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These methods are based on an <jats:italic>O<\/jats:italic>(<jats:italic>N<\/jats:italic>) solution to the constrained equations of motion. The benefits of this approach are that (1) the system constraints are solved exactly at each time step, (2) the solution algorithm is noniterative, (3) the algorithm is recursive and scales as <jats:italic>O<\/jats:italic>(<jats:italic>N<\/jats:italic>), (4) the algorithm is numerically stable, (5) the algorithm is highly amenable to parallel processing, and (6) potentially greater integration step sizes are possible. It is anticipated that application of this methodology will provide a 10\u2010 to 100\u2010improvement in the speed of a large molecular trajectory as compared with the time required to run a conventional atomistic unconstrained simulation. 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