Recursive computation of matrix triangular factorizations

Recursion is a new effective method for computing dense linear algebra. It brings about automatic matrix blocking and allows for efficient utilization of memory hierarchies of today's high-performance computers, and hence it improves the efficiency of computation. The recursive algorithms for matrix triangular factorizations that are used to solve linear systems of equations are studied in this paper, including LU factorization for a general matrix and Cholesky factorization for a symmetric positive-definite matrix. Detailed derivation and description of the recursive LU and Cholesky algorithms are presented. The recursive algorithms are implemented based on calling the efficient level-3 BLAS routines. FORTRAN90 language that supports recursion is used to write the program codes. The developed algorithms and programs are tested and compared with the LAPACK algorithms of LU and Cholesky factorizations. The results show that the presented recursive LU and Cholesky algorithms are 15% to 20% faster than the corresponding LAPACK algorithms for larger matrices.