5. Dynamic Programming Approaches to Optimizing the Blocking Strategy for Basic Matrix Decompositions

In this chapter, we survey several approaches to optimizing the blocking strategy for basic matrix decompositions, such as LU, Cholesky, and QR. Conventional blocking strategies such as fixed-size blocking and recursive blocking are widely used to optimize the performance of these decompositions. However, these strategies have only a small number of parameters such as the block size or the level of recursion and are not sufficiently flexible to exploit the performance of modern high-performance architectures. As such, several attempts have been made to define a much larger class of strategies and to choose the best strategy among them according to the target machine and the matrix size. The number of candidate strategies is usually exponential in the size of the matrix. However, with the use of dynamic programming, the cost of optimization can be reduced to a realistic level. As representatives of such approaches, we survey variable-size blocking, generalized recursive blocking, and the combination of variable-size blocking and the TSQR algorithm. Directions for future research are also discussed.