Direct methods for sparse matrices
Direct methods for sparse matrices
Solving tridiagonal systems on ensemble architectures
SIAM Journal on Scientific and Statistical Computing
Static and Dynamic Evaluation of Data Dependence Analysis Techniques
IEEE Transactions on Parallel and Distributed Systems
A Fast Direct Solution of Poisson's Equation Using Fourier Analysis
Journal of the ACM (JACM)
An Efficient Parallel Algorithm for the Solution of a Tridiagonal Linear System of Equations
Journal of the ACM (JACM)
The Solution of Tridiagonal Linear Systems on the CDC STAR 100 Computer
ACM Transactions on Mathematical Software (TOMS)
A Parallel Method for Tridiagonal Equations
ACM Transactions on Mathematical Software (TOMS)
A unifying graph model for designing parallel algorithms for tridiagonal systems
Parallel Computing - Linear systems and associated problems
Dependence graphs and compiler optimizations
POPL '81 Proceedings of the 8th ACM SIGPLAN-SIGACT symposium on Principles of programming languages
Computer Solution of Large Sparse Positive Definite
Computer Solution of Large Sparse Positive Definite
Integrating loop and data transformations for global optimization
Journal of Parallel and Distributed Computing
Designing Parallel Sparse Matrix Algorithms beyond Data Dependence Analysis
ICPPW '01 Proceedings of the 2001 International Conference on Parallel Processing Workshops
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Algorithms are often parallelized based on data dependence analysis manually or by means of parallel compilers. Some vector/matrix computations such as the matrix-vector products with simple data dependence structures (data parallelism) can be easily parallelized. For problems with more complicated data dependence structures, parallelization is less straightforward. The data dependence graph is a powerful means for designing and analyzing parallel algorithms. However, for sparse matrix computations, parallelization based on solely exploiting the existing parallelism in an algorithm does not always give satisfactory results. For example, the conventional Gaussian elimination algorithm for the solution of a tri-diagonal system is inherently sequential, so algorithms specially for parallel computation has to be designed. After briefly reviewing different parallelization approaches, a powerful graph formalism for designing parallel algorithms is introduced. This formalism will be discussed using a tri-diagonal system as an example. Its application to general matrix computations is also discussed. Its power in designing parallel algorithms beyond the ability of data dependence analysis is shown by means of a new algorithm called ACER (Alternating Cyclic Elimination and Reduction algorithm).