Unified frameworks for high order Newton-Schulz and Richardson iterations: a computationally efficient toolkit for convergence rate improvement

Convergence rate and robustness improvement together with reduction of computational complexity are required for solving the system of linear equations A θ ∗ = b in many applications such as system identification, signal and image processing, network analysis, machine learning and many others. Two u...

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Bibliographic Details
Published in:Journal of applied mathematics & computing Vol. 60; no. 1-2; pp. 605 - 623
Main Author: Stotsky, Alexander
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2019
Springer Nature B.V
Subjects:
Algorithms
Applied mathematics
Brief Reports
Computational efficiency
Computational Mathematics and Numerical Analysis
Computer simulation
Convergence
Image processing
Iterative methods
Linear equations
Machine learning
Markov analysis
Mathematical and Computational Engineering
Mathematics
Mathematics and Statistics
Mathematics of Computing
Network analysis
Richardson iteration · Neumann series · High order Newton-Schulz algorithm · Least squares estimation · Harmonic regressor · Strictly Diagonally Dominant Matrix · Symmetric positive definite matrix · Ill-conditioned matrix · Polynomial preconditioning · Matrix power series factorization · Computationally efficient matrix inversion algorithm · Simultaneous calculations
Robustness (mathematics)
Signal processing
System identification
Theory of Computation
Computationally efficient matrix inversion algorithm
Richardson iteration
Strictly Diagonally Dominant Matrix
Ill-conditioned matrix
Polynomial preconditioning
Symmetric positive definite matrix
Simultaneous calculations
Least squares estimation
Neumann series
High order Newton-Schulz algorithm
Matrix power series factorization
Harmonic regressor
ISSN:1598-5865, 1865-2085
Online Access:Get full text
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Summary:Convergence rate and robustness improvement together with reduction of computational complexity are required for solving the system of linear equations A θ ∗ = b in many applications such as system identification, signal and image processing, network analysis, machine learning and many others. Two unified frameworks (1) for convergence rate improvement of high order Newton-Schulz matrix inversion algorithms and (2) for combination of Richardson and iterative matrix inversion algorithms with improved convergence rate for estimation of θ ∗ are proposed. Recursive and computationally efficient version of new algorithms is developed for implementation on parallel computational units. In addition to unified description of the algorithms the frameworks include explicit transient models of estimation errors and convergence analysis. Simulation results confirm significant performance improvement of proposed algorithms in comparison with existing methods.
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ISSN:1598-5865
1865-2085
DOI:10.1007/s12190-018-01229-8