  As the Cosmic Microwave Background (CMB) radiation is observed to higher and
higher angular resolution the size of the resulting datasets becomes a serious
constraint on their analysis. In particular current algorithms to determine the
location of, and curvature at, the peak of the power spectrum likelihood
function from a general $N_{p}$-pixel CMB sky map scale as $O(N_{p}^{3})$.
Moreover the current best algorithm --- the quadratic estimator --- is a
Newton-Raphson iterative scheme and so requires a `sufficiently good' starting
point to guarantee convergence to the true maximum. Here we present an
algorithm to calculate bounds on the likelihood function at any point in
parameter space using Gaussian quadrature and show that, judiciously applied,
it scales as only $O(N_{p}^{7/3})$. Although it provides no direct curvature
information we show how this approach is well-suited both to estimating
cosmological parameters directly and to providing a coarse map of the power
spectrum likelihood function from which to select the starting point for more
refined techniques.
