  The Rational Hybrid Monte Carlo (RHMC) algorithm extends the Hybrid Monte
Carlo algorithm for lattice QCD simulations to situations involving fractional
powers of the determinant of the quadratic Dirac operator. This avoids the
updating increment ($dt$) dependence of observables which plagues the Hybrid
Molecular-dynamics (HMD) method. The RHMC algorithm uses rational
approximations to fractional powers of the quadratic Dirac operator. Such
approximations are only available when positive upper and lower bounds to the
operator's spectrum are known. We apply the RHMC algorithm to simulations of 2
theories for which a positive lower spectral bound is unknown: lattice QCD with
staggered quarks at finite isospin chemical potential and lattice QCD with
massless staggered quarks and chiral 4-fermion interactions ($\chi$QCD). A
choice of lower bound is made in each case, and the properties of the RHMC
simulations these define are studied. Justification of our choices of lower
bounds is made by comparing measurements with those from HMD simulations, and
by comparing different choices of lower bounds.
