API Reference
This section contains the complete API reference for BallArithmetic.jl, organized by functionality.
Pages
- Core Types - Ball, BallArray, BallMatrix, BallVector and the
Baseinterface - Arithmetic & Rounding - the directed operations, the error-free transformations, the transforms
- Norm Bounds & Matrix Classes - the rigorous norm and singular value bounds, and the classifiers
- Matrix Decompositions - LU, QR, Cholesky, polar, Takagi and the singular value routines
- Linear Systems - Verified linear system solvers
- Eigenvalues & SVD - Eigenvalue and singular value computation
- Algorithm Steps - the steps of the published algorithms, each named for its statement
- Pseudospectra - the resolvent bounds and the cells proved inside or outside
- CertifScripts - Pseudospectra certification
- NumericalTest - Testing utilities
BallArithmetic.BallArithmeticBallArithmetic.CouplingARSolveBallArithmetic.CouplingNoneBallArithmetic.CouplingOffDirectBallArithmetic.FrobeniusNormBallArithmetic.IdentityPreconditionerBallArithmetic.LDLTFactorizationBallArithmetic.LUFactorizationBallArithmetic.MidpointInverseBallArithmetic.NeumannCollatz2BallArithmetic.NeumannOneInfBallArithmetic.NumericalTest.ARRAY_ROUNDING_OKBallArithmetic.NumericalTest.BLAS_ROUNDING_OKBallArithmetic.OneInfNormBallArithmetic.TriBacksubBallArithmetic.ηBallArithmetic.ϵpBallArithmetic.AdaptiveSVDResultBallArithmetic.AlmostInvariantBasisBallArithmetic.BallBallArithmetic.BallArrayBallArithmetic.BallComplexF64BallArithmetic.BallF64BallArithmetic.BallFFTPlanBallArithmetic.BallMatrixBallArithmetic.BallVectorBallArithmetic.BlockResolventFloorBallArithmetic.CertifScripts.CertificationCircleBallArithmetic.CertifScripts.GramTransformBallArithmetic.CouplingEstimatorBallArithmetic.DInverseEstimatorBallArithmetic.DenseBallMatrixBallArithmetic.DenseBallVectorBallArithmetic.DeterminantResultBallArithmetic.EnclosureBallArithmetic.GEVResultBallArithmetic.GaussianEliminationResultBallArithmetic.HBRResultBallArithmetic.HermitianBallMatrixBallArithmetic.IterativeResultBallArithmetic.KrawczykResultBallArithmetic.Miyajima2014aSchurNewtonResultBallArithmetic.MiyajimaAutoBallArithmetic.MiyajimaGEVResultBallArithmetic.MiyajimaM1BallArithmetic.MiyajimaM3BallArithmetic.MiyajimaM4BallArithmetic.NKEigenpairResultBallArithmetic.NKEigenpairsResultBallArithmetic.NormEstimatorBallArithmetic.OgitaSVDRefinementResultBallArithmetic.Oishi2023ResultBallArithmetic.OverdeterminedResultBallArithmetic.ParametricResolventResultBallArithmetic.PolarIterationResultBallArithmetic.PolyRangeBallArithmetic.PreconditionerResultBallArithmetic.PreconditionerTypeBallArithmetic.PseudospectrumCellBallArithmetic.PseudospectrumCellsBallArithmetic.RegularityResultBallArithmetic.ResolventBoundConfigBallArithmetic.RigorousBlockSchurResultBallArithmetic.RigorousEigenvaluesResultBallArithmetic.RigorousGeneralizedEigenvaluesResultBallArithmetic.RigorousSVDResultBallArithmetic.RumpOishi2024ResultBallArithmetic.SVDFrameFloorBallArithmetic.SVDMethodBallArithmetic.SVDWarmStartBallArithmetic.SchurGershgorinResultBallArithmetic.SchurRefinementResultBallArithmetic.SchurSpectralProjectorResultBallArithmetic.ShavingResultBallArithmetic.SpectralCoefficientResultBallArithmetic.SylvesterResolventResultBallArithmetic.SymmetricEigenRefinementResultBallArithmetic.VerifiedCholeskyResultBallArithmetic.VerifiedLUResultBallArithmetic.VerifiedLinearSystemResultBallArithmetic.VerifiedPolarResultBallArithmetic.VerifiedQRResultBallArithmetic.VerifiedTakagiResultBallArithmetic.VerifyEigAllResultBallArithmetic.VerifyLssResultBallArithmetic.VerifySvdAllResultBallArithmetic._ComplexDiscsBallArithmetic._RealIntervalsBallArithmetic.:±BallArithmetic.CertifScripts._bf_ogita_cache_statsBallArithmetic.CertifScripts._clear_bf_ogita_cache!BallArithmetic.CertifScripts._clear_ogita_cache!BallArithmetic.CertifScripts._clear_parametric_cache!BallArithmetic.CertifScripts._compute_schur_bigfloat_directBallArithmetic.CertifScripts._diagonal_gram_factorBallArithmetic.CertifScripts._diagonal_inverseBallArithmetic.CertifScripts._evaluate_sample_ogita_bigfloatBallArithmetic.CertifScripts._evaluate_sample_parametricBallArithmetic.CertifScripts._evaluate_sample_with_ogita_cacheBallArithmetic.CertifScripts._gram_convertBallArithmetic.CertifScripts._gram_promoteBallArithmetic.CertifScripts._gram_residual_boundBallArithmetic.CertifScripts._lower_boundBallArithmetic.CertifScripts._ogita_cache_statsBallArithmetic.CertifScripts._parametric_cache_statsBallArithmetic.CertifScripts._prepare_gramBallArithmetic.CertifScripts._set_center_svd_cache!BallArithmetic.CertifScripts._triangular_inverseBallArithmetic.CertifScripts._upper_boundBallArithmetic.CertifScripts._verify_inverseBallArithmetic.CertifScripts.adaptive_arcs!BallArithmetic.CertifScripts.apply_gram_transformBallArithmetic.CertifScripts.bound_res_originalBallArithmetic.CertifScripts.bound_resolvent_schurBallArithmetic.CertifScripts.choose_snapshot_to_loadBallArithmetic.CertifScripts.circle_resolvent_boundBallArithmetic.CertifScripts.compute_schur_and_errorBallArithmetic.CertifScripts.configure_certification!BallArithmetic.CertifScripts.doworkBallArithmetic.CertifScripts.dowork_ogitaBallArithmetic.CertifScripts.dowork_ogita_bigfloatBallArithmetic.CertifScripts.dowork_parametricBallArithmetic.CertifScripts.gram_transformBallArithmetic.CertifScripts.points_onBallArithmetic.CertifScripts.poly_from_rootsBallArithmetic.CertifScripts.polyconvBallArithmetic.CertifScripts.polygon_sagittaBallArithmetic.CertifScripts.run_certificationBallArithmetic.CertifScripts.run_certification_ogitaBallArithmetic.CertifScripts.run_certification_parametricBallArithmetic.CertifScripts.save_snapshot!BallArithmetic.CertifScripts.schur_to_original_resolventBallArithmetic.CertifScripts.schur_to_original_resolvent_defectsBallArithmetic.CertifScripts.set_parametric_config!BallArithmetic.CertifScripts.set_schur_matrix!BallArithmetic.NumericalTest.array_rounding_testBallArithmetic.NumericalTest.rounding_testBallArithmetic._accurate_product_sumBallArithmetic._apply_block_structureBallArithmetic._ball_from_boundsBallArithmetic._build_augmented_jacobian_ballBallArithmetic._bujanovicetal2023_alg4BallArithmetic._bujanovicetal2023_alg4_truncatedBallArithmetic._bujanovicetal2023_alg4_unorderedBallArithmetic._ccrBallArithmetic._ccrprodBallArithmetic._ccrprod_primeBallArithmetic._certify_ballBallArithmetic._check_axesBallArithmetic._ciprodBallArithmetic._ciprod_primeBallArithmetic._compute_left_eigenvectorBallArithmetic._compute_off_diagonal_normBallArithmetic._cprodBallArithmetic._crBallArithmetic._diagonal_scale_leftBallArithmetic._dot2err_ogita_rump_oishi_2005BallArithmetic._fast_gamma_boundBallArithmetic._fftw_transformBallArithmetic._forward_substitutionBallArithmetic._frobenius_normBallArithmetic._has_distinct_diagonalBallArithmetic._inverse_diagonal_normBallArithmetic._iprodBallArithmetic._iprod_rightBallArithmetic._lange_rump_two_termsBallArithmetic._miyajima2014_thm10BallArithmetic._miyajima2014_thm10_boundsBallArithmetic._miyajima2014_thm10_weylBallArithmetic._miyajima2014_thm11BallArithmetic._miyajima2014_thm4BallArithmetic._miyajima2014_thm7BallArithmetic._miyajima2014a_alg1BallArithmetic._ogita_svd_refine_implBallArithmetic._oishi_MMul_up_loBallArithmetic._opnorm_boundBallArithmetic._product_down_up_dot2errBallArithmetic._refine_schur_implBallArithmetic._refine_symmetric_eigen_implBallArithmetic._rigorous_svd_bigfloatBallArithmetic._rump2010_alg10_7BallArithmetic._rump2011_thm3_1BallArithmetic._rump2022aBallArithmetic._rump2022a_conditionedBallArithmetic._rump2022a_discclustersBallArithmetic._rump2022a_pencilBallArithmetic._rump2022a_pencil_transformBallArithmetic._rump2022a_prodKBallArithmetic._rump2022a_prodK_pencilBallArithmetic._rump2022a_schurBallArithmetic._rump2022a_schur_step6BallArithmetic._rump2022a_transformBallArithmetic._rump2022aneumannBallArithmetic._rump2022aneumann_transformBallArithmetic._rumplange2023BallArithmetic._rumplange2023_eigBallArithmetic._rumpogita2024_alg3_2BallArithmetic._rumpogita2024_choleskyBallArithmetic._rumpogita2024_luBallArithmetic._rumpogita2024_polarBallArithmetic._rumpogita2024_qrBallArithmetic._rumpogita2024_schurBallArithmetic._rumpogita2024_takagiBallArithmetic._shaving_lpBallArithmetic._solve_ball_systemBallArithmetic._solve_ball_system_transposeBallArithmetic._solve_sylvester_triangular!BallArithmetic._solve_triangular_equation_direct!BallArithmetic._solve_triangular_equation_recursive!BallArithmetic._spectral_norm_boundBallArithmetic._strilBallArithmetic._svd_auto_theoremBallArithmetic._sylvester_triangular_columnsBallArithmetic._sylvester_triangular_direct_ballBallArithmetic._two_term_product_sumBallArithmetic._unverified_resultBallArithmetic._unverified_result_with_diagnosticsBallArithmetic._up_bound_Linf_normBallArithmetic._up_bound_Linf_opnormBallArithmetic._vbd_solveBallArithmetic._vbd_unitary_basisBallArithmetic._zero_radiusBallArithmetic.abs_downBallArithmetic.abs_upBallArithmetic.adaptive_ogita_svdBallArithmetic.apply_preconditionerBallArithmetic.backward_substitutionBallArithmetic.ball_fftBallArithmetic.ball_hornerBallArithmetic.ball_hullBallArithmetic.bdd_R2BallArithmetic.block_enclosureBallArithmetic.block_resolvent_floorBallArithmetic.bound_resolventBallArithmetic.certify_eigenpairBallArithmetic.certify_eigenpairsBallArithmetic.certify_schur_decompositionBallArithmetic.certify_svdBallArithmetic.check_enclosureBallArithmetic.check_enclosureBallArithmetic.cholesky_qr2BallArithmetic.clear_svd_cache!BallArithmetic.collatz_upper_boundBallArithmetic.collatz_upper_bound_L2_opnormBallArithmetic.compare_all_configsBallArithmetic.comparison_matrixBallArithmetic.compensated_termsBallArithmetic.compensated_terms2BallArithmetic.compute_beta_boundBallArithmetic.compute_enclosureBallArithmetic.compute_preconditionerBallArithmetic.compute_spectral_coefficientBallArithmetic.compute_spectral_projector_hermitianBallArithmetic.compute_spectral_projector_schurBallArithmetic.config_v1BallArithmetic.config_v2BallArithmetic.config_v2p5BallArithmetic.config_v3BallArithmetic.contains_zeroBallArithmetic.det_cofactorBallArithmetic.det_cramerBallArithmetic.det_gershgorinBallArithmetic.det_hadamardBallArithmetic.diagonal_abs_lower_boundBallArithmetic.dist_downBallArithmetic.dist_upBallArithmetic.eigencount_in_discBallArithmetic.eigencount_outsideBallArithmetic.estimate_2normBallArithmetic.estimate_block_separationBallArithmetic.evboxBallArithmetic.extract_cluster_blockBallArithmetic.find_optimal_splitBallArithmetic.forward_substitutionBallArithmetic.gamma_boundBallArithmetic.gevboxBallArithmetic.hbr_methodBallArithmetic.hbr_method_simpleBallArithmetic.in0BallArithmetic.inclusion_discsBallArithmetic.infBallArithmetic.intersect_ballBallArithmetic.interval_detBallArithmetic.interval_gauss_seidelBallArithmetic.interval_gaussian_eliminationBallArithmetic.interval_gaussian_elimination_detBallArithmetic.interval_initial_enclosureBallArithmetic.interval_jacobiBallArithmetic.interval_least_squaresBallArithmetic.interval_shavingBallArithmetic.is_H_matrixBallArithmetic.is_M_matrixBallArithmetic.is_regularBallArithmetic.is_regular_diagonal_dominanceBallArithmetic.is_regular_gaussian_eliminationBallArithmetic.is_regular_gershgorinBallArithmetic.is_regular_spectral_radiusBallArithmetic.is_regular_sufficient_conditionBallArithmetic.is_singular_sufficient_conditionBallArithmetic.is_well_preconditionedBallArithmetic.iscertifiedBallArithmetic.isvalid_enclosureBallArithmetic.krawczyk_linear_systemBallArithmetic.machine_epsilonBallArithmetic.magBallArithmetic.midBallArithmetic.midtypeBallArithmetic.migBallArithmetic.miyajima2010_algorithm1_steps1to3BallArithmetic.miyajima2014a_schurnewtonBallArithmetic.miyajima_gev_enclosureBallArithmetic.mmul_ogita_rump_oishi_2005BallArithmetic.multi_jacobi_methodBallArithmetic.newton_schulz_orthogonalize!BallArithmetic.off_diagonal_absBallArithmetic.ogita_iterations_for_precisionBallArithmetic.ogita_svd_refineBallArithmetic.oishi_2023_optimal_block_sizeBallArithmetic.oishi_2023_sigma_min_boundBallArithmetic.oishi_MMulBallArithmetic.ordschur_ballBallArithmetic.ordschur_bigfloatBallArithmetic.orthonormal_invariant_basisBallArithmetic.overlap_componentsBallArithmetic.parametric_resolvent_boundBallArithmetic.pi_normBallArithmetic.plan_ball_fftBallArithmetic.polar_newtonBallArithmetic.polar_newton_schulzBallArithmetic.polar_qdwhBallArithmetic.pow_downBallArithmetic.pow_upBallArithmetic.project_vector_spectralBallArithmetic.pseudospectrum_cellsBallArithmetic.psi_schur_factorBallArithmetic.psi_squaredBallArithmetic.radBallArithmetic.radtypeBallArithmetic.range_bisectBallArithmetic.range_criticalBallArithmetic.refine_schur_decompositionBallArithmetic.refine_symmetric_eigenBallArithmetic.resolvent_boundBallArithmetic.rigorous_block_schurBallArithmetic.rigorous_eigenvaluesBallArithmetic.rigorous_generalized_eigenvaluesBallArithmetic.rigorous_schur_bigfloatBallArithmetic.rigorous_svdBallArithmetic.rigorous_svd_gpuBallArithmetic.rigorous_svd_m4BallArithmetic.root_upBallArithmetic.rump_oishi_2024_optimal_block_sizeBallArithmetic.rump_oishi_2024_sigma_min_boundBallArithmetic.rump_oishi_2024_triangular_boundBallArithmetic.schur_gershgorin_enclosureBallArithmetic.schur_sylvester_midpointBallArithmetic.select_subsquaresBallArithmetic.sep_clustersBallArithmetic.set_svd_cache!BallArithmetic.sherman_morrison_inverse_updateBallArithmetic.sigma_min_floorBallArithmetic.sigma_min_upperBallArithmetic.similarity_condition_numberBallArithmetic.solve_sylvester_oracleBallArithmetic.solve_triangular_matrix_equationBallArithmetic.spectral_projector_error_boundBallArithmetic.split_veltkampBallArithmetic.subnormal_minBallArithmetic.subsquares_methodBallArithmetic.sum_upBallArithmetic.supBallArithmetic.svd_bigfloatBallArithmetic.svd_bound_L2_opnormBallArithmetic.svd_bound_L2_opnorm_inverseBallArithmetic.svd_bound_L2_resolventBallArithmetic.svd_boundsBallArithmetic.svd_cache_statsBallArithmetic.svd_frame_floorBallArithmetic.svd_lower_bound_sigma_minBallArithmetic.svdboxBallArithmetic.sylvester_miyajima_enclosureBallArithmetic.sylvester_resolvent_precomputeBallArithmetic.triangular_eigenvectorsBallArithmetic.triangular_inverse_inf_norm_boundBallArithmetic.triangular_inverse_one_norm_boundBallArithmetic.triangular_inverse_two_norm_boundBallArithmetic.triangular_sylvester_miyajima_enclosureBallArithmetic.two_productBallArithmetic.two_sumBallArithmetic.unit_roundoffBallArithmetic.upper_absBallArithmetic.upper_bound_L1_opnormBallArithmetic.upper_bound_L2_opnormBallArithmetic.upper_bound_L_inf_opnormBallArithmetic.upper_bound_frobeniusBallArithmetic.upper_bound_normBallArithmetic.verified_choleskyBallArithmetic.verified_linear_solve_hmatrixBallArithmetic.verified_luBallArithmetic.verified_polarBallArithmetic.verified_qrBallArithmetic.verified_svdBallArithmetic.verified_sylvester_enclosureBallArithmetic.verified_takagiBallArithmetic.verify_block_schur_propertiesBallArithmetic.verify_generalized_eigenpairsBallArithmetic.verify_spectral_projector_propertiesBallArithmetic.verifyeigallBallArithmetic.verifylssBallArithmetic.verifysvdallBase.:*Base.:*Base.:*Base.:*Base.:*Base.:*Base.:*Base.:*Base.:*Base.:*Base.:*Base.:*Base.:*Base.:*Base.:*Base.:*Base.:*Base.:*Base.:*Base.:+Base.:+Base.:+Base.:+Base.:+Base.:+Base.:+Base.:+Base.:+Base.:-Base.:-Base.:-Base.:-Base.:-Base.:-Base.:-Base.:-Base.:-Base.:-Base.:/Base.:<Base.:<=Base.:>Base.:>=Base.absBase.adjointBase.conjBase.convertBase.convertBase.copyBase.fillBase.getindexBase.imagBase.inBase.inBase.inBase.invBase.invBase.islessBase.islessBase.lengthBase.onesBase.realBase.setindex!Base.sizeBase.sqrtBase.transposeBase.zerosRoundingEmulator.add_downRoundingEmulator.add_upRoundingEmulator.add_upRoundingEmulator.div_downRoundingEmulator.div_upRoundingEmulator.mul_downRoundingEmulator.mul_upRoundingEmulator.mul_upRoundingEmulator.sqrt_downRoundingEmulator.sqrt_upRoundingEmulator.sub_downRoundingEmulator.sub_upRoundingEmulator.sub_upBallArithmetic.@downBallArithmetic.@up