High-performance computation of the exponential of a large sparse matrix

TL;DR

This paper introduces a novel, efficient algorithm for computing the exponential of large sparse matrices, significantly reducing memory and computational costs while improving accuracy over existing methods.

Contribution

The paper proposes a new filtering technique and sparsity analysis based on real bandwidth, enabling a more efficient computation of large sparse matrix exponentials.

Findings

Outperforms MATLAB's expm in accuracy and efficiency

Handles matrices larger than 2 million dimensions

Reduces memory usage and computational cost

Abstract

Computation of the large sparse matrix exponential has been an important topic in many fields, such as network and finite-element analysis. The existing scaling and squaring algorithm (SSA) is not suitable for the computation of the large sparse matrix exponential as it requires greater memories and computational cost than is actually needed. By introducing two novel concepts, i.e., real bandwidth and bandwidth, to measure the sparsity of the matrix, the sparsity of the matrix exponential is analyzed. It is found that for every matrix computed in the squaring phase of the SSA, a corresponding sparse approximate matrix exists. To obtain the sparse approximate matrix, a new filtering technique in terms of forward error analysis is proposed. Combining the filtering technique with the idea of keeping track of the incremental part, a competitive algorithm is developed for the large sparse matrix exponential. The proposed method can primarily alleviate the over-scaling problem due to the filtering technique. Three sets of numerical experiments, including one large matrix with a dimension larger than 2e6 , are conducted. The numerical experiments show that, compared with the expm function in MATLAB, the proposed algorithm can provide higher accuracy at lower computational cost and with less memory.