  The Lorentz equations describe the motion of electrically charged particles
in electric and magnetic fields and are used widely in plasma physics. The most
popular numerical algorithm for solving them is the Boris method, a variant of
the St\"ormer-Verlet algorithm. Boris' method is phase space volume conserving
and simulated particles typically remain near the correct trajectory. However,
it is only second order accurate. Therefore, in scenarios where it is not
enough to know that a particle stays on the right trajectory but one needs to
know where on the trajectory the particle is at a given time, Boris method
requires very small time steps to deliver accurate phase information, making it
computationally expensive. We derive an improved version of the high-order
Boris spectral deferred correction algorithm (Boris-SDC) by adopting a
convergence acceleration strategy for second order problems based on the
Generalised Minimum Residual (GMRES) method. Our new algorithm is easy to
implement as it still relies on the standard Boris method. Like Boris-SDC it
can deliver arbitrary order of accuracy through simple changes of runtime
parameter but possesses better long-term energy stability. We demonstrate for
two examples, a magnetic mirror trap and the Solev'ev equilibrium, that the new
method can deliver better accuracy at lower computational cost compared to the
standard Boris method. While our examples are motivated by tracking ions in the
magnetic field of a nuclear fusion reactor, the introduced algorithm can
potentially deliver similar improvements in efficiency for other applications.
