A new method for unconstrained optimization is presented. It consists of a modification of Powell's 1970 dogleg strategy with the approximate Hessian given by Davidson's 1975 updating scheme which uses the projections of $△x$ and $△g$ in updating H and G and optimizes the condition number of $H-1H+$. This new algorithm performs well without Powell's special iterations and singularity safeguards. Only symmetric and positive definite updates to the Hessian are used.