The block Kaczmarz method is an iterative algorithm for solving large-scale linear sys-tems.At each iteration,the current iteration points will be orthogonally projected to the solution space of the constraint subset.In the paper,according to the principle of the block Kaczmarz method,we propose the randomized block Kaczmarz method with the Gaussian mixture model(GMM-RBK(k)).The partition of blocks is carried out by the Gaussian mixture model.In order to avoid the computation of pseudo-inverse or the solution of the least-squares problem,we propose the randomized average block Kaczmarz method with the Gaussian mixture model(GMM-RABK(k)).We prove that the two algorithms converge when the linear system is consistent,and give the corresponding formula of convergence rate.The effectiveness of the GMM-RBK(k)method and GMM-RABK(k)method is also verified in the final numerical experiments,and avoiding the calculation of pseudo-inverse,the GMM-RABK(k)method is superior to the GMM-RBK(k)method.
In this paper,based on the active set identification,we propose a Proximal Newton method for l1 problems.One advantage of this method is that it uses the good support identification property of the ISTA algorithm to determine the free variables and active set variables,and another advantage is that it uses the information of the partial Hessian matrix to update the free variables.Under appropriate conditions,we demonstrate the global convergence of the algorithm with a nonmonotonic line search strategy.The results of numerical experiments show that our proposed algorithm is effective.
Gegenbauer's addition theorem is an important theory of Bessel functions.This paper studies the truncation error of Gegenbauer's addition theorem.Based on the limiting forms of Bessel and Neumann functions as the argument tend to zero,an explicit bound of the truncation error and its convergence order are derived.Numerical experiments show that the estimates are valid and sharp.
The two-subspace projection algorithm and the multi-step inertial randomized Kaczmarz algorithm are effective methods for solving the coherent linear system.By adding the soft shrinkage in the two-subspace projection algorithm and the multi-step inertial randomized Kaczmarz algorithm,this paper introduces the sparse two-subspace projection algorithm and the sparse multi-step inertial randomized Kaczmarz algorithm.The linear convergence rate in expectation of the proposed algorithms in noise and noiseless cases is presented.Finally,we give some numerical experiments to illustrate the effectiveness and advantage of our algorithms.
In the paper,for solving large sparse vertical linear complementarity problems,a two-step modulus-based matrix synchronous multisplitting parallel iteration method is construct-ed by using two-step multisplitting technique.The new method can be viewed as a gener-alization of the modulus-based matrix synchronous multisplitting iterative method and the two-step modulus-based matrix splitting iterative method in the existing literatures.Fur-thermore,under the assumption that the system matrices are H+-matrices,the convergence analysis of the proposed method is given,and the convergence domain of the parameter matrix is obtained,which extends the convergence results of the existing methods.Finally,numerical experiments are carried out under the OpenMP framework for two numerical ex-amples in the existing literatures.Numerical results show that the two-step multisplitting technique can improve the computational efficiency of the existing methods.
