  We discuss an automated computational methodology for computing
two-dimensional spectral submanifolds (SSMs) in autonomous nonlinear mechanical
systems of arbitrary degrees of freedom. In our algorithm, SSMs, the smoothest
nonlinear continuations of modal subspaces of the linearized system, are
constructed up to arbitrary orders of accuracy, using the parameterization
method. An advantage of this approach is that the construction of the SSMs does
not break down when the SSM folds over its underlying spectral subspace. A
further advantage is an automated a posteriori error estimation feature that
enables a systematic increase in the orders of the SSM computation until the
required accuracy is reached. We find that the present algorithm provides a
major speed-up, relative to numerical continuation methods, in the computation
of backbone curves, especially in higher-dimensional problems. We illustrate
the accuracy and speed of the automated SSM algorithm on lower- and
higher-dimensional mechanical systems.
