  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.
