Schur Complement Factorizations and Parallel O(Log N) Algorithms for Computation of Operational Space Mass Matrix and Its Inverse

ing this paper new factorization techniques for computation of the Operational Space Mass Matrix (A) and its inverse (A-l) are developed. Starting with a new factorization of the inverse of mass matrix (M-l) in the form of Schur Complement as M-l = G’ %)TsI-lB, where d and 8 are block tridlagonal matrices and G is a tridiagonal matrix, similar factorization for A and A-i are derived. Specifically, the Schur Complement factorizatlons of A-l and A are derived -1 as A = D &Td-i& and A = S RT$’-lR, where .5 and R are sparse matrices and D and S are 6x6 matrices. The Schur Complement factorization provides a unified framework for computation of M -1 , A-l, and A. It also provides a deeper physical insight as well as simple physical interpretations of these factorization. However, the main advantage of these new factorizatlons 1s that they are highly efficient for parallel computation. With O(N) processors, the computation of A-l and A as well as their operator applications can be performed in O(Log N) steps. This represents both timeand processoroptlmal parallel algorithms for their computations. To our knowledge, these are the first parallel algorithms that achieve the time lower bound of O(Log N) in the computation,