A higher order iterative method for $A^{(2)}_{T,S}$

Srivastava, Shwetabh; Gupta, Dharmendra K.

doi:10.1007/s12190-013-0743-4

Cited by 10 publications

(4 citation statements)

References 18 publications

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“…Richardson iteration plays a role of integration loop for recursive matrix inversion algorithms and increases the order of convergence by one, which can be seen by comparison of the following pairs of the error models: (11), (14) and (16),(45) and (31), (47) and (32), (49), see also Table 4. Notice that the same integration effect is present in the additional loop of modified high order Newton-Schulz algorithm proposed in [35]. Combination of high order Newton-Schulz and Richardson framework is preferable for more accurate and robust parameter calculation with improved convergence rate.…”

Section: Integration Property Of Richardson Iteration and Convergencementioning

confidence: 97%

“…which is more robust in finite-digit calculations (in some cases) compared to (34), (35). Notice also that the computational complexity of the recursive realization T k does not depend on the order h, but requires order dependent precalculations, which are also used as initial condition G 0 in (17) and calculated only once.…”

Section: Recursive Calculations For Part Imentioning

confidence: 99%

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

Stotsky

2019

J. Appl. Math. Comput.

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Convergence rate and robustness improvement together with reduction of computational complexity are required for solving the system of linear equations Aθ * = 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. Keywords 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

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Section: Integration Property Of Richardson Iteration and Convergencementioning

confidence: 97%

Section: Recursive Calculations For Part Imentioning

confidence: 99%

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

Stotsky

2019

J. Appl. Math. Comput.

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“…Remark 2. The algorithm similar to (34) -( 39) was proposed in [51]. The algorithm written in the following form:…”

Section: Double Newton-schulz Algorithm With High Order Residual As C...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). Notice that the matrix-by-vector product Aθ k−1 in (51) can be easily calculated in parallel via methods described for example in [36].…”

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

confidence: 99%

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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A two-step iterative method and its acceleration for outer inverses

Srivastava

Gupta

2016

Sādhanā

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A higher order iterative method for $A^{(2)}_{T,S}$

Cited by 10 publications

References 18 publications

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

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

Convergence Rate Improvement of Richardson and Newton-Schulz Iterations

A two-step iterative method and its acceleration for outer inverses

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