We derive conditions for the existence and investigate representations of $\{2,4\}$ and $\{2,3\}$-inverses with prescribed range $T$ and null space $S$. A general computational algorithm for $\{2,4\}$ and $\{2,3\}$ generalized inverses with given rank and prescribed range and null space is derived. The algorithm is derived generating the full-rank representations of these generalized inverses by means of various complete orthogonal matrix factorizations. More precisely, computational algorithm for $\{2,4\}$ and $\{2,3\}$ inverses of a given matrix $A$ is defined using an unique approach on SVD, QR and URV matrix decompositions of appropriately selected matrix $W$.