An efficient implementation of the block Gram--Schmidt method

Yoichi Matsuo, Takashi Nodera


The block Gram--Schmidt method computes the QR factorisation rapidly, but this is dependent on block size \(m\). We endeavor to determine the optimal \(m\) automatically during one execution. Our algorithm determines \(m\) through observing the relationship between computation time and complexity. Numerical experiments show that our proposed algorithms compute approximately twice as fast as the block Gram--Schmidt method for some block sizes, and is a viable option for computing the QR factorisation in a more stable and rapid manner.

  • Bjorck, A., Numerical Methods for Least Squares Problems, SIAM, (1996).
  • Elden, L., and Park, H., Block Downdating of Least Squares Solutions, SIAM J. Matrix Anal. Appl., 15:1018--1034 (1994). doi:10.1137/S089547989223691X
  • Runger, G., and Schwind, M., Comparison of Different Parallel Modified Gram--Schmidt Algorithms, Euro-Par 2005, LNCS 3648:826--836 (2005). doi:10.1007/11549468_90
  • Katagiri, T., Performance Evaluation of Parallel Gram--Schmidt Re-orthogonalization Methods, VECPAR 2002, LNCS 2565:302--314 (2003). doi:10.1007/3-540-36569-9_19
  • Matrix Market, Mathematical and Computational Sciences Division, Information Technology Laboratory of the National Institute of Standards and Technology, USA.
  • Matsuo, Y. and Nodera, T., The Optimal Block-Size for the Block Gram--Schmidt Orthogonalization, J. Sci. Tech, 49:348--354 (2011).
  • Moriya, K. and Nodera, T., The DEFLATED-GMRES(m, k) Method with Switching the Restart Frequency Dynamically, Numer. Linear Alg. Appl., 7:569--584 (2000). doi:10.1002/1099-1506(200010/12)7:7/8<569::AID-NLA213>3.0.CO;2-8
  • Moriya, K. and Nodera, T., Usage of the convergence test of the residual norm in the Tsuno--Nodera version of the GMRES algorithm, ANZIAM J., 49:293--308 (2007). doi:10.1017/S1446181100012852
  • Liu, Q., Modified Gram--Schmidt-based Methods for Block Downdating the Cholesky Factorization, J. Comput. Appl. Math., 235:1897--1905 (2011). doi:10.1016/
  • Saad, Y. and Schultz, M. H., GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems, SIAM J. Sci. Stat. Comput., 7:856--869 (1986). doi:10.1137/0907058
  • Shiroishi, J. and Nodera, T., A GMRES($m$) Method with Two Stage Deflated Preconditioners, ANZIAM J., 52:C222--C236 (2011).
  • Leon, S. J., Bjorck, A., and Gander, W., Gram--Schmidt Orthogonalization: 100 years and more, Numer. Linear Algebra Appl., 20:492--532 (2013). doi:10.1002/nla.1839
  • Stewart, G. W., Block Gram--Schmidt Orthogonalization, SIAM J. Sci. Comput., 31:761--775 (2008). doi:10.1137/070682563
  • Vanderstraeten, D., An Accurate Parallel Block Gram-Schmidt Algorithm without Reorthogonalization, Numer. Lin. Alg. Appl., 7:219--236 (2000). doi:10.1002/1099-1506(200005)7:4<219::AID-NLA196>3.0.CO;2-L
  • Yokozawa, T., Takahashi, T., Boku, T. and Sato, M., Efficient Parallel Implementation of Classical Gram-Schmidt Orthogonalization Using Matrix Multiplication, (in Japanese) Information Processing Society of Japan (IPSJ), Computing System, 1:61--72 (2008).


block Gram-Schmidt algorithm; optimal block size; parallel computing

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