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The Jacobi method for real symmetric matrices
Numerische Mathematik, 1966As is well known, a real symmetric matrix can be transformed iteratively into diagonal form through a sequence of appropriately chosen elementary orthogonal transformations (in the following called Jacobi rotations): $${A_k} \to {A_{k + 1}} = U_k^T{A_k}{U_k}{\text{ (}}{A_0}{\text{ = given matrix),}}$$ where U k = U k(p,q, φ) is an orthogonal ...
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On the Convergence of Cyclic Jacobi Methods
1991Jacobi’s method is an iterative algorithm for diagonalizing a symmetric matrix A = A (1), in which at iteration k a plane rotation J k =J k (i k ,j k ,θ k ) is chosen so as to annihilate a pair of off diagonal elements in the matrix A (k) $$ \left( {\begin{array}{*{20}{c}} c&{ - s} \\ s&c \end{array}} \right)\left( {\begin{array}{*{20}{c}} {a_{ii}^{(
Gautam M. Shroff, Robert Schreiber
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On the convergence of the euler-jacobi method
Numerical Functional Analysis and Optimization, 1992The Euler-Jacobi method for the solution of the symmetric eigenvalue problem uses as basic transformations Euler rotations which diagonalize exactly 3 × 3 submatrices. We prove that the cyclic Euler-Jacobi method is quadratically convergent for matrices with distinct eigenvalues.
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PRECONDITIONING OF SEQUENTIAL AND PARALLEL JACOBI-DAVIDSON METHOD
Parallel Computing, 2002We exploit an optimization method, called DACG, which sequentially computes the smallest eigenpairs of a symmetric, positive definite, generalized eigenproblem, by CG minimizations of the Rayleigh quotient over subspaces of decreasing size. In this paper we analyze the effectiveness of the approximate inverse preconditioners, AINV and FSAI as DACG ...
BERGAMASCHI, LUCA +2 more
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An extended Hamilton — Jacobi method
Regular and Chaotic Dynamics, 2012We develop a new method for solving Hamilton’s canonical differential equations. The method is based on the search for invariant vortex manifolds of special type. In the case of Lagrangian (potential) manifolds, we arrive at the classical Hamilton — Jacobi method.
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Accelerating the SVD Block-Jacobi Method
Computing, 2005The paper discusses how to improve performance of the one-sided block-Jacobi algorithm for computing the singular value decomposition of rectangular matrices. In particular, it is shown how cosine-sine decomposition of orthogonal matrices can be used to accelerate the slowest part of the algorithm – updating the block-columns.
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Accelerating Block-Jacobi Methods
2003Until recently QR methods and their variations have been considered as the fastest methods for computing the singular value and the symmetric eigenvalue problems. Lately, Divide and Conquer methods have developed to a stage at which for larger matrices they overcome the performance of the QR methods. However, the both types of methods, require reducing
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Generalized eigensolutions by Jacobi methods
Zeitschrift für angewandte Mathematik und Mechanik, 1996Generalized eigenvalue problem ; Jacobi ...
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Improving Block-Jacobi Methods
1904A way how to exploit memory hierarchy to improve performance of some block methods, is explained for the case of a one-sided block-Jacobi method for computing SVD of rectangular matrices. At each step of that method, an orthogonal matrix U must be applied to two block-columns of the iterated matrix, $ [G_i, G_j]U$.
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