Practical quasi-Newton methods are usually used to solve unconstrained opti mization problems successfully see for example Fletcher (22]). Since these meth ods require the storage of matrices, they cannot be applied to large-scale problems with limited memory storage. Thus, for this type of problems, the methods are implemented in a way without storing any matrix explicitly. A particular method will be considered, which is known as the limited memory L-BFGS method of No cedal (for detail, see for example Nocedal and Wright [6]). Although the L-BFGS Hessian consists of details less than those of the BFGS Hessian, it usually works well in practice except for certain type of problems. To rectify this difficulty, for various line searches, we consider the possibility of improving the L-BFGS Hessian based on the following. We introduce the damped technique of AlBaali-Powell to the L-BFGS method and use a measure for selecting the stored vector pairs ap propriately. The proposed improved algorithms will be tested, by applying them to a set of standard test problems and the numerical results will be analyzed to illustrate the behaviour of the algorithms. We will propose a procedure for accelerating the L-BFGS in certain cases substantially.