Developing Kaczmarz method for solving Sylvester matrix equations

Shafiei, Soheila Ghoroghi; Hajarian, Masoud

doi:10.1016/j.jfranklin.2022.09.028

Cited by 12 publications

(3 citation statements)

References 44 publications

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“…Therefore, according to Lemma 2 of [33], w(t) is fixed-time convergent to the equilibrium point w(t) = 0. This fact, together with w(t) = w(t) − w * = vec(X(t)−X * ), demonstrates that neural state X(t) is fixedtime convergent to the exact solution X * of the Sylvester equation (1). Moreover, according to Lemma 2 of [33], the upper bound of fixed-time convergence is estimated as…”

Section: And Matrixmentioning

confidence: 83%

“…I N the domain of scientific research and engineering, a considerable number of applications [1]- [6], such as observer design, state estimation, commutative rings, and pole placement, are germane to the solution to the Sylvester equation. Mathematically, the Sylvester equation to be investigated can be formulated as the following form [6]:…”

Section: Introductionmentioning

confidence: 99%

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Robustness Analysis of a Fixed-Time Convergent GNN for Online Solving Sylvester Equation and Its Application to Robot Path Tracking

Tan,

Yang

2023

IEEE Access

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It is shown in a recently-published work that the GNN (gradient-based neural network) model activated by the msbp (modified sign-bi-power) function exhibits superior fixed-time convergence for solving the Sylvester equation in the noise-free case. Encouraged by this point, in this paper, we study its robust performance when it is disturbed by three kinds of noises, i.e., dynamic bounded vanishing noise, dynamic bounded non-vanishing noise, and constant noise. Detailed mathematical analyses are conducted to show the robustness properties (e.g., convergence property, the upper bound of steady-state solution error) of the GNN model, which are further verified by a simulation example. Ultimately, the GNN model is applied to the path tracking of a planar four-link redundant robot arm. INDEX TERMS Gradient-based neural network; Sylvester equation; Robustness analysis; Robot path tracking.

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Section: And Matrixmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 99%

Robustness Analysis of a Fixed-Time Convergent GNN for Online Solving Sylvester Equation and Its Application to Robot Path Tracking

Tan,

Yang

2023

IEEE Access

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“…When the matrices A and B are small and dense, direct methods based on QR fractions are attractive [5,6]. However, for large A and B matrices, iterative methods have attracted a lot of attention [7][8][9][10][11]. Recently, Du et al proposed the randomized block coordinate descent (RBCD) method for solving the matrix least-squares problem min…”

Section: Introductionmentioning

confidence: 99%

On the Convergence of the Randomized Block Kaczmarz Algorithm for Solving a Matrix Equation

Xing,

Bao,

2023

Mathematics

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A randomized block Kaczmarz method and a randomized extended block Kaczmarz method are proposed for solving the matrix equation AXB=C, where the matrices A and B may be full-rank or rank-deficient. These methods are iterative methods without matrix multiplication, and are especially suitable for solving large-scale matrix equations. It is theoretically proved that these methods converge to the solution or least-square solution of the matrix equation. The numerical results show that these methods are more efficient than the existing algorithms for high-dimensional matrix equations.

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A randomized block Douglas–Rachford method for solving linear matrix equation

Huang,

Peng

2024

Calcolo

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Developing Kaczmarz method for solving Sylvester matrix equations

Cited by 12 publications

References 44 publications

Robustness Analysis of a Fixed-Time Convergent GNN for Online Solving Sylvester Equation and Its Application to Robot Path Tracking

Robustness Analysis of a Fixed-Time Convergent GNN for Online Solving Sylvester Equation and Its Application to Robot Path Tracking

On the Convergence of the Randomized Block Kaczmarz Algorithm for Solving a Matrix Equation

A randomized block Douglas–Rachford method for solving linear matrix equation

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