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Jacobi Decomposition and Eigenvalues of Symmetric Matrices

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Discrete and Computational Geometry

Part of the book series: Algorithms and Combinatorics ((AC,volume 25))

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Abstract

We show that every n-dimensional orthogonal matrix can be factored into O(n 2) Jacobi rotations (also called Givens rotations in the literature). It is well known that the Jacobi method,wh ich constructs the eigen-decomposition of a symmetric matrix through a sequence of Jacobi rotations,is slower than the eigenvalue algorithms currently used in practice,but is capable of computing eigenvalues, particularly tiny ones,t o a high relative accuracy. The above decomposition, which to the best of our knowledge is new,s hows that the infinite-precision nondeterministic Jacobi method (in which an oracle is invoked to obtain the correct rotation angle in each Jacobi rotation) can construct the eigen-decomposition with O(n 2) Jacobi rotations. The complexity of the nondeterministic Jacobi algorithm motivates the efforts to narrow the gap between the complexity of the known Jacobi algorithms and that of the nondeterministic version. Speeding up the Jacobi algorithm while retaining its excellent numerical properties would be of considerable interest. In that direction,we describe,as an example,a variant of the Jacobi method in which the rotation angle for each Jacobi rotation is computed (in closed-form) through 1-dimensional optimization. We also show that the computation of the closed-form optimal solution of the 1-dimensional problems guarantees convergence of the new method.

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Authors and Affiliations

  1. School of Industrial Engineering, Purdue University, West Lafayette, IN, 47907, USA

    W. He & N. Prabhu

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  1. W. He
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  2. N. Prabhu
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Editors and Affiliations

  1. Department of Computer and Information Science, Polytechnic University, Six Metro Tech Center, Brooklyn, NY, 11201, USA

    Boris Aronov

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA

    Saugata Basu

  3. City College and Courant Institute, 251 Mercer St., New York, NY, 10012, USA

    János Pach

  4. School of Computer Science, Tel Aviv University, Tel Aviv, 69978, Israel

    Micha Sharir

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He, W., Prabhu, N. (2003). Jacobi Decomposition and Eigenvalues of Symmetric Matrices. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds) Discrete and Computational Geometry. Algorithms and Combinatorics, vol 25. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55566-4_24

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  • DOI: https://doi.org/10.1007/978-3-642-55566-4_24

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-62442-1

  • Online ISBN: 978-3-642-55566-4

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