On sharp quadratic convergence bounds for the serial Jacobi methods

Hari, Vjeran

doi:10.1007/bf01385728

Cited by 44 publications

(51 citation statements)

References 18 publications

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“…The symmetric Jacobi algorithm with rowcyclic strategy is quadratically convergent. This is a well known fact, and the proof of the general case of multiple eigenvalues is given by Hari [15]. Using the off-norm Ω(·) = · −diag(·) F , the quadratic convergence is stated as follows: …”

Section: Review Of Asymptotic Convergencementioning

confidence: 99%

New Fast and Accurate Jacobi SVD Algorithm. II

Drmač¹,

Veselić²

2008

SIAM J. Matrix Anal. & Appl.

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Abstract. This paper presents new implementation of one-sided Jacobi SVD for triangular matrices and its use as the core routine in a new preconditioned Jacobi SVD algorithm, recently proposed by the authors. New pivot strategy exploits the triangular form and uses the fact that the input triangular matrix is the result of rank revealing QR factorization. If used in the preconditioned Jacobi SVD algorithm, it delivers superior performance leading to the currently fastest method for computing SVD decomposition with high relative accuracy. Furthermore, the efficiency of the new algorithm is comparable to the less accurate bidiagonalization based methods. The paper also discusses underflow issues in floating point implementation, and shows how to use perturbation theory to fix the imperfectness of machine arithmetic on some systems.

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Section: Review Of Asymptotic Convergencementioning

confidence: 99%

New Fast and Accurate Jacobi SVD Algorithm. II

Drmač¹,

Veselić²

2008

SIAM J. Matrix Anal. & Appl.

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“…It was later shown in [9,10] that for matrix with well separated eigenvalues, the CJT has a quadratic convergence rate 3 . This result was extended in [11] to a more general case which includes identical eigenvalues and clusters of eigenvalues (that is, very close eigenvalues). Studies have shown that in practice the number of iterations that is required for the CJT to reach its asymptotic quadratic convergence rate is a small number, but this has not been proven rigorously.…”

Section: The One-bit Blind Null Space Learning Algorithm (Obnsla)mentioning

confidence: 94%

“…The theorem enables the SU to solve the optimization problems in (11) and (14) via line searches based on {x(n), q(n)} T n=1 . This is because under the OC, the SU can extracth(x(n),x(n − m)), which indicates whether S(G,x(n)) > S(G,x(n − m)) is true or false.…”

Section: A the One-bit Line Searchmentioning

confidence: 99%

“…The complexity of such a search is at least O(1/η) since each point in the grid must be compared to a different point at least once. The two line searches in (11) and (14) would be carried out much more efficiently if binary searches could be invoked. However, a binary search is feasible only if the objective function has a unique local minimum point, which is not the case in (11) and (14) because…”

Section: A the One-bit Line Searchmentioning

confidence: 99%

“…Thus, before invoking the binary search, a single-minimum interval (SMI) must be determined; i.e, an interval in which the target function in (11) or (14) …”

Section: A the One-bit Line Searchmentioning

confidence: 99%

See 2 more Smart Citations

One-Bit Null Space Learning for MIMO underlay cognitive radio

Noam

Goldsmith

2013

2013 Information Theory and Applications Workshop (ITA)

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Abstract-We present a new algorithm, called the One-Bit Null Space Learning Algorithm (OBNSLA), for MIMO cognitive radio Secondary Users (SU) to learn the null space of the interference channel to the Primary User (PU). The SU observes a binary function that indicates whether the interference it inflicts on the PU has increased or decreased in comparison to the SU's previous transmitted signal. This function is obtained by listening to the PU's transmitted signal or control channel and extracting information from it about whether the PU's Signal to Interference plus Noise power Ratio has increased or decreased. In addition to introducing the OBNSLA, the paper provides a thorough convergence analysis of this algorithm. The OBNSLA is shown to have a linear convergence rate and an asymptotic quadratic convergence rate.

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Sort‐Jacobi and generalizations of the symmetric eigenvalue problem

Kleinsteuber

2007

Proc Appl Math and Mech

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In this paper, it is sketched how the Sort-Jacobi method for the symmetric eigenvalue problem extends to the (−1)-eigenspace of the Cartan involution on an arbitrary semisimple Lie algebra. The proposed method is independent of the representation of the underlying Lie algebra and generalizes well-known normal form problems such as the symmetric, Hermitian, skewsymmetric, symmetric and skew-symmetric R-Hamiltonian eigenvalue problem and the singular value decomposition. It allows a unified treatment of the above eigenvalue methods, including local convergence analysis. *

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On sharp quadratic convergence bounds for the serial Jacobi methods

Cited by 44 publications

References 18 publications

New Fast and Accurate Jacobi SVD Algorithm. II

New Fast and Accurate Jacobi SVD Algorithm. II

One-Bit Null Space Learning for MIMO underlay cognitive radio

Sort‐Jacobi and generalizations of the symmetric eigenvalue problem

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