A predictor-corrector iterative method for solving linear least squares problems and perturbation error analysis

Buranay, Suzan Cival; Iyikal, Ovgu Cidar

doi:10.1186/s13660-019-2154-z

Cited by 5 publications

(8 citation statements)

References 22 publications

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“…An iterative method is said to be a p th‐order method, for some p >1, if there is a positive constant c such that

‖T_{m_{r} + 1}^{(r)}‖ \leq c {‖T_{m_{r}}^{(r)}‖}^{p}, m_{r} = 0, 1, 2, \dots,

for any multiplicative matrix norm. (See section 7 of chapter 7 in Israel and Greville 23 for iterative methods of computing the Moore‐Penrose inverses and for a version of a predictor‐corrector iterative method for solving linear least squares problems in Buranay and Iyikal 24 ).…”

Section: Incomplete Block‐matrix Factorization Using Two‐step Iterative Methodsmentioning

confidence: 99%

Incomplete block‐matrix factorization of M‐matrices using two‐step iterative method for matrix inversion and preconditioning

Buranay

Iyikal

2020

Math Methods in App Sciences

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Using the general method of Owe Axelsson given in 1986 for incomplete factorization of M‐matrices in block‐matrix form, we give a recursive approach to construct incomplete block‐matrix factorization of M‐matrices by proposing a two‐step iterative method for the approximation of the inverse of diagonal pivoting block matrices at each stage of the recursion. For various predescribed tolerances in the accuracy of the approximation of the inverses, the obtained incomplete block‐matrix factorizations are used to precondition the iterative methods as one‐step stationary iterative (OSSI) method and biconjugate gradient stabilized method (BI‐CGSTAB). Certain applications are conducted on M‐matrices occurring from the discretization of two Dirichlet boundary value problems of Laplace's equation on a rectangle using finite difference method. Numerical results justify that the given incomplete block‐matrix factorization of M‐matrices using the two‐step iterative method to approximate the inverse of diagonal pivoting block matrices at each stage of the recursion give robust preconditioners. The obtained results are presented through tables and figures.

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“…An iterative method is said to be a p th‐order method, for some p >1, if there is a positive constant c such that

‖T_{m_{r} + 1}^{(r)}‖ \leq c {‖T_{m_{r}}^{(r)}‖}^{p}, m_{r} = 0, 1, 2, \dots,

Section: Incomplete Block‐matrix Factorization Using Two‐step Iterative Methodsmentioning

confidence: 99%

Incomplete block‐matrix factorization of M‐matrices using two‐step iterative method for matrix inversion and preconditioning

Buranay

Iyikal

2020

Math Methods in App Sciences

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“…Example: Nested algorithm as unification of the hyperpower iteration method (described in [35]) of order 45 that requires 10 mmm only. Newton-Schulz iteration of the order 45 can be factorized in a step-wise way as follows:…”

Section: Reduction Of Computational Complexity Via Unified Factorizat...mentioning

confidence: 99%

“…2) The second part is associated with calculations of L k and Γ n k in (35) and (36) respectively using n−1 j=0 Γ j k . The results of both parts are merged in (52) to be included in the Richardson iteration (51).…”

Section: Reduction Of Computational Complexity Via Recursive and Simu...mentioning

confidence: 99%

“…Reduction of the computational complexity of high order Newton-Schulz algorithm is one the most important challenges in this area. The computational complexity can be reduced via factorizations of the power series, see for example [17], [21], [26] - [35] and references therein. Practical applications of these factorizations (excepting Horner's rule) are hampered by the lack of general unified description.…”

Section: Introductionmentioning

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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“…Perturbation analysis of least squares problems is a major topic in numerical linear algebra. Related work has focused on establishing various error bounds [32][33][34]. We combine the basic techniques of perturbation analysis with multivariate statistics [35] to quantitatively evaluate the estimators for a nonlinear estimation problem.…”

Section: Introductionmentioning

confidence: 99%

Design and Analysis of a Non-Iterative Estimator for Target Location in Multistatic Sonar Systems with Sensor Position Uncertainties

Wang

Yang

et al. 2020

Mathematics

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Target location is the basic application of a multistatic sonar system. Determining the position/velocity vector of a target from the related sonar observations is a nonlinear estimation problem. The presence of possible sensor position uncertainties turns this problem into a more challenging hybrid parameter estimation problem. Conventional gradient-based iterative estimators suffer from the problems of initialization difficulties and local convergence. Even if there is no problem with initialization and convergence, a large computational cost is required in most cases. In view of these drawbacks, we develop a computationally efficient non-iterative position/velocity estimator. The main numerical computation involved is the weighted least squares optimization, which makes the estimator computationally efficient. Parameter transformation, model linearization and two-stage processing are exploited to prevent the estimator from iterative computation. Through performance analysis and experimental verification, we find that the proposed estimator reaches the hybrid Cramér–Rao bound and has linear computational complexity.

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A predictor-corrector iterative method for solving linear least squares problems and perturbation error analysis

Cited by 5 publications

References 22 publications

Incomplete block‐matrix factorization of M‐matrices using two‐step iterative method for matrix inversion and preconditioning

Incomplete block‐matrix factorization of M‐matrices using two‐step iterative method for matrix inversion and preconditioning

Convergence Rate Improvement of Richardson and Newton-Schulz Iterations

Design and Analysis of a Non-Iterative Estimator for Target Location in Multistatic Sonar Systems with Sensor Position Uncertainties

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