Schur complement factorizations and parallel O(log N) algorithms for computation of operational space mass matrix and its inverse

In this paper new factorization techniques for computation of the operational space mass matrix (Λ) and its inverse (Λ^-1) are developed. Starting with a new factorization of the inverse of mass matrix (M^-1) in the form of Schur complement as M^-1=C-B^TA^-1B, where A and B are block tridiagonal matrices and C is a tridiagonal matrix, similar factorizations for Λ and Λ^-1 are derived. Specifically, the Schur complement factorizations of Λ^-1 and Λ are derived as Λ^-1=D-E^TA^-1E and Λ=G-R ^TS^-1R, where E and R are sparse matrices and D and G are 6×6 matrices. The Schur complement factorization provides a unified framework for computation of M^-1, Λ^-1 , and Λ. The main advantage of these new factorizations is that they are highly efficient for parallel computation. With O(N) processors, the computation of Λ^-1 and Λ as well as their operator applications can be performed in O(log N) steps