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LU Factorization

LU factorization is a variant of Gaussian elimination, and has numerous variants that depend on the order of calculations in addition to other implementation factors. There any numerous algorithms for LU factorization of dense matrices, that have three nested for loops around the statement:

 

In this statement, the indices run:

Sparse matrix LU factorization can mirror any dense LU factorization algorithm, although generally a sparse matrix algorithm has only one explicit for loop, which can be for any single index in the dense case. The remaining indices are examined only for non-zero values in the original matrix or for non-zero values that will occur from fillin in the matrix. Sparse matrix fillin occurs when a value that formally was zero becomes non-zero in the process of factoring the matrix. Fillin can be controlled in sparse LU factorization of a matrix by ordering the matrix before factorization [5]. Fillin is discussed in greater detail below.

The most significant aspect of parallel sparse LU factorization is that the sparsity structure can be exploited to offer more parallelism than is available with dense matrix solvers. Parallelism in dense matrix factorization is achieved by distributing the data in a manner that the calculations in one of the for loops in equation 4 can be performed in parallel. Due to precedence relationships in the algorithm, this is generally the inner most for loop. Sparse factorization algorithms have inadequate calculations using the inner most index for efficient parallelism; however, sparse matrices have additional parallelism as a result of the nature of the data and the precedence rules governing the order of calculations. Instead of just parallelizing the inner most for loop as in parallel dense matrix factorization, entire independent portions of a sparse matrix can be factored in parallel --- especially when the sparse matrix has been ordered into block-diagonal-bordered form.



next up previous
Next: Precedence in LU Up: Background Previous: Background



David P. Koester
Sun Oct 22 16:27:33 EDT 1995