Efficient Computation of Exponential Matrices for Large Symmetric Negative Semidefinite Matrices

Abstract

The exponential of a matrix is a fundamental mathematical operation with numerous applications in various fields, including numerical linear algebra. The computation of e^Av for large symmetric negative semidefinite matrices presents significant challenges due to computational complexity and memory requirements. This research paper introduces an innovative iterative approach that combines Krylov subspace methods with projection techniques to compute e^Av efficiently. The Krylov subspace iteration constructs an orthogonal basis capturing essential information for the matrix exponential. Through projection techniques, the problem's dimensionality is reduced, enabling efficient computations. A comprehensive step-by-step description of the approach is provided, highlighting its benefits, such as reduced computational complexity, improved memory efficiency, scalability to large matrices, and high accuracy. The proposed approach introduces new possibilities for efficient approximation of e^Av in diverse scientific and engineering applications involving large symmetric negative semidefinite matrices. Experimental results validate the approach's effectiveness and accuracy, illustrating its potential to revolutionize computations involving exponential matrices in high-dimensional systems.