Abstract
Typically, parametrization captures the essence of a class of matrices, and its potential advantage is to make accurate computations possible. But, in general, parametrization suitable for accurate computations is not always easy to find. In this paper, we introduce a parametrization of a class of negative matrices to accurately solve the singular value problem. It is observed that, given a set of parameters, the associated nonsingular negative matrix can be orthogonally transformed into a totally nonnegative matrix in an implicit and subtraction-free way, which implies that such a set of parameters determines singular values of the associated negative matrix accurately. Based on this observation, a new O(n3) algorithm is designed to compute all the singular values, large and small, to high relative accuracy.
Similar content being viewed by others
References
Alfa, C. A. S., Xue, J., Ye, Q.: Entrywise perturbation theory for diagonally dominant M-matrices with applications. Numer. Math. 90, 401–141 (2002)
Alfa, C. A. S., Xue, J., Ye, Q.: Accurate computation of the smallest eigenvalue of a diagonally dominant M-matrix. Math. Comp. 71, 217–236 (2002)
Ando, Y.: Totally positive matrices. Linear Algebra Appl. 90, 165–219 (1987)
Bevilacqua, C.R., Bozzo, E., Corso, G. M. D.: Qd-type methods for quasiseparable matrices. SIAM J. Matrix Anal. Appl. 32, 722–747 (2011)
Castro-González, N., Geballos, J., Dopico, F.M., Molera, J.M.: Accurate solution of structured least squares problems via rank-revealing decompositions. SIAM J. Matrix Anal. Appl. 34, 1112–1128 (2013)
Dailey, C. M., Dopico, F. M., Ye, Q.: Relative perturbation theory for diagonally dominant matrices. SIAM J. Matrix Anal. Appl. 35, 1303–1328 (2014)
Dailey, C. M., Dopico, F. M., Ye, Q.: A new perturbation bound for the LDU factorization of diagonally dominant matrices. SIAM J. Matrix Anal. Appl. 35, 904–930 (2014)
Delgado, C. J., Peña, J. M.: Accurate computations with collocation matrices of q-Bernstein polynomials. SIAM J. Matrix Anal. Appl. 36, 880–893 (2015)
Demmel, J.: Accurate singular value decompositions of structured matrices. SIAM J. Matrix Anal. Appl. 21, 562–580 (1999)
Demmel, C. J., Gu, M., Eisenstat, S., Slapničar, I., Veselić, K., Drmač, Z.: Computing the singular value decomposition with high relative accuracy. Linear Algebra Appl. 299, 21–80 (1999)
Demmel, C. J., Dumitriu, I., Holtz, O., Koev, P.: Accurate and efficient expression evaluation and linear algebra. Acta Numer. 17, 87–145 (2008)
Dopico, F.M., Molera, J.M.: Accurate solution of structured linear systems via rank-revealing decompositions. IMA J. Numer Anal. 32, 1096–1116 (2012)
Dopico, C. F. M., Molera, J. M., Moro, J.: An orthogonal high relative accuracy algorithm for the symmetric eigenproblem. SIAM J. Matrix Anal. Appl. 25, 301–351 (2003)
Dopico, C. F. M., Pomés, K.: Structured eigenvalue condition numbers for parameterized quasiseparable matrices. Numer. Math. 134, 473–512 (2016)
Gasca, C. M., Peña, J. M.: Total positivity and Neville elimination. Linear Algebra Appl. 165, 25–44 (1992)
Gasca, C. M., Peña, J. M.: On factorizations of totally positive matrices, in Total positivity and its Applications. Kluwer Academic Publishers, Dordrecht (1996)
Huang, R.: A periodic qd-type reduction for computing eigenvalues of structured matrix products to high relative accuracy. J. Sci. Comput. 75, 1229–1261 (2018)
Huang, R.: Accurate solutions of weighted least squares problems associated with rank-structured matrices. Appl. Numer. Math. 146, 416–435 (2019)
Huang, R.: Accurate eigenvalues of some generalized sign regular matrices via relatively robust representations. J. Sci. Comput. 82, 78 (2020). https://doi.org/10.1007/s10915-020-01182-4
Huang, C. R.: A qd-type method for computing generalized singular values of BF matrix pairs with sign regularity to high relative accuracy. Math. Comp. 89, 229–252 (2020)
Koev, C. P.: Accurate eigenvalues and SVDs of totally nonnegative matrices. SIAM J Matrix Anal. Appl. 27, 1–23 (2005)
Koev, P.: Accurate computations with totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 29, 731–751 (2007)
Liu, Q.H., Li, X.X., Yan, J.: On the large time behavior of solutions for a class of time-dependent Hamilton-Jacobi equations. Sci. China Math. 59, 875–890 (2016)
Marco, C. A., Martínez, J. -J.: A fast and accurate algorithm for solving Bernstein-Vandermonde linear systems. Linear Algebra Appl. 422, 616–628 (2007)
Marco, C. A., Martínez, J. -J.: Accurate computations with Said-Ball-Vandemonde matrices. Linear Algebra Appl. 432, 2894–2908 (2010)
Marco, C. A., Martínez, J. -J.: Polynomial least squares fitting in the Bernstein basis. Linear Algebra Appl. 433, 1254–1264 (2010)
Marco, A., Martínez, J-J., Peña, J.M.: Accurate bidiagonal decomposition of totally positive Cauchy-Vandermonde matrices and applications. Linear Algebra Appl. 517, 63–84 (2017)
Vandebril, C. R., Van Barel, M., Mastronardi, N.: Matrix Computations and Semiseparable Matrices. Vol. I: Linear systems. Johns Hopkins University Press, Baltimore (2008)
Ye, C. Q.: Computing singular values of diagonally dominant matrices to high relative accuracy. Math. Comp. 77, 2195–2230 (2008)
Ye, Q.: Relative perturbation bounds for eigenvalues of symmetric positive definite diagonally dominant matrices. SIAM J. Matrix Anal. Appl. 31, 11–17 (2009)
Acknowledgements
The authors would like to thank the Editor and the anonymous referees for their valuable comments and suggestions which have helped improve the overall presentation of the paper.
Funding
The work of the first author was supported by the National Natural Science Foundation of China (Grant No. 11871020), the Natural Science Foundation for Distinguished Young Scholars of Hunan Province (Grant No. 2017JJ1025) and the Research Foundation of Education Bureau of Hunan Province (Grant No. 18A198). The work of the second author was supported by the National Science Foundation of China (Grant No. 11771100) and Laboratory of Mathematics for Nonlinear Science, Fudan University.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Raymond H. Chan
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Huang, R., Xue, J. Accurate singular values of a class of parameterized negative matrices. Adv Comput Math 47, 73 (2021). https://doi.org/10.1007/s10444-021-09898-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10444-021-09898-z