Efficient numerical diagonalization of hermitian 3x3 matrices

Abstract

A very common problem in science is the numerical diagonalization of symmetric or hermitian 3x3 matrices. Since standard "black box" packages may be too inefficient if the number of matrices is large, we study several alternatives. We consider optimized implementations of the Jacobi, QL, and Cuppen algorithms and compare them with an analytical method relying on Cardano's formula for the eigenvalues and on vector cross products for the eigenvectors. Jacobi is the most accurate, but also the slowest method, while QL and Cuppen are good general purpose algorithms. The analytical algorithm outperforms the others by more than a factor of 2, but becomes inaccurate or may even fail completely if the matrix entries differ greatly in magnitude. This can mostly be circumvented by using a hybrid method, which falls back to QL if conditions are such that the analytical calculation might become too inaccurate. For all algorithms, we give an overview of the underlying mathematical ideas, and present detailed benchmark results. C and Fortran implementations of our code are available for download from http://www.mpi-hd.mpg.de/~globes/3x3/ .

13 pages, no figures, new hybrid algorithm added, matches published version, typo in Eq. (39) corrected; software library available at http://www.mpi-hd.mpg.de/~globes/3x3/

References in corpus (1)

• Simulation of long-baseline neutrino oscillation experiments with GLoBES

Cited by in corpus (91)

• First measurement of electron neutrino appearance in NOvA
• Combined analysis of ${\nu }_{\mu }$ disappearance and ${\nu }_{\mu }\to {\nu }_e$ appearance in MINOS using accelerator and atmospheric neutrinos
• Modeling the formation and evolution of star cluster populations in galaxy simulations
• Degeneracies in long-baseline neutrino experiments from nonstandard interactions
• Fuzzy Dark Matter and Non-Standard Neutrino Interactions
• The shape of the proton at high energies
• Fully compressive tides in galaxy mergers
• Electroweak and Flavour Structure of a Warped Extra Dimension with Custodial Protection
• Phase-Field Modeling of Crack Branching and Deflection in Heterogeneous Media
• Non-standard interactions and the resolution of ordering of neutrino masses at DUNE and other long baseline experiments
• Non-standard interactions spoiling the CP violation sensitivity at DUNE and other long baseline experiments
• Decaying Sterile Neutrinos and the Short Baseline Oscillation Anomalies
• Calibration artefacts in radio interferometry. I. Ghost sources in WSRT data
• Superconducting order of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ from a three-dimensional microscopic model
• Neutrinophilic Axion-Like Dark Matter
• Can nonstandard interactions jeopardize the hierarchy sensitivity of DUNE ?
• Quantum and Thermal Phase Transitions of the Triangular SU(3) Heisenberg Model under Magnetic Fields
• Imprints of a light Sterile Neutrino at DUNE, T2HK and T2HKK
• Constraining the non-standard interaction parameters in long baseline neutrino experiments
• CP-Violation and Non-Standard Interactions at the MOMENT
• Challenges posed by non-standard neutrino interactions in the determination of ${\delta }_{CP}$ at DUNE
• Neutrino oscillation probabilities through the looking glass
• Correlations and degeneracies among the NSI parameters with tunable beams at DUNE
• A DPF analysis yields accurate analytic potentials for Li${}_2(a{}^3{\Sigma }_u^{+})$ and Li${}_2(1{}^3{\Sigma }_g^{+})$ that incorporate 3-state mixing near the $1{}^3{\Sigma }_g^{+}$-state asymptote
• Sterile neutrinos beyond LSND at the Neutrino Factory
• Exploring the effects of Scalar Non Standard Interactions on the CP violation sensitivity at DUNE
• Measuring the Sterile Neutrino CP Phase at DUNE and T2HK
• Exact and Approximate Formulas for Neutrino Mixing and Oscillations with Non-Standard Interactions
• Sensitivities to charged-current nonstandard neutrino interactions at DUNE
• Constraints on Sterile Neutrino Oscillations using DUNE Near Detector
• Quasi-Dirac neutrino oscillations at DUNE and JUNO
• Two experiments for the price of one? -- The role of the second oscillation maximum in long baseline neutrino experiments
• A statistical inference approach to time-delay interferometry for gravitational-wave detection
• Constraining Super-light Sterile Neutrino Scenario by JUNO and RENO-50
• Imprints of scalar NSI on the CP-violation sensitivity using synergy among DUNE, T2HK and T2HKK
• Implications of lepton flavour violation on long baseline neutrino oscillation experiments
• Can Lorentz Invariance Violation affect the Sensitivity of Deep Underground Neutrino Experiment?
• Non-Abelian chiral instabilities at high temperature on the lattice
• Neutrino Oscillations with MINOS and MINOS+
• New physics vs new paradigms: distinguishing CPT violation from NSI
• Non-Minimal Chaotic Inflation, Peccei-Quinn Phase Transition and non-Thermal Leptogenesis
• Topological States with Broken Translational and Time-Reversal Symmetries in a Honeycomb-Triangular Lattice
• Distinguishing Non-Standard Interaction and Lorentz Invariance Violation at Protvino to Super-ORCA experiment
• Study of light sterile neutrino at the long-baseline experiment options at KM3NeT
• Why Is The Neutrino Oscillation Formula Expanded In $\Delta m_{21}^2/\Delta m_{31}^2$ Still Accurate Near The Solar Resonance In Matter?
• Non-standard neutrino oscillations: perspective from unitarity triangles
• Probing muonic charged current nonstandard interactions at decay-at-rest facilities in conjunction with T2HK
• On the systematic uncertainties in DUNE and their role in New Physics studies
• Poissonian cellular Potts models reveal nonequilibrium kinetics of cell sorting
• A fast Chebyshev method for the Bingham closure with application to active nematic suspensions
• Optimal configuration of Protvino to ORCA experiment for hierarchy and non-standard interactions
• Nonstandard neutrino interactions and mu-tau reflection symmetry
• Investigating Lorentz Violation with the long baseline experiment P2O
• Probing Source and Detector NSI parameters at the DUNE Near Detector
• Inflation, Leptogenesis, and Yukawa Quasi-Unification within a Supersymmetric Left-Right Model
• Effect of final state interactions on neutrino energy reconstruction at DUNE
• Effective Theory of Higgs Sector Vacuum States
• Impact of scalar NSI on the neutrino mass ordering sensitivity at DUNE, HK and KNO
• Nuclear Effects and CP Sensitivity at DUNE
• Non-standard interactions versus planet-scale neutrino oscillations
• Fibonacci Fast Convergence for Neutrino Oscillations in Matter
• Charged Higgs effects in IceCube: PeV events and NSIs
• Exploring New Physics from nu_tau events in OPERA
• Lectures on neutrino phenomenology
• Precision measurements and tau neutrino physics in a future accelerator neutrino experiment
• A plethora of long-range neutrino interactions probed by DUNE and T2HK
• Analysis of structural correlations in a model binary 3D liquid through the eigenvalues and eigenvectors of the atomic stress tensors
• Impact of lepton flavour universality violation on CP violation sensitivity of long baseline neutrino oscillation experiments
• Exploring the new physics phases in 3+1 scenario in neutrino oscillation experiments
• Entropy methods for CMB analysis of anisotropy and non-Gaussianity
• Magnetized matter effects on dilaton photon mixing
• Fast Exact Algorithm for Neutrino Oscillation in Constant Matter Density
• Vector leptoquark $U_3$ and CP violation at T2K, NOvA experiments
• Multiple breaking patterns in the Brout-Englert-Higgs effect beyond perturbation theory
• From ray to spray: augmenting amplitudes and taming fast oscillations in fully numerical neutrino codes
• Neutrino Lorentz invariance violation from the CPT-even SME coefficients through a tensor interaction with cosmological scalar fields
• Thermal Ising transition in two-dimensional SU(3) Fermi lattice gases with population imbalance
• Sterile Neutrino Sensitivity with Wrong-Sign Muon Appearance at nuSTORM
• Constraining the isotropic CPT-odd coefficients of the Standard Model Extension by a combined DUNE and ESSnuSB analysis
• Sterile neutrino searches at tagged kaon beams
• The effect of non-standard interactions and environmental decoherence at DUNE
• Working towards an optimal sampling of the 21 cm signal parameter space
• Vorticity Statistics and the Time Scales of Turbulent Strain
• Impact of CP violation searches at MOMENT experiment with sterile neutrinos
• MRRR-based Eigensolvers for Multi-core Processors and Supercomputers
• Copycat process in the early stages of einselection
• The Sensitivity of DUNE in Presence of Off-Diagonal Scalar NSI Parameters
• Probing new physics scenarios using high energy events at NOvA far detector
• The mass-to-light ratio of rich star clusters
• Dynamics of mode entanglement induced by particle-tunneling in the extended Bose-Hubbard dimer model
• Symbolic spectral decomposition of 3x3 matrices