Abstract
In the last years, much effort has been devoted to high relative accuracy algorithms for the singular value problem. However, such algorithms have been constructed only for a few classes of matrices with certain structure or properties. In this paper, we study a different class of matrices—parameterized matrices with total nonpositivity. We develop a new algorithm to compute singular value decompositions of such matrices to high relative accuracy. Our numerical results confirm the high relative accuracy of our algorithm.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Dr. Xiaowei Zhang did all numerical experiments in this section. The authors thank him for his kind assistance.
References
Alfa, A.S., Xue, J., Ye, Q.: Accurate computation of the smallest eigenvalue of a diagonally dominant M-matrix. Math. Comput. 71, 217–236 (2002)
Ando, T.: Totally positive matrices. Linear Algebra Appl. 90, 165–219 (1987)
Castro-González, N., Ceballos, 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)
Cortés, V., Peña, J.M.: A stable test for strict sign regularity. Math. Comput. 77, 2155–2171 (2008)
Dailey, 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, 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, J., Gragg, W.: On computing accurate singular values and eigenvalues of acyclic matrices. Linear Algebra Appl. 185, 203–218 (1993)
Demmel, 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, J., Kahan, W.: Accurate singular values of bidiagonal matrices. SIAM J. Sci Stat. Comput. 11, 873–912 (1990)
Demmel, J., Koev, P.: Accurate SVDs of weakly diagonally dominant \(M\)-matrices. Numer. Math. 98, 99–104 (2004)
Dhilion, I.S., Parlett, B.N.: Orthogonal eigenvectors and relative gaps. SIAM J. Matrix Anal. Appl. 25, 858–899 (2004)
Dopico, F.M., Koev, P., Molera, J.M.: Implicit standard Jacobi Givens high relative accuracy. Numer. Math. 113, 519–553 (2009)
Fallat, S.M., Johnson, C.R.: Totally nonnegative matrices. Princeton University Press, Princeton (2011)
Gasca, M., Peña, J.M.: Totally positivity, QR factorization and Neville elimination. SIAM J. Matrix Anal. Appl. 14, 1132–1140 (1993)
Gasca, M., Peña, J.M.: A matricial description of Neville elimination with applications to total positivity. Linear Algebra Appl. 202, 33–35 (1994)
Gasca, M., Peña, J.M.: On factorizations of totally positive matrices. In: Total Positivity and its Applications. Kluwer Academic Publishers, Dordrecht (1996)
Huang, R.: A test and bidiagonal factorization for certain sign regular matrices. Linear Algebra Appl. 438, 1240–1251 (2013)
Huang, R., Chu, D.: Total nonpositivity of nonsingular matrices. Linear Algebra Appl. 432, 2931–2941 (2010)
Huang, R., Chu, D.: Relative perturbation analysis for eigenvalues and singular values of totally nonpositive matrices. SIAM J. Matrix Anal. Appl. 36, 476–495 (2015)
Huang, R., Liu, J.Z.: On Schur complements of sign regular matrices of order \(k\). Linear Algebra Appl. 433, 143–148 (2010)
Karlin, S.: Total Positivity. Stanford University Press, Stanford (1968)
Koev, P.: Accurate eigenvalues and SVDs of totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 27, 1–23 (2005)
Peláez, M.J., Moro, J.: Accurate factorization and eigenvalues algorithms for symmetric DSTU and TSC matrices. SIAM J. Matrix Anal. Appl. 28, 1173–1198 (2006)
Peña, J.M.: Shape Preserving Representations in Computer-Aided Geometric Design. Nova Science Publishers, Commack, New York (1999)
Peña, J.M.: On nonsinuglar sign regular matrices. Linear Algebra Appl. 359, 91–100 (2003)
Vandebril, R., Barel, M.V., Mastronardi, N.: Matrix Computations and Simiseparable Matrices. Vol. II: Eigenvalue and Singular Value Methods. Johns Hopkins University Press, Baltimore (2008)
Xue, J., Xu, S., Li, R.-C.: Accurate solutions of M-Matrix algebraic riccati equations. Numer. Math. 120, 671–700 (2012)
Ye, Q.: Computing singular values of diagonally dominant matrices to high relative accuracy. Math. Comput. 77, 2195–2230 (2008)
Acknowledgements
The authors would like to thank the Associate Editor and the anonymous referees for their valuable comments and suggestions which have helped to improve the overall presentation of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author was supported by the National Natural Science Foundation of China (Grant No. 11471279) and the Research Foundation of Education Bureau of Hunan Province (Grant No. 14B178). The second author was supported in part by NUS research Grant R-146-000-187-112.
Rights and permissions
About this article
Cite this article
Huang, R., Chu, D. Computing Singular Value Decompositions of Parameterized Matrices with Total Nonpositivity to High Relative Accuracy. J Sci Comput 71, 682–711 (2017). https://doi.org/10.1007/s10915-016-0315-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-016-0315-5