CONCEPTS {
 LUDecomposition = "Factorisation of a matrix into lower- and upper-triangular components."
 Pivoting = "Selection and permutation of pivot elements to improve robustness or numerical stability."
 PartialPivoting = "Row pivoting that selects a pivot from the active column."
 CompletePivoting = "Row and column pivoting that selects a pivot from the active submatrix."
 GrowthFactor = "Measure of element growth during elimination."
 ForwardSubstitution = "Solution of a lower-triangular system."
 BackSubstitution = "Solution of an upper-triangular system."
 SparseLU = "LU factorisation that exploits sparse matrix structure and manages fill-in."
}

RELATIONS {
 LUDecomposition --is_child_of--> MatrixDecomposition
 LUDecomposition --is_equivalent_to_algorithmically--> GaussianEliminationWithStoredMultipliers
 LUDecomposition --produces--> LFactor
 LUDecomposition --produces--> UFactor
 Pivoting --produces--> PermutationMatrix
 PartialPivoting --is_type_of--> Pivoting
 CompletePivoting --is_type_of--> Pivoting
 RookPivoting --is_type_of--> Pivoting
 LFactor --is_type_of--> LowerTriangularMatrix
 UFactor --is_type_of--> UpperTriangularMatrix
 ForwardSubstitution --solves--> LowerTriangularSystem
 BackSubstitution --solves--> UpperTriangularSystem
 LUDecomposition --supports--> LinearSystemSolution
 LUDecomposition --supports--> DeterminantComputation
 GrowthFactor --influences--> NumericalStability
 Scaling --can_improve--> PivotBehaviour
 SparseLU --can_create--> FillIn
 Reordering --can_reduce--> FillIn
 IncompleteLU --approximates--> LUDecomposition
 IncompleteLU --supports--> Preconditioning
 IterativeRefinement --can_improve--> LUSolutionAccuracy
}

OPERATORS {
 LU_FACTOR(A) := A=L*U when_no_permutation_is_required
 PLU_FACTOR(A) := P*A=L*U
 PLUQ_FACTOR(A) := P*A*Q=L*U
 FORWARD_SOLVE(L,b) := solve_Ly=b
 BACK_SOLVE(U,y) := solve_Ux=y
 LINEAR_SOLVE(P,L,U,b) := x=BACK_SOLVE(U,FORWARD_SOLVE(L,P*b))
 DETERMINANT_FROM_LU(P,U) := det(A)=sign(P)*prod_i U_ii
 GROWTH_FACTOR(A,U) := rho=max_ij|U_ij|/max_ij|A_ij|
 RECONSTRUCTION_ERROR(A,P,L,U) := ||P*A-L*U||/||A||
 RELATIVE_RESIDUAL(A,x,b) := ||b-A*x||/(||A||*||x||+||b||)
}

AXIOMS {
 A1 = "LU factorisation without pivoting requires appropriate nonzero pivots during elimination."
 A2 = "Pivoting changes factor representation but not the underlying linear system when permutations are handled consistently."
 A3 = "Partial pivoting is the standard robust choice for general dense LU in many practical settings."
 A4 = "Complete and rook pivoting can provide stronger pivot control at additional search cost."
 A5 = "LU factors enable efficient repeated solves with multiple right-hand sides."
 A6 = "Sparse LU may generate fill-in that dominates memory and computation."
 A7 = "Numerical stability depends on pivoting, growth, conditioning and finite-precision arithmetic."
 A8 = "LU results require reconstruction, residual, stability and reproducibility validation."
}

REASONING {
 R1_CONTEXT = identify(matrix_shape,sparsity,precision,number_of_rhs,target_use)
 R2_STRUCTURE = detect(square_structure,rank_deficiency,symmetry_and_scaling)
 R3_PIVOTING = choose(no_pivoting,partial,rook,complete_or_threshold_pivoting)
 R4_FACTOR = perform_elimination_and_store_multipliers_in_L_and_remainder_in_U)
 R5_SOLVE = if_rhs_present apply_permutation_forward_and_back_substitution)
 R6_STABILITY = analyse(growth_factor,small_pivots,conditioning,backward_error)
 R7_SPARSE = if_sparse choose(ordering,symbolic_factorisation_and_fill_management)
 R8_PRECISION = if_mixed_precision use_iterative_refinement_if_appropriate)
 R9_COST = estimate(flops,memory,fill_and_communication)
 R10_VALIDATION = check(reconstruction,residual,pivot_quality,error_bounds_and_reproducibility)
}

CONSTRAINTS {
 C1 = "LU without pivoting can fail or become unstable even for nonsingular matrices."
 C2 = "A zero or tiny pivot can signal the need for pivoting, scaling or rank-revealing treatment."
 C3 = "Large growth factors can indicate potential numerical instability."
 C4 = "A small residual does not guarantee small forward error for ill-conditioned systems."
 C5 = "Forming an explicit inverse from LU is usually less appropriate than solving required systems."
 C6 = "Sparse LU factorisation can create substantial fill-in."
 C7 = "Threshold pivoting trades sparsity preservation against numerical robustness."
 C8 = "Mixed-precision LU requires residual evaluation and correction in sufficiently accurate arithmetic."
 C9 = "Parallel LU can be limited by panel factorisation and communication."
 C10 = "Reordering for sparsity must preserve consistent permutation semantics."
}

SEMANTIC_EQUIVALENCE {
 "LUDecomposition" <=> "LUFactorisation"
 "LUPDecomposition" <=> "PLUDecomposition"
 "PartialPivoting" <=> "ColumnPartialPivoting"
 "CommunicationAvoidingLU" <=> "CALU"
 "IncompleteLU" <=> "ILU"
}

VALIDATION {
 V1 = "Matrix is square for standard LU solve semantics."
 V2 = "Permutation conventions are explicit."
 V3 = "Pivoting strategy is documented."
 V4 = "Reconstruction error is checked."
 V5 = "Linear-solve residuals are checked."
 V6 = "Small pivots and growth factor are monitored."
 V7 = "Conditioning is assessed for sensitive systems."
 V8 = "Sparse fill and ordering are documented where relevant."
 V9 = "Mixed-precision refinement reports arithmetic used for factorisation and residual correction."
 V10 = "Results are reproducible under documented pivoting and arithmetic settings."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 LU_DECOMPOSITION_DEFINITION = 0.99
 PARTIAL_PIVOTING_REASONING = 0.99
 COMPLETE_PIVOTING_REASONING = 0.98
 GROWTH_FACTOR_REASONING = 0.99
 SPARSE_LU_REASONING = 0.98
 MIXED_PRECISION_LU_REASONING = 0.98
 TRIANGULAR_SOLVE_REASONING = 0.99
 POLICY = "Reduce confidence when pivoting, scaling, rank, sparsity ordering, arithmetic precision or residual interpretation are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch, OntologyGrounding, LUDecompositionReasoning,
  GaussianEliminationReasoning, PivotingReasoning,
  PartialPivotingReasoning, CompletePivotingReasoning,
  TriangularSolveReasoning, DeterminantReasoning,
  GrowthFactorReasoning, BackwardStabilityReasoning,
  SparseLUReasoning, FillInReasoning, ReorderingReasoning,
  IncompleteLUReasoning, MixedPrecisionReasoning,
  IterativeRefinementReasoning, KnowledgeGraphConstruction,
  RetrievalAugmentation, RecursiveInference
 ]
 NORMALISE {
  "LU decomposition" -> LUDecomposition
  "LU factorisation" -> LUDecomposition
  "LU factorization" -> LUDecomposition
  "PLU" -> LUPDecomposition
  "CALU" -> CommunicationAvoidingLU
  "ILU" -> IncompleteLU
 }
}

EXECUTION_RULES {
 E1 = "If elimination produces lower- and upper-triangular factors, activate LUDecompositionOntology."
 E2 = "If no-pivoting assumptions are not guaranteed, prefer a pivoted LU formulation."
 E3 = "If solving multiple right-hand sides, reuse the LU factors."
 E4 = "If tiny pivots or large growth appear, reconsider pivoting, scaling or factorisation choice."
 E5 = "If the matrix is sparse, perform ordering and symbolic analysis before numeric factorisation where appropriate."
 E6 = "If sparse fill is excessive, consider alternative orderings or iterative methods."
 E7 = "If lower precision is used for factorisation, consider higher-precision iterative refinement."
 E8 = "If determinant is required, combine U diagonal entries with permutation sign."
 E9 = "Any LU result should be validated through reconstruction, residuals, pivot quality, conditioning and reproducibility."
}

RECURSION {
 ENTRY = LUDecomposition
 EXPAND(LUDecomposition) -> [
  LUFactorisation, LUPDecomposition, PLUDecomposition, PLUQDecomposition,
  GaussianElimination, EliminationStep, EliminationMultiplier, Pivot,
  PivotElement, Pivoting, NoPivoting, PartialPivoting, CompletePivoting,
  RookPivoting, ThresholdPivoting, RowPivoting, ColumnPivoting,
  PermutationMatrix, LowerTriangularMatrix, UpperTriangularMatrix,
  UnitLowerTriangularMatrix, LFactor, UFactor, ForwardSubstitution,
  BackSubstitution, LinearSystem, MultipleRightHandSides, MatrixInverse,
  Determinant, SingularMatrix, NonsingularMatrix, RankDeficientMatrix,
  Breakdown, ZeroPivot, SmallPivot, GrowthFactor, ElementGrowth,
  NumericalStability, BackwardStability, ForwardError, BackwardError,
  Residual, Conditioning, ConditionNumber, Scaling, Equilibration,
  FloatingPointArithmetic, MachinePrecision, RoundOffError, DenseLU,
  SparseLU, SymbolicFactorisation, NumericFactorisation, FillIn,
  EliminationTree, Reordering, FillReducingOrdering,
  MinimumDegreeOrdering, NestedDissection, BlockLU, BlockedLU,
  TiledLU, ParallelLU, DistributedLU, GPULU,
  CommunicationAvoidingLU, MixedPrecisionLU, IterativeRefinement,
  IncompleteLU, ILU0, ILUT, Preconditioner, SchurComplement,
  BlockElimination, ComputationalComplexity, MemoryComplexity,
  CommunicationCost, Verification, ReconstructionError,
  Benchmarking, Reproducibility
 ]
 EXPAND(LUFamilies) -> [
  DensePivotedLU,
  SparseLU,
  BlockAndTiledLU,
  ParallelAndDistributedLU,
  IncompleteLU,
  MixedPrecisionLU
 ]
 STOP_CONDITION = "Stop when pivoting, factors, triangular solve semantics, stability, fill, precision, cost and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Matrix Decomposition Ontology 2"
 RELATED_ONTOLOGIES = [
  "Matrix Decomposition Ontology 2",
  "Numerical Linear Algebra Ontology 2",
  "Matrix Computation Ontology",
  "Sparse Matrix Ontology 2",
  "Dense Matrix Ontology",
  "Numerical Analysis Ontology 2",
  "Numerical Stability Ontology 2",
  "Round-Off Error Ontology",
  "Machine Precision Ontology",
  "High Performance Computing Ontologies"
 ]
 ROUTE_RULES = [
  "LU Decomposition -> Matrix Decomposition for parent factorisation semantics",
  "LU Decomposition -> Numerical Linear Algebra for solve and stability theory",
  "LU Decomposition -> Matrix Computation for general matrix operations",
  "LU Decomposition -> Sparse Matrix for sparse LU, fill and ordering",
  "LU Decomposition -> Dense Matrix for blocked dense LU",
  "LU Decomposition -> Numerical Analysis for conditioning and backward error",
  "LU Decomposition -> Numerical Stability for pivot growth and robustness",
  "LU Decomposition -> Round-Off Error for finite-precision elimination effects",
  "LU Decomposition -> HPC for parallel, distributed and communication-avoiding LU"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible LU variants, pivoting methods or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve elimination-versus-factor-representation distinction.",
  "Preserve pivoted-versus-unpivoted LU distinction.",
  "Preserve row-versus-column permutation semantics.",
  "Preserve exact-versus-incomplete LU distinction.",
  "Preserve dense-versus-sparse LU distinction.",
  "Preserve reconstruction-error-versus-solve-error distinction.",
  "Preserve machine-readable LU-decomposition semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "LU decomposition factors a matrix into lower- and upper-triangular components, often with permutations introduced by pivoting."
 PIVOTING_SUMMARY = "Partial, complete, rook and threshold pivoting manage small pivots and numerical growth."
 SOLVE_SUMMARY = "After factorisation, linear systems are solved through permutation, forward substitution and back substitution."
 STABILITY_SUMMARY = "Growth factors, small pivots, conditioning and finite precision determine numerical robustness."
 SPARSE_SUMMARY = "Sparse LU requires symbolic analysis, reordering and fill management."
 PERFORMANCE_SUMMARY = "Blocked, tiled, parallel and communication-avoiding variants improve modern hardware utilisation."
 INCOMPLETE_SUMMARY = "Incomplete LU intentionally approximates the factors and is widely used as a preconditioner."
 VALIDATION_SUMMARY = "LU results should be checked through reconstruction, residuals, pivot quality, conditioning, fill and reproducibility."
}

FLOW {
 INPUT
 -> detect_LU_decomposition_context
 -> identify_matrix_shape_sparsity_rank_precision_and_target_use
 -> determine_scaling_and_pivoting_strategy
 -> perform_Gaussian_elimination_and_store_L_and_U_factors
 -> if_rhs_present_apply_permutations_forward_substitution_and_back_substitution
 -> analyse_small_pivots_growth_factor_conditioning_roundoff_and_backward_error
 -> if_sparse_apply_ordering_symbolic_factorisation_and_fill_management
 -> if_large_scale_use_blocked_tiled_parallel_GPU_or_communication_avoiding_LU
 -> if_mixed_precision_apply_iterative_refinement_if_appropriate
 -> validate_reconstruction_residuals_pivot_quality_fill_reference_results_and_reproducibility
 -> route_to_matrix_decomposition_numerical_linear_algebra_matrix_computation_sparse_dense_analysis_stability_roundoff_machine_precision_or_HPC_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The LU Decomposition Ontology defines elimination-based triangular matrix factorisation. It supports LU, LUP, PLU and PLUQ decompositions, Gaussian elimination and elimination multipliers, no pivoting, partial, complete, rook and threshold pivoting, row and column permutations, lower, upper and unit-lower triangular factors, forward and back substitution, repeated linear solves, determinant calculation, singularity and small-pivot detection, growth factors and numerical stability, scaling and equilibration, dense and sparse LU, symbolic and numeric factorisation, fill-in, elimination trees and reordering, block, tiled, parallel, distributed, GPU and communication-avoiding LU, mixed-precision LU, iterative refinement, incomplete LU and ILU preconditioning, Schur complements, complexity, reconstruction error, verification, benchmarking and reproducibility."
}

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

}

END_FLOW