exptrAB

exptrAB is a name commonly used to denote a computational routine or operator that evaluates the action of a matrix exponential on a matrix or set of vectors, i.e., computes exp(A) times B (often exp(tA) B for a scalar t) without explicitly forming the dense matrix exp(A). The formulation is especially useful when A is large and sparse and B has relatively few columns, since forming exp(A) directly can be prohibitively expensive in memory and time.

Algorithms for exptrAB typically combine scaling and squaring, truncated Taylor series, and Krylov subspace or polynomial approximation techniques to approximate the product with controlled error. Krylov-based methods build a low-dimensional basis from A and B and compute the exponential of a small projected matrix, while other approaches use rational approximations or restarted iterations to manage storage and convergence. Error estimates and adaptivity are important components to ensure numerical stability and efficiency.

Applications include time integration of linear ordinary differential equations, model reduction, control and signal processing, and any context requiring repeated evaluation of exp(A) applied to multiple right-hand sides. Related routines and concepts are matrix exponential (expm), expmv (action on a vector), and expm_multiply in scientific computing libraries. Performance depends on matrix properties (sparsity, spectrum) and the number of columns in B; selecting an appropriate algorithm balances accuracy, runtime, and memory.