The WY representation for products of householder matrices
SIAM Journal on Scientific and Statistical Computing - Papers from the Second Conference on Parallel Processing for Scientific Computin
Block reflectors: theory and computation
SIAM Journal on Numerical Analysis
The algebraic eigenvalue problem
The algebraic eigenvalue problem
A block QMR method for computing multiple simultaneous solutions to complex symmetric systems
SIAM Journal on Scientific Computing - Special issue on iterative methods in numerical linear algebra; selected papers from the Colorado conference
The computation of elementary unitary matrices
ACM Transactions on Mathematical Software (TOMS)
A Stabilized QMR Version of Block BiCG
SIAM Journal on Matrix Analysis and Applications
ABLE: An Adaptive Block Lanczos Method for Non-Hermitian Eigenvalue Problems
SIAM Journal on Matrix Analysis and Applications
A Lanczos-type method for multiple starting vectors
Mathematics of Computation
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
An iterative block lanczos method for the solution of large sparse symmetric eigenproblems.
An iterative block lanczos method for the solution of large sparse symmetric eigenproblems.
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We present an efficient block-wise update scheme for the QR decomposition of block tridiagonal and block Hessenberg matrices. For example, such matrices come up in generalizations of the Krylov space solvers MinRes, SymmLQ, GMRes, and QMR to block methods for linear systems of equations with multiple right-hand sides. In the non-block case it is very efficient (and, in fact, standard) to use Givens rotations for these QR decompositions. Normally, the same approach is also used with column-wise updates in the block case. However, we show that, even for small block sizes, block-wise updates using (in general, complex) Householder reflections instead of Givens rotations are far more efficient in this case, in particular if the unitary transformations that incorporate the reflections determined by a whole block are computed explicitly. Naturally, the bigger the block size the bigger the savings. We discuss the somewhat complicated algorithmic details of this block-wise update, and present numerical experiments on accuracy and timing for the various options (Givens vs. Householder, block-wise vs. column-wise update, explicit vs. implicit computation of unitary transformations). Our treatment allows variable block sizes and can be adapted to block Hessenberg matrices that do not have the special structure encountered in the above mentioned block Krylov space solvers.