Dynamic constraints are a natural way of describing many interesting physical systems. There are two general categories of dynamic constraints discussed in the computer graphics literature: reduced coordinate formulations and Lagrange multiplier dynamic constraints.

Reduced coordinate formulations involve re-parameterizing an
unconstrained system with *n* degrees of freedom as a constrained
system with *i* < *n* degrees of freedom. This can be quite difficult
for arbitrary system of constraints, since there is often no obvious
way of re-parameterizing systems involving constraint loops
[5,3].

A more flexible technique is to use Lagrange multiplier dynamic
constraints. Lagrange multiplier approaches maintain constraints by
introducing constraint forces into the original *n* degrees of freedom
system. Re-parameterization is not involved, and the constraints are
described in a modular way which makes the construction of arbitrary
systems of constraints straightforward. However, Lagrange multiplier
techniques involve solving for a vector of Lagrange multipliers
each time the constraint forces are evaluated, which
requires the solution of a system of linear equations. For a system
of *n* constraints, this generally requires solving an equation of the
form

Recently, Baraff [3] has described an *O*(*n*) Lagrange
multiplier solution for acyclic constraint problems. Central to this
technique is an always sparse formulation, and a relatively simple
*O*(*n*) technique for solving the resulting linear system.

Over the last year and a half, I have developed an iterative algorithm
for solving arbitrary, cyclic, Lagrange multiplier constraint
problems. This algorithm (which is central to my as-yet unpublished
M.S. Visualization Science thesis) appears to be *O*(*n*) with respect to
the number of constraints. However, I do not yet fully understand the
convergence properties of this algorithm, which (I believe) is
essentially a relaxation process. Gaining a better understanding of
this process would be a great benefit to my research.