  A new approach to analyzing intrinsic properties of the Josephus function,
$J_{_k}$, is presented in this paper. The linear structure between extreme
points of $J_{_k}$ is fully revealed, leading to the design of an efficient
algorithm for evaluating $J_{_k}(n)$. Algebraic expressions that describe how
recursively compute extreme points, including fixed points, are derived. The
existence of consecutive extreme and also fixed points for all $k\geq 2$ is
proven as a consequence, which generalizes Knuth result for $k=2$. Moreover, an
extensive comparative numerical experiment is conducted to illustrate the
performance of the proposed algorithm for evaluating the Josephus function
compared to established algorithms. The results show that the proposed scheme
is highly effective in computing $J_{_k}(n)$ for large inputs.
