Error analysis of two algorithms for the computation of the matrix exponential (Q1207776)

This paper investigates the roundoff and truncation errors for the Taylor series of a matrix exponential. If the norm of the matrix \(A\) is less than one, the roundoff error grows linearly with respect to its size and norm; otherwise it grows exponentially. Hence for \(A\) with \(\| A\|\geq 1\) a scaling and squaring algorithm is introduced for \(m^{-1}A\) where \(m\) is a sufficiently large integer to give \(\| m^{-1}A\| < 1\): Compute the truncated Taylor series of \(m^{-1}A\) and raise it to the \(m\)-th power, where for ease of computations, \(m\) is chosen as a power of two. In the final analysis, the scaling and squaring algorithm is shown to be stable for all \(A\).