mathlib-fp 1.6.0

Release date: 2026-07-27

Version 1.6.0 completes the first typed dense decomposition workflow on the 1.5 contiguous matrix foundation. It is a native Object Pascal release with no third-party runtime dependency.

Highlights

• Reusable Householder QR and column-pivoted QR factors support full-rank and rank-revealing least-squares solves for tall and square matrices.
• Reusable compact one-sided Jacobi SVD factors support tall, square, and wide matrices plus rank-deficient and underdetermined minimum-norm solves.
• Full real symmetric and complex Hermitian eigensystems return ascending eigenvalues, normalized column eigenvectors, and inspectable convergence sweep counts.
• Reusable triangular solves cover lower/upper, unit/non-unit, ordinary, transposed, and conjugate-transposed systems.
• LU and Cholesky keep their 1.5 behavior and gain additive condition indicators and residual/backward-error diagnostic solves.
• Every applicable operation has matching single/double real/complex entry points, supports vector or multiple right-hand sides, and never forms an inverse to solve or diagnose a system.

Start with the dense solver-selection guide and run 16_dense_solver_selection.pas.

Public API

AlgebraLib.DenseDecompositions adds:

• SolveTriangular with TDenseTriangle, TDenseDiagonal, and TDenseTranspose;
• FactorQR, FactorPivotedQR, LeastSquares, and RankRevealingLeastSquares;
• FactorSVD and MinimumNormSolve;
• FactorSymmetricEigen and FactorHermitianEigen;
• reusable IDenseSingleQR, IDenseDoubleQR, IDenseSingleComplexQR, IDenseComplexQR, matching four SVD handles, and the real-symmetric/complex-Hermitian eigen handles; and
• TDenseSolveDiagnostics, reporting numerical rank, rank deficiency, selected tolerance, condition indicator, residual norm, and normalized backward error.

AlgebraLib.DenseSolvers adds ConditionIndicator and SolveWithInfo to existing LU/Cholesky handles, plus SolveWithInfo(A,B,Info) and SolvePositiveDefinite(A,B,Info) convenience paths.

Factors own immutable snapshots. Coefficients and right-hand sides are never overwritten; factor outputs and permutation/eigen/singular arrays are copies. Compact conventions, phase/sign freedoms, tolerances, ordering, allocation, thread safety, and error behavior are specified in the 1.6 design record.

Solver guidance

• General square: pivoted LU.
• Positive-definite symmetric/Hermitian: Cholesky.
• Tall and full rank: Householder QR.
• Tall with uncertain rank: column-pivoted QR.
• Rank deficient or underdetermined when minimum norm matters: SVD.
• Full real symmetric or complex Hermitian spectrum: the matching Jacobi eigensystem.

SVD is deliberately explicit rather than an automatic expensive fallback. CPQR's rank-deficient solve is a basic solution; use SVD for a minimum-norm guarantee.

Numerical and maintenance evidence

The focused tests cover reconstruction, orthogonality/unitarity, permutations, multiple right-hand sides, residuals/backward errors, numerical rank, singular-value ordering, real and complex eigenpair residuals, factor reuse, immutable source snapshots, empty and singleton-compatible shapes, repeated spectra, tiny scale, near rank deficiency, non-finite input, invalid shape/structure, and single/double scalar parity.

Independent exact fixtures include a published two-parameter least-squares fit, a known underdetermined Moore-Penrose solution, analytic 2-by-2 real and complex spectra, diagonal singular values, and triangular systems. Acceptance budgets are precision and algorithm specific; the qualification report records the configurations and normalized measures.

The deterministic benchmark reports QR factor reuse against allocating convenience calls, compact SVD/minimum-norm work, symmetric eigen work, checksums, factor-build/result-allocation counts, and estimated peak scalar working storage. These are reproducibility measurements, not cross-library speed claims.

Compatibility

No IMatrix, TMatrixKit, IVector, or typed 1.5 symbol is removed or deprecated. Legacy LU, QR, SVD, Cholesky, pseudoinverse, and eigen methods remain the compatibility API. They are not silently rerouted where shape, ordering, tolerance, error, or ownership contracts differ. Migration remains an explicit copy into typed storage and is documented in the migration guide.

Known limitations and deferred families

The following are explicitly unsupported in the typed 1.6 API:

• sparse, packed, and other structured storage/factors;
• LDLT and specialized structure-only factor families;
• CG, MINRES, GMRES, BiCGSTAB, LSQR, preconditioners, callbacks, and matrix-free solves;
• nonsymmetric, generalized, polynomial, partial, and Schur eigensystems;
• mutable update/downdate factors, destructive factorisation, and a general public workspace API;
• automatic dispatch, parallel/SIMD/GPU decomposition kernels, and external BLAS/LAPACK bindings; and
• wholesale migration of fitting, statistics, machine-learning, or other higher-level domains.

The pure Pascal portable paths are the complete stable implementation, not a fallback for a foreign binary.