Efficient Iterative Solvers in the Least Squares Method

Stotsky, Alexander

doi:10.1016/j.ifacol.2020.12.847

Cited by 3 publications

(6 citation statements)

References 12 publications

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“…Consider the following combination of two Newton-Schulz iterations associated with two-step iterative method, [27] :…”

Section: Convergence Rate Improvement Via Combination Of High Order N...mentioning

confidence: 99%

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Systematic Review of Newton-Schulz Iterations with Unified Factorizations : Integration in the Richardson Method and Application to Robust Failure Detection in Electrical Networks

Stotsky¹

2022

Preprint

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Systematic overview of Newton-Schulz and Durand iterations with convergence analysis and factorizations is presented in the chronological sequence in unified framework. Practical recommendations for the choice of the order and factorizations of the algorithms and integration into Richardson iteration are given. The simplest combination of Newton-Schulz and Richardson iteration is applied to the parameter estimation problem associated with the failure detection via evaluation of the frequency content of the signals in electrical network. The detection is performed on real data for which the software failure was simulated, which resulted in the rank deficient information matrix. Robust preconditioning for rank deficient matrices is proposed and the efficiency of the approach is demonstrated by simulations via comparison with standard LU decomposition method.

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“…Consider the following combination of two Newton-Schulz iterations associated with two-step iterative method, [27] :…”

Section: Convergence Rate Improvement Via Combination Of High Order N...mentioning

confidence: 99%

“…Notice that Durand iteration (13) can be derived from ( 16) - (19) with n = 1. The error model ( 20) can be proved via multiplication of ( 19) by A and explicit evaluation of the error F k (similar to Section 2.2, see also [21], [27]) as follows:…”

Section: Convergence Rate Improvement Via Combination Of High Order N...mentioning

confidence: 99%

Systematic Review of Newton-Schulz Iterations with Unified Factorizations : Integration in the Richardson Method and Application to Robust Failure Detection in Electrical Networks

Stotsky¹

2022

Preprint

Self Cite

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“…has also two Newton-Schulz loops, where both Z k and G k are the estimates of the matrix inverse and p = 2, 3, ... is the order. Algorithm ( 40), (41) has the following error model…”

Section: Double Newton-schulz Algorithm With High Order Residual As C...mentioning

confidence: 99%

“…The algorithm ( 34) -( 39) has faster convergence due to the high order error F n k−1 in the error model ( 38) compared to algorithm (40), (41) which has the error model (43) with the first order error F k−1 .…”

Section: Double Newton-schulz Algorithm With High Order Residual As C...mentioning

confidence: 99%

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Convergence Rate Improvement of Richardson and Newton-Schulz Iterations

Stotsky

2020

Preprint

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Fast convergent, accurate, computationally efficient, parallelizable, and robust matrix inversion and parameter estimation algorithms are required in many time-critical and accuracy-critical applications such as system identification, signal and image processing, network and big data analysis, machine learning and in many others. This paper introduces new composite power series expansion with optionally chosen rates (which can be calculated simultaneously on parallel units with different computational capacities) for further convergence rate improvement of high order Newton-Schulz iteration. New expansion was integrated into the Richardson iteration and resulted in significant convergence rate improvement. The improvement is quantified via explicit transient models for estimation errors and by simulations. In addition, the recursive and computationally efficient version of the combination of Richardson iteration and Newton-Schulz iteration with composite expansion is developed for simultaneous calculations. Moreover, unified factorization is developed in this paper in the form of tool-kit for power series expansion, which results in a new family of computationally efficient Newton-Schulz algorithms.

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Recursive versus nonrecursive Richardson algorithms: systematic overview, unified frameworks and application to electric grid power quality monitoring

Stotsky

2022

Automatika

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Efficient Iterative Solvers in the Least Squares Method

Cited by 3 publications

References 12 publications

Systematic Review of Newton-Schulz Iterations with Unified Factorizations : Integration in the Richardson Method and Application to Robust Failure Detection in Electrical Networks

Systematic Review of Newton-Schulz Iterations with Unified Factorizations : Integration in the Richardson Method and Application to Robust Failure Detection in Electrical Networks

Convergence Rate Improvement of Richardson and Newton-Schulz Iterations

Recursive versus nonrecursive Richardson algorithms: systematic overview, unified frameworks and application to electric grid power quality monitoring

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