Expmmultiply

Expmmultiply is a term used in numerical linear algebra to denote the operation of computing the product of a matrix exponential with a vector or another matrix, without forming the matrix exponential explicitly. In mathematical form, for a square matrix A and a right-hand side B, expmmultiply returns exp(A) B. The naming mirrors the combination of the matrix exponential with a subsequent multiplication, and it is used in contexts where forming exp(A) directly would be inefficient or impractical.

Purpose and use cases include solving linear dynamical systems and time-evolution problems described by du/dt = A u, where the solution involves e^{tA} u0. When A is large or sparse, expmmultiply enables efficient computation by avoiding the dense exp(A) and instead applying approximation techniques that operate on B or multiple right-hand sides at once.

Algorithms and methods commonly employed include Krylov subspace approaches (such as Arnoldi or Lanczos iterations) to approximate exp(A) B, often in combination with scaling and squaring and Padé approximants, or other polynomial and rational approximations tailored for matrix-vector products. The choice of method typically depends on the size and sparsity of A and on whether B is a vector or a full matrix.

Relationship to related concepts: Expmmultiply is closely related to routines like expm_multiply or expmv in numerical libraries, which implement the same idea under different naming conventions. The underlying goal across implementations is to deliver accurate results for exp(A) B while reducing memory usage and computational cost.

Advantages and limitations: The primary advantage is efficiency for large-scale problems. Limitations include sensitivity to the conditioning of A and the need for appropriate algorithm selection to balance speed and accuracy.