The tensor absolute value equation is a class of interesting structured multilinear systems. In this article, from the perspective of pure numerical algebra, we first consider a tensor-type successive over-relaxation method (SOR) (called TSOR) and tensor-type accelerated over-relaxation method (AOR) (called TAOR) for solving tensor absolute value equations. Furthermore, one type of preconditioned tensor splitting method is also applied for solving structured multilinear systems. Numerical experiments adequately demonstrate the efficiency of the presented methods.

In this paper, a new multi-parameter iterative algorithm is proposed to address the PageRank problem based on the multi-splitting iteration method. The proposed method solves two linear subsystems at each iteration by splitting the coefficient matrix, considering therefore inner and outer iteration to find the approximate solutions of these linear subsystems. It can be shown that the iterative sequence generated by the multi-parameter iterative algorithm finally converges to the PageRank vector when the parameters satisfy certain conditions. Numerical experiments show that the proposed algorithm has better convergence and numerical stability than the existing algorithms.

In this paper, the Newton-type splitting iterative method for a class of coupled Sylvester-like absolute value equation is proposed. Some sufficient conditions for the existence of the unique solution of the coupled Sylvester-like absolute value equation are given and sufficient conditions for the nonexistence of solution is discussed. The Newton-base bimatrix splitting iteration method, the Newton-base generalized Gauss-Seidel bimatrix splitting iteration method and the inexact relaxed generalized Newton bimatrix splitting method are proposed to solve the coupled Sylvester-like absolute value equation. Numerical experiments confirm the conclusions proposed in this paper.

In this paper, a new multi parameter iterative algorithm is proposed to address the PageRank problem based on the multi-splitting iteration method described by Gu et al . The proposed method in each iteration needs to solve two linear subsystems by splitting the coefficient matrix, therefore, we consider inner and outer iteration to find the approximate solutions of these linear subsystems. It can be shown that the iterative sequence generated by the multi parameter iterative algorithm finally converges to the PageRank vector when the parameters satisfy the certain conditions. Numerical experiments show that the proposed algorithm has better convergence and numerical stability than the existing algorithms.

As chip interconnect density increases, routing problems become increasingly complex. The routing scheme significantly impacts key performance indicators such as chip delay, power consumption, and area. In Very Large-Scale Integration (VLSI) routing, the rectilinear Steiner minimal tree is an excellent interconnect model for multi-pin nets. However, modern VLSI designs require multi-layer obstacle-avoiding routing, where wires must traverse multiple metal layers while avoiding obstacles to ensure connectivity and performance. This makes the Multi-Layer Obstacle-Avoiding Rectilinear Steiner Minimal Tree (ML-OARSMT) problem highly challenging in VLSI physical design. To address this issue, this paper proposes an ML-OARSMT construction algorithm based on deep reinforcement learning. First, a multi-layer obstacle-avoiding spanning graph is constructed by introducing vertex mapping, which connects different layers to handle the multi-layer obstacle-avoiding routing problem. Then, an agent is designed to learn edge selection for constructing the Multi-Layer Obstacle-Avoiding Steiner Tree (ML-OAST) using Double Deep Q-Network (DDQN). Finally, a post-processing stage is applied to further shorten the total wirelength through fast pruning and local optimization. Experimental results demonstrate that the proposed algorithm achieves better wirelength quality compared to state-of-the-art heuristic algorithms. Additionally, an ablation study confirms the effectiveness of DDQN in routing optimization.