A Geometric Gaussian Kaczmarz Method for Large Scaled Consistent Linear Equations

Li Wen,Yimou Liao,Feng Yin,Guangxin Huang

doi:10.4236/jamp.2021.911189

Li Wen, Yimou Liao + Show 2 more

Open Access

https://doi.org/10.4236/jamp.2021.911189

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Abstract

This paper presents a geometric Gaussian Kaczmarz (GGK) method for solving the large-scaled consistent linear systems of equation. The GGK method improves the geometric probability randomized Kaczmarz method in [1] by introducing a new block set strategy and the iteration process. The GGK is proved to be of linear convergence. Several numerical examples show the efficiency and effectiveness of the GGK method.

Highlights

A Geometric Gaussian Kaczmarz AlgorithmThis section describes a geometric Gaussian Kaczmarz (GGK) algorithm to compute the solution of (1)
This paper presents a geometric Gaussian Kaczmarz (GGK) method for solving the large-scaled consistent linear systems of equation
It can be measured by the speed-up of GGK against geometric probability randomized Kaczmarz algorithm (GPRK) (SU) is defined by SU = computing time (CPU) of GPRK

Summary

A Geometric Gaussian Kaczmarz Algorithm

This section describes a geometric Gaussian Kaczmarz (GGK) algorithm to compute the solution of (1). Steps 5 and 6 give the iteration process of the GGK method. The following results show the convergence of Algorithm 1. Assume the linear system (1) is consistent, and the iterative sequence { }xk k≥0 generated by Algorithm 1 converges to the least-norm solution x∗ = A†b of linear systems (1). Let Ek ∈ m×τk denote the matrix whose columns orderly are constituted of all the vector ei ∈ m with i ∈τ k , Aτk = EkT A. Aτk F ( ) λ A AT max τ k τ k λmin AT A ≤ 1 −η λmin AT A < 1

Numerical Experiments

Experiments with Sparse Matrix

Experiments with Dense Matrix

Conclusion