Block LU factorization

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 founded in terms of the condition number kappa(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.

[1]  James R. Bunch,et al.  BLOCK METHODS FOR SOLVING SPARSE LINEAR SYSTEMS , 1976 .

[2]  N. Higham,et al.  COMPONENTWISE ERROR ANALYSIS FOR STATIONARY ITERATIVE METHODS , 1993 .

[3]  D. Sorensen,et al.  Block reduction of matrices to condensed forms for eigenvalue computations , 1990 .

[4]  Martin H. Schultz,et al.  Numerical Algorithms for Modern Parallel Computer Architectures , 1988 .

[5]  R.M.M. Mattheij.  The stability of LU-decompositions of block tridiagonal matrices , 1984, Bulletin of the Australian Mathematical Society.

[6]  Nicholas J. Higham,et al.  Algorithm 694: a collection of test matrices in MATLAB , 1991, TOMS.

[7]  Jack J. Dongarra,et al.  Solving linear systems on vector and shared memory computers , 1990 .

[8]  James Demmel,et al.  On a Block Implementation of Hessenberg Multishift QR Iteration , 1989, Int. J. High Speed Comput..

[9]  Jack J. Dongarra,et al.  Algorithm 679: A set of level 3 basic linear algebra subprograms: model implementation and test programs , 1990, TOMS.

[10]  Christian H. Bischof,et al.  The WY representation for products of householder matrices , 1985, PPSC.

[11]  B. Polman.  Incomplete blockwise factorizations of (block) H-matrices , 1987 .

[12]  Robert J. Plemmons,et al.  Linear algebra, Markov chains, and queueing models , 1993 .

[13]  N. Higham.  How Accurate is Gaussian Elimination , 1989 .

[14]  R. Varga,et al.  Block diagonally dominant matrices and generalizations of the Gerschgorin circle theorem , 1962 .

[15]  D. Rose,et al.  Marching Algorithms for Elliptic Boundary Value Problems. I: The Constant Coefficient Case , 1977 .

[17]  Robert Schreiber,et al.  Block Algorithms for Parallel Machines , 1988 .

[18]  L. Trefethen,et al.  Average-case stability of Gaussian elimination , 1990 .

[19]  K. A. Gallivan,et al.  Parallel Algorithms for Dense Linear Algebra Computations , 1990, SIAM Rev..

[20]  J. Varah,et al.  On the Solution of Block-Tridiagonal Systems Arising from Certain Finite-Difference Equations* , 1972 .

[21]  N. Higham.  Iterative refinement enhances the stability ofQR factorization methods for solving linear equations , 1991 .

[22]  James Demmel,et al.  Stability of block algorithms with fast level-3 BLAS , 1992, TOMS.

[23]  H Woz╠üniakowski.  Round-off error analysis of iterations for large linear systems , 1977 .

[24]  R.M.M. Mattheij.  Stability of Block $LU$-Decompositions of Matrices Arising from BVP , 1984 .

[25]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[26]  N. Higham,et al.  Stability of methods for matrix inversion , 1992 .

[27]  William Jalby,et al.  Impact of Hierarchical Memory Systems On Linear Algebra Algorithm Design , 1988 .

[28]  J. Varah.  A lower bound for the smallest singular value of a matrix , 1975 .

[29]  Jack J. Dongarra,et al.  A set of level 3 basic linear algebra subprograms , 1990, TOMS.

[30]  James Hardy Wilkinson,et al.  Error Analysis of Direct Methods of Matrix Inversion , 1961, JACM.