Pivot element

Abstract: We investigate several ways to improve the performance of sparse LU factorization with partial pivoting, as used to solve unsymmetric linear systems. We introduce the notion of unsymmetric supernodes to perform most of the numerical computation in dense matrix kernels. We introduce unsymmetric supernode-panel updates and two-dimensional data partitioning to better exploit the memory hierarchy. We use Gilbert and Peierls's depth-first search with Eisenstat and Liu's symmetric structural reductions to speed up symbolic factorization. We have developed a sparse LU code using all these ideas. We present experiments demonstrating that it is significantly faster than earlier partial pivoting codes. We also compare its performance with UMFPACK, which uses a multifrontal approach; our code is very competitive in time and storage requirements, especially for large problems.

Abstract: We present the main algorithmic features in the software package SuperLU_DIST, a distributed-memory sparse direct solver for large sets of linear equations. We give in detail our parallelization strategies, with a focus on scalability issues, and demonstrate the software's parallel performance and scalability on current machines. The solver is based on sparse Gaussian elimination, with an innovative static pivoting strategy proposed earlier by the authors. The main advantage of static pivoting over classical partial pivoting is that it permits a priori determination of data structures and communication patterns, which lets us exploit techniques used in parallel sparse Cholesky algorithms to better parallelize both LU decomposition and triangular solution on large-scale distributed machines.

Abstract: Although Gaussian elimination with partial pivoting is a robust algorithm to solve unsymmetric sparse linear systems of equations, it is difficult to implement efficiently on parallel machines because of its dynamic and somewhat unpredictable way of generating work and intermediate results at run time. In this paper, we present an efficient parallel algorithm that overcomes this difficulty. The high performance of our algorithm is achieved through (1) using a graph reduction technique and a supernode-panel computational kernel for high single processor utilization, and (2) scheduling two types of parallel tasks for a high level of concurrency. One such task is factoring the independent panels in the disjoint subtrees of the column elimination tree of $A$. Another task is updating a panel by previously computed supernodes. A scheduler assigns tasks to free processors dynamically and facilitates the smooth transition between the two types of parallel tasks. No global synchronization is used in the algorithm. The algorithm is well suited for shared memory machines (SMP) with a modest number of processors. We demonstrate 4- to 7-fold speedups on a range of 8 processor SMPs, and more on larger SMPs. One realistic problem arising from a 3-D flow calculation achieves factorization rates of 1.0, 2.5, 0.8, and 0.8 gigaflops on the 12 processor Power Challenge, 8 processor Cray C90, 16 processor Cray J90, and 8 processor AlphaServer 8400.

Abstract: Fast O(n 2 ) implementation of Gaussian elimination with partial pivoting is designed for matrices possessing Cauchy-like displacement structure. We show how Toeplitz-like, Toeplitz-plus-Hankel-like and Vandermonde-like matrices can be transformed into Cauchy-like matrices by using Discrete Fourier, Cosine or Sine Transform matrices. In particular this allows us to propose a new fast O(n 2 ) Toeplitz solver GKO, which incorporates partial pivoting. A large set of numerical examples showed that GKO demonstrated stable numerical behavior and can be recommended for solving linear systems, especially with nonsymmetric, indefinite and ill-conditioned positive definite Toeplitz matrices. It is also useful for block Toeplitz and mosaic Toeplitz ( Toeplitz-block ) matrices. The algorithms proposed in this paper suggest an attractive alternative to look-ahead approaches, where one has to jump over ill-conditioned leading submatrices, which in the worst case requires O(n 3 ) operations.