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
A storage-efficient WY representation for products of householder transformations
SIAM Journal on Scientific and Statistical Computing
A set of level 3 basic linear algebra subprograms
ACM Transactions on Mathematical Software (TOMS)
LAPACK's user's guide
ScaLAPACK user's guide
A framework for symmetric band reduction
ACM Transactions on Mathematical Software (TOMS)
The Shifted Hessenberg System Solve Computation
The Shifted Hessenberg System Solve Computation
LAPACK Working Note 33: Robust Incremental Condition Estimation
LAPACK Working Note 33: Robust Incremental Condition Estimation
LAPACK Working Note 32: Generalizing Incremental Condition Estimation
LAPACK Working Note 32: Generalizing Incremental Condition Estimation
On the Failure of Rank-Revealing QR Factorization Software -- A Case Study
ACM Transactions on Mathematical Software (TOMS)
A note on shifted Hessenberg systems and frequency response computation
ACM Transactions on Mathematical Software (TOMS)
Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis
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This article proposes an efficient algorithm for reducing matrices to generalized Hessenberg form by unitary similarity, and recommends using it as a preprocessor in a variety of applications. To illustrate its usefulness, two cases from control theory are analyzed in detail: a solution procedure for a sequence of shifted linear systems with multiple right hand sides (e.g. evaluating the transfer function of a MIMO LTI dynamical system at many points) and computation of the staircase form. The proposed algorithm for the generalized Hessenberg reduction uses two levels of aggregation of Householder reflectors, thus allowing efficient BLAS 3-based computation. Another level of aggregation is introduced when solving many shifted systems by processing the shifts in batches. Numerical experiments confirm that the proposed methods have superior efficiency.