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The results of a parallel implementation of a randomized vector algorithm for solving systems of linear equations are presented in the paper. The solution is represented in the form of a Neumann series. The stochastic method computes this series by sampling only random columns, avoiding multiplication of matrix by matrix and matrix by vector. We consider the case when the matrix is too large to fit in random-access memory (RAM). We use two approaches to solve this problem. In the first approach, the matrix is divided into parts that are distributed among MPI processes and stored in the available RAM of the cluster nodes. In the second approach, the entire matrix is stored on each node's hard drive, loaded into RAM, and processed in parts. Independent Monte Carlo experiments for random column indices are distributed among MPI processes or OpenMP threads for both approaches to matrix storage. The efficiency of parallel implementations is analyzed. Results are given for a system governed by dense matrices of size 10 4 and 10 5 . [ABSTRACT FROM AUTHOR]

The Jacobi iterative algorithm has the characteristic of low computational load, and multiple components of the solution can be solved independently. This paper applies these characteristics to the ternary optical computer, which can be used for parallel optimization because it has a large number of data bits and reconfigurable processor bits. Therefore, a new parallel design scheme is constructed to solve the problem of slow efficiency in solving large linear equations. And the elaborate experiment is used to verify. The experimental method is to simulate the calculation on the ternary optical computer experimental platform. Then, the resource consumption is numerically calculated and summarized to measure the feasibility of the parallel design. Eventually, the results show that the parallel design has obvious advantages in computing speed. The Jacobi iterative algorithm is optimized in parallel on ternary optical processor for the first time. There are two parallel highlights of the scheme. First, the n components are calculated in full parallel. Second, the modified signed-digit (MSD) multiplier based on the minimum module and one-step MSD adder are used to calculate each component to eliminate the impact of large amount of data on calculation time. The research provides a new method for fast solution of large linear equations. [ABSTRACT FROM AUTHOR]