  We describe a three precision variant of Newton's method for nonlinear
equations. We evaluate the nonlinear residual in double precision, store the
Jacobian matrix in single precision, and solve the equation for the Newton step
with iterative refinement with a factorization in half precision. We analyze
the method as an inexact Newton method. This analysis shows that, except for
very poorly conditioned Jacobians, the number of nonlinear iterations needed is
the same that one would get if one stored and factored the Jacobian in double
precision. In many ill-conditioned cases one can use the low precision
factorization as a preconditioner for a GMRES iteration. That approach can
recover fast convergence of the nonlinear iteration. We present an example to
illustrate the results.
