### Abstract:

Magnus' expansion approximates the solution of a linear, nonconstant-coefficient system of ordinary differential equations (ODEs) as the exponential of an infinite series of integrals of commutators of the matrix-valued coefficient function. It generalizes a standard technique for solving first-order, scalar, linear ODEs. However, much about the convergence of Magnus' expansion and its efficient computation is not known. This thesis describes in detail the derivation of Magnus' expansion and reviews Iserles' ordering for efficient calculation. Convergence of the expansion is explored and known convergence estimates are applied. Finally, Magnus' expansion is applied to several numerical examples, keeping track of convergence as it depends on parameters. These examples demonstrate the failure of current convergence estimates to correctly account for the degree of commutativity of the matrix-valued coefficient function.