The main objective of the presented work is the development of an efficient method for the calculation of the inverse kinematic of anthropomorphic robotic hands. Our algorithms is able to treat all fingers as one set when generating a given grasp configuration or in-hand manipulation trajectories, resulting in a simultaneous solution for the wrist and the five fingertips. To reach this objective a reduced computation cost of the Extended Selectively Damped Least Squares method was developed. The high cost of the original method arises from computing the Singular Value Decomposition of the extended Jacobian matrix. To avoid this problem, we adopt a new estimation algorithm that computes only the relevant smallest singular values and corresponding vectors. Unlike the original damping least squares approach, our approach associates a specific damping factor for each estimated singular value and uses a Cholesky decomposition for the joint angles calculations. In addition to avoiding singularities, the proposed method supports joint limits avoidance and can be applied to any redundant robotic system. The computational aspects of different approaches are first discussed through the implementation on the one finger case. The obtained results are then extended to cover all degrees of freedom of the whole hand as a single multibody system. The approach is tested in simulation and experimentally on a real anthropomorphic robotic hand, where the inverse kinematics was used to grasp and turn a cylindrical object, a knob of a mixer.