Block LU Factorization
James Weldon Demmel, Nicholas John Higham, Robert Schreiber · 1992
. Many of the currently popular "block algorithms" are scalar algorithms in which the operations have been grouped and reordered into matrix operations. One genuine block algorithm in practical use is block LU factorization, and this has recently been shown by Demmel and Higham to be unstable in general. It is shown here that block LU factorization is stable if A is block diagonally dominant by columns. Moreover, for a general matrix the level of instability in block LU factorization can be bounded in terms of the condition number (A) and the growth factor for Gaussian elimination without pivoting. A consequence is that block LU factorization is stable for a matrix A that is symmetric positive definite or point diagonally dominant by rows or columns as long as A is well-conditioned. Key words: block algorithm, LAPACK, level 3 BLAS, iterative refinement, LU factorization, backward error analysis, block diagonal dominance. AMS(MOS) subject classifications. primary 65F05, 65F25, 65G05. 1...