  In this paper, an efficient modified Newton type algorithm is proposed for
nonlinear unconstrianed optimization problems. The modified Hessian is a convex
combination of the identity matrix (for steepest descent algorithm) and the
Hessian matrix (for Newton algorithm). The coefficients of the convex
combination are dynamically chosen in every iteration. The algorithm is proved
to be globally and quadratically convergent for (convex and nonconvex)
nonlinear functions. Efficient implementation is described. Numerical test on
widely used CUTE test problems is conducted for the new algorithm. The test
results are compared with those obtained by MATLAB optimization toolbox
function {\tt fminunc}. The test results are also compared with those obtained
by some established and state-of-the-art algorithms, such as a limited memory
BFGS, a descent and conjugate gradient algorithm, and a limited memory and
descent conjugate gradient algorithm. The comparisons show that the new
algorithm is promising.
