  A single-step high-order implicit time integration scheme for the solution of
transient and wave propagation problems is presented. It is constructed from
the Pad\'e expansions of the matrix exponential solution of a system of
first-order ordinary differential equations formulated in the state-space. A
computationally efficient scheme is developed exploiting the techniques of
polynomial factorization and partial fractions of rational functions, and by
decoupling the solution for the displacement and velocity vectors. An important
feature of the novel algorithm is that no direct inversion of the mass matrix
is required. From the diagonal Pad\'e expansion of order $M$ a time-stepping
scheme of order $2M$ is developed. Here, each elevation of the accuracy by two
orders results in an additional system of real or complex sparse equations to
be solved. These systems are comparable in complexity to the standard Newmark
method, i.e., the effective system matrix is a linear combination of the static
stiffness, damping, and mass matrices. It is shown that the second-order scheme
is equivalent to Newmark's constant average acceleration method, often also
referred to as trapezoidal rule. The proposed time integrator has been
implemented in MATLAB using the built-in direct linear equation solvers. In
this article, numerical examples featuring nearly one million degrees of
freedom are presented. High-accuracy and efficiency in comparison with common
second-order time integration schemes are observed. The MATLAB-implementation
is available from the authors upon request or from the GitHub repository (to be
added).
