dorsal/arxiv
View SchemaEfficient numerical diagonalization of hermitian 3x3 matrices
| Authors | Joachim Kopp |
|---|---|
| Categories | |
| ArXiv ID | physics/0610206 |
| URL | https://arxiv.org/abs/physics/0610206 |
| DOI | 10.1142/S0129183108012303 |
| Journal | Int.J.Mod.Phys.C19:523-548,2008 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
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/ .
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"abstract": "A very common problem in science is the numerical diagonalization of\nsymmetric or hermitian 3x3 matrices. Since standard \"black box\" packages may be\ntoo inefficient if the number of matrices is large, we study several\nalternatives. We consider optimized implementations of the Jacobi, QL, and\nCuppen algorithms and compare them with an analytical method relying on\nCardano\u0027s formula for the eigenvalues and on vector cross products for the\neigenvectors. Jacobi is the most accurate, but also the slowest method, while\nQL and Cuppen are good general purpose algorithms. The analytical algorithm\noutperforms the others by more than a factor of 2, but becomes inaccurate or\nmay even fail completely if the matrix entries differ greatly in magnitude.\nThis can mostly be circumvented by using a hybrid method, which falls back to\nQL if conditions are such that the analytical calculation might become too\ninaccurate. For all algorithms, we give an overview of the underlying\nmathematical ideas, and present detailed benchmark results. C and Fortran\nimplementations of our code are available for download from\nhttp://www.mpi-hd.mpg.de/~globes/3x3/ .",
"arxiv_id": "physics/0610206",
"authors": [
"Joachim Kopp"
],
"categories": [
"physics.comp-ph",
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"math.NA"
],
"doi": "10.1142/S0129183108012303",
"journal_ref": "Int.J.Mod.Phys.C19:523-548,2008",
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "Efficient numerical diagonalization of hermitian 3x3 matrices",
"url": "https://arxiv.org/abs/physics/0610206"
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