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Updating the QR decomposition of block tridiagonal and block Hessenberg matrices generated by block Krylov space methods

by M. H. Gutknecht and T. Schmelzer

(Report number 2005-10)

Abstract
For MinRes and SymmLQ it is essential to compute the QR decompositions of tridiagonal coefficient matrices gained in the Lanczos process. Likewise, for GMRes one has to find those of Hessenberg matrices. These QR decompositions are computed by an update scheme where in every step a single Givens rotation is constructed. Generalizing this approach we introduce a block-wise update scheme for the QR decomposition of the block tridiagonal and block Hessenberg matrices that come up in generalizations of MinRes, SymmLQ, GMRes, and QMR to block methods for systems with multiple right-hand sides. Using (in general, complex) Householder reflections instead of Givens rotations is seen to be much more efficient in the block case. Some implementation details and numerical experiments on accuracy and timing are given. In particular, we compare our method with the one based on Givens rotations that has been used in a version of block QMR. Our treatment includes the option of deflation, that is, the successive reduction of the block size due to linear dependencies.

Keywords: block Arnoldi process, block Lanczos process, block Krylov space method, block MinRes, block SymmLQ, block GMRes, block QMR, block tridiagonal matrix, block Hessenberg matrix, QR decomposition, QR factorization

@Techreport{GS05_349,
  author = {M. H. Gutknecht and T. Schmelzer},
  title = {Updating the QR decomposition of block tridiagonal and block Hessenberg matrices generated by block Krylov space methods },
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2005-10},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2005/2005-10.pdf },
  year = {2005}
}

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