Solving Sparse Linear Least Squares Problems on Some Supercomputers by Using a Sequence of Large Dense Blocks

It is relatively easy to obtain good computational speed on many high-speed computers when computations with dense matrix techniques are performed. This explains why very eecient subroutines have been developed for dense matrix computations. On some computers, as on the new CRAY models, the speed obtained in this situation is near the top-performance given by the manifactures. However, both the number of arithmetic operations and the number of locations needed in the computer memory grow very quickly when pure dense matrix technique is used and when the involved matrices become large. These two facts may cause problems; also when the new modern computers, which are very fast and very big, are used. Assume that the matrices involved are sparse, and that the sparsity is exploited. Then both the number of arithmetic operations and the number of locations in the computer memory are reduced very considerably, because one operates only with the non-zero elements and stores only the non-zero elements. The reductions could be made more considerable by dropping "small" elements (some iterative procedure has to be used in the latter case in an attempt to regain the accuracy lost by dropping small elements). However, some price is to be paid for the reduction of the number of arithmetic operations and for the reduction of the storage needed. The price is a great degradation of the speed of computations when supercomputers are used (which is due to the use of indirect addresses, to the need to insert new non-zero elements in the sparse storage scheme, to the lack of data locality, etc.). On many high-speed computers the dense matrix technique is more preferrable than the sparse matrix technique when the matrices are not very large, because the high computational speed compensates fully the disadvantages of using more arithmetic operations and more storage. If the sparse matrices are very large, one can still use dense matrix technique, but the computations must be organized as a sequence of tasks in each of which a dense block is treated. The blocks in this sequence must be large enough to achieve a high computational speed, but not too large, because too large blocks will lead to a very quick increase of both the computing time and the storage). A special algorithm, LORA, must be used to reorder the matrix (before the start of the computations) to a form that allows us to construct a sequence …