Exact analytical solutions of the master equation are limited to special cases and exact numerical methods are
inefficient. Even the generic one-dimensional, one-step master equation has evaded exact solution, aside from
the steady-state case. This type of master equation describes the dynamics of a continuous-time Markov process
whose range consists of positive integers and whose transitions are allowed only between adjacent sites. The
solution of any master equation can be written as the exponential of a (typically huge) matrix, which requires
the calculation of the eigenvalues and eigenvectors of the matrix. Here we propose a linear algebraic method
for simplifying this exponential for the general one-dimensional, one-step process. In particular, we prove that
the calculation of the eigenvectors is actually not necessary for the computation of exponential, thereby we
dramatically cut the time of this calculation. We apply our new methodology to examples from birth-death
processes and biochemical networks. We show that the computational time is significantly reduced compared to
existing methods