  The Worldvolume Hybrid Monte Carlo method (WV-HMC method) [arXiv:2012.08468]
is a reliable and versatile algorithm towards solving the sign problem.
Similarly to the tempered Lefschetz thimble method, this method removes the
ergodicity problem inherent in algorithms based on Lefschetz thimbles. In
addition to this advantage, the WV-HMC method significantly reduces the
computational cost because one needs not compute the Jacobian of deformation in
generating configurations. A crucial step in this method is the RATTLE
algorithm, where the Newton method is used at each molecular dynamics step to
project a transported configuration onto a submanifold (worldvolume) in the
complex space. In this paper, we simplify the RATTLE algorithm by employing a
simplified Newton method (the fixed-point method) along with iterative solvers
for orthogonal decompositions of vectors, and show that this algorithm further
reduces the computational cost. We also apply this algorithm to the HMC
algorithm for the generalized thimble method (GT-HMC method). We perform a
numerical test for the convergence of the simplified RATTLE algorithm, and show
that the convergence depends on the system size only weakly. The application of
this simplified algorithm to various models will be reported in subsequent
papers.
