Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access September 3, 2019

On computing the minimum singular value of a tensor sum

  • A. Ohashi EMAIL logo and T. Sogabe
From the journal Special Matrices


Recently, the Lanczos bidiagonalization method over tensor space has been proposed for computing the maximum and minimum singular values of a tensor sum T. The method over tensor space is practical in memory and has a simple implementation due to recent developments in tensor computations; however, there is still room for improvement in the convergence to the minimum singular value. This study reconstructed an invert Lanczos bidiagonalization method from vector space to tensor space. The resulting algorithm requires solving linear systems at each iteration step. Using standard direct methods, such as the LU decomposition for solving the linear systems requires a huge memory of O(n6). Therefore, this paper proposes a tensor-structure-preserving direct methods of T whose memory requirements are of O(n3), which is equivalent to the order of iterative methods. Numerical examples indicate that the number of iterations tends to be much smaller than that of the conventional method.


[1] A. Ohashi, T. Sogabe, On computing maximum/minimum singular values of a generalized tensor sum, Electron. T. Numer. Ana., 43(1), 244–254 (2015)Search in Google Scholar

[2] J. Ballani, L. Grasedyck, A projection method to solve linear systems in tensor format, Numer. Linear Algebr., 20(1), 27–43 (2013)10.1002/nla.1818Search in Google Scholar

[3] D. Kressner, C. Tobler, Krylov subspace methods for linear systems with tensor product structure, SIAM J. Matrix Anal. A., 31(4), 1688–1714 (2010)10.1137/090756843Search in Google Scholar

[4] T.G. Kolda, B.W. Bader, Tensor decompositions and applications, SIAM Rev., 51(3), 455–500 (2009)10.1137/07070111XSearch in Google Scholar

[5] G. Golub, W. Kahan, Calculating the singular values and pseudo-inverse of a matrix, SIAM J. Numer. Anal., 2(2), 205–224 (1965)10.1137/0702016Search in Google Scholar

[6] B.W. Li, S. Tian, Y.S. Sun, Z.M. Hu, Schur-decomposition for 3D matrix equations and its application in solving radiative discrete ordinates equations discretized by Chebyshev collocation spectral method, J. Comput. Phys., 229(4), 1198–1212 (2010)10.1016/ in Google Scholar

[7] E. Kokiopoulou, C. Bekas, E. Gallopoulos, Computing smallest singular triplets with implicitly restarted Lanczos bidiagonalization, Appl. Numer. Math., 49(1), 39–61 (2004)10.1016/j.apnum.2003.11.011Search in Google Scholar

[8] Z. Jia, D. Niu, Calculating the singular values and pseudo-inverse of a matrix, SIAM J. Sci. Comput., 32(2), 714–744 (2010)10.1137/080733383Search in Google Scholar

[9] D. Niu, X. Yuan, A harmonic Lanczos bidiagonalization method for computing interior singular triplets of large matrices, Appl. Math. Comput., 218(14), 7459–7467 (2012)10.1016/j.amc.2012.01.013Search in Google Scholar

[10] J. Baglama, L. Reichel, An implicitly restarted block Lanczos bidiagonalization method using Leja shifts, BIT, 53(2), 285–310 (2013)10.1007/s10543-012-0409-xSearch in Google Scholar

[11] D. Niu, X. Yuan, An implicitly restarted Lanczos bidiagonalization method with refined harmonic shifts for computing smallest singular triplets, J. Comput. Appl. Math., 260(14), 208-217 (2014)10.1016/ in Google Scholar

[12] S.C. Hawkins, K. Chen, An implicit wavelet sparse approximate inverse preconditioner, SIAM J. Sci. Comput., 27(2), 667-686 (2005)10.1137/S1064827503423500Search in Google Scholar

[13] A. Imakura, T. Sogabe, S.L. Zhang, An implicit wavelet sparse approximate inverse preconditioner using block finger pattern, Numer. Linear Algebr., 16(11–12), 915-928 (2009)10.1002/nla.657Search in Google Scholar

Received: 2018-11-22
Accepted: 2019-07-17
Published Online: 2019-09-03

© 2019 A. Ohashi et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 9.6.2023 from
Scroll to top button