This paper focused on accurate singular value computation for fg-Vandermonde-type (fg-VT) matrices, which are widely used in interpolation and other applications. We first derived explicit expressions for the initial minors of fg-VT matrices, then presented parametrization formulas for their corresponding matrices. Based on these results, we proposed an algorithm with high relative accuracy for computing all singular values. Additionally, we developed perturbation theory and error analysis to further assess the accuracy of our approach. Finally, numerical experiments were conducted to verify high accuracy of the algorithm.
Citation: Jiaxian Huang, Jiaqi Shu. Accurate computations of singular values for fg-Vandermonde-type matrices[J]. AIMS Mathematics, 2026, 11(7): 21235-21250. doi: 10.3934/math.2026862
This paper focused on accurate singular value computation for fg-Vandermonde-type (fg-VT) matrices, which are widely used in interpolation and other applications. We first derived explicit expressions for the initial minors of fg-VT matrices, then presented parametrization formulas for their corresponding matrices. Based on these results, we proposed an algorithm with high relative accuracy for computing all singular values. Additionally, we developed perturbation theory and error analysis to further assess the accuracy of our approach. Finally, numerical experiments were conducted to verify high accuracy of the algorithm.
| [1] |
P. Koev, Accurate eigenvalues and exact zero Jordan blocks of totally nonnegative matrices, Numer. Math., 141 (2019), 693–713. https://doi.org/10.1007/s00211-019-01022-0 doi: 10.1007/s00211-019-01022-0
|
| [2] |
X. X. Ma, Y. Q. Xiao, Z. Yang, Computing eigenvalues for products of two classes of sign regular matrices to high relative accuracy, Linear Algebra Appl., 707 (2025), 80–106. https://doi.org/10.1016/j.laa.2024.11.006 doi: 10.1016/j.laa.2024.11.006
|
| [3] |
A. Marco, J. J. Martínez, J. M. Peña, Accurate bidiagonal decomposition of totally positive Cauchy–Vandermonde matrices and applications, Linear Algebra Appl., 517 (2017), 63–84. https://doi.org/10.1016/j.laa.2016.12.003 doi: 10.1016/j.laa.2016.12.003
|
| [4] |
R. Huang, D. Chu, Relative perturbation analysis for eigenvalues and singular values of totally nonpositive matrices, SIAM J. Matrix Anal. Appl., 36 (2015), 476–495. https://doi.org/10.1137/140995702 doi: 10.1137/140995702
|
| [5] |
R. Huang, D. Chu, Computing singular value decompositions of parameterized matrices with total nonpositivity to high relative accuracy, J. Sci. Comput., 71 (2017), 682–711. https://doi.org/10.1007/s10915-016-0315-5 doi: 10.1007/s10915-016-0315-5
|
| [6] |
R. Huang, A qd-type method for computing generalized singular values of BF matrix pairs with sign regularity to high relative accuracy, Math. Comp., 89 (2020), 229–252. https://doi.org/10.1090/mcom/3444 doi: 10.1090/mcom/3444
|
| [7] |
P. Koev, F. Dopico, Accurate eigenvalues of certain sign regular matrices, Linear Algebra Appl., 424 (2007), 435–447. https://doi.org/10.1016/j.laa.2007.02.012 doi: 10.1016/j.laa.2007.02.012
|
| [8] |
Z. Yang, T. Chen, The accurate and efficient solutions of linear systems for generalized sign regular matrices with certain signature, J. Comput. Appl. Math., 431 (2023), 115280. https://doi.org/10.1016/j.cam.2023.115280 doi: 10.1016/j.cam.2023.115280
|
| [9] |
R. Huang, Accurate solutions of weighted least squares problems associated with rank-structured matrices, Appl. Numer. Math., 146 (2019), 416–435. https://doi.org/10.1016/j.apnum.2019.08.001 doi: 10.1016/j.apnum.2019.08.001
|
| [10] |
Z. Yang, R. Huang, W. Zhu, Accurate computations for eigenvalues of products of Cauchy-polynomial-Vandermonde matrices, Numer. Algor., 85 (2020), 329–351. https://doi.org/10.1007/s11075-019-00816-5 doi: 10.1007/s11075-019-00816-5
|
| [11] |
R. Huang, J. Xue, Accurate singular values of a class of parameterized negative matrices, Adv. Comput. Math., 47 (2021), 73. https://doi.org/10.1007/s10444-021-09898-z doi: 10.1007/s10444-021-09898-z
|
| [12] |
A. Marco, J. J. Martínez, Accurate computations with totally positive Bernstein-Vandermonde matrices, Electron. J. Linear Algebra, 26 (2013), 357–380. https://doi.org/10.13001/1081-3810.1658 doi: 10.13001/1081-3810.1658
|
| [13] |
E. Mainar, J. M. Peña, Accurate computations with collocation matrices of a general class of bases, Numer. Linear Algebra Appl., 25 (2018), e2184. https://doi.org/10.1002/nla.2184 doi: 10.1002/nla.2184
|
| [14] | V. Y. Pan, Structured matrices and polynomials: unified superfast algorithms, Birkhäuser Boston, MA, 2012. https://doi.org/10.1007/978-1-4612-0129-8 |
| [15] | G. M. Phillips, Interpolation and approximation by polynomials, Springer New York, NY, 2003. https://doi.org/10.1007/b97417 |
| [16] |
M. Dailey, F. M. Dopico, Q. Ye, Relative perturbation theory for diagonally dominant matrices, SIAM J. Matrix Anal. Appl., 35 (2014), 1303–1328. https://doi.org/10.1137/130943613 doi: 10.1137/130943613
|
| [17] |
M. S. Arif, K. Abodayeh, Y. Nawaz, Numerical schemes for fractional energy balance model of climate change with diffusion effects, Emerging Sci. J., 7 (2023), 808–820. https://doi.org/10.28991/ESJ-2023-07-03-011 doi: 10.28991/ESJ-2023-07-03-011
|
| [18] |
Z. Yang, S. Liu, C. Cao, Accurate computations of singular values and linear systems for Polynomial-Vandermonde-type matrices, Calcolo, 61 (2024), 62. https://doi.org/10.1007/s10092-024-00615-y doi: 10.1007/s10092-024-00615-y
|
Figures(3) / Tables(1)
Jiaxian Huang, Jiaqi Shu. Accurate computations of singular values for fg-Vandermonde-type matrices[J]. AIMS Mathematics, 2026, 11(7): 21235-21250. doi: 10.3934/math.2026862
DownLoad: