Linear Convergence of the Collatz Method for Computing the Perron Eigenpair of a Primitive Dual Number Matrix

doi:10.1007/s42967-024-00426-6

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (1): 232-247.doi: 10.1007/s42967-024-00426-6

Linear Convergence of the Collatz Method for Computing the Perron Eigenpair of a Primitive Dual Number Matrix

Yongjun Chen, Liping Zhang

Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

收稿日期:2024-02-11 修回日期:2024-05-01 出版日期:2026-02-20 发布日期:2026-02-11
通讯作者: Liping Zhang,E-mail:lipingzhang@tsinghua.edu.cn E-mail:lipingzhang@tsinghua.edu.cn
作者简介:Yongjun Chen,E-mail:chenzhang@tsinghua.edu.cn
基金资助:
This work is supported by the National Natural Science Foundation of China (Grant no. 12171271).

Linear Convergence of the Collatz Method for Computing the Perron Eigenpair of a Primitive Dual Number Matrix

Yongjun Chen, Liping Zhang

Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

Received:2024-02-11 Revised:2024-05-01 Online:2026-02-20 Published:2026-02-11
Contact: Liping Zhang,E-mail:lipingzhang@tsinghua.edu.cn E-mail:lipingzhang@tsinghua.edu.cn
Supported by:
This work is supported by the National Natural Science Foundation of China (Grant no. 12171271).

摘要/Abstract

摘要： Very recently, Qi and Cui extended the Perron-Frobenius theory to dual number matrices with primitive and irreducible nonnegative standard parts and proved that they have a Perron eigenpair and a Perron-Frobenius eigenpair. The Collatz method was also extended to find the Perron eigenpair. Qi and Cui proposed two conjectures. One is the k-order power of a dual number matrix, which tends to zero if and only if the spectral radius of its standard part is less than one, and another is the linear convergence of the Collatz method. In this paper, we confirm these conjectures and provide the theoretical proof. The main contribution is to show that the Collatz method R-linearly converges with an explicit rate.

关键词: Dual numbers, Dual primitive matrices, Eigenvalues, Collatz method, Linear convergence

Abstract: Very recently, Qi and Cui extended the Perron-Frobenius theory to dual number matrices with primitive and irreducible nonnegative standard parts and proved that they have a Perron eigenpair and a Perron-Frobenius eigenpair. The Collatz method was also extended to find the Perron eigenpair. Qi and Cui proposed two conjectures. One is the k-order power of a dual number matrix, which tends to zero if and only if the spectral radius of its standard part is less than one, and another is the linear convergence of the Collatz method. In this paper, we confirm these conjectures and provide the theoretical proof. The main contribution is to show that the Collatz method R-linearly converges with an explicit rate.

Key words: Dual numbers, Dual primitive matrices, Eigenvalues, Collatz method, Linear convergence

中图分类号:

Yongjun Chen, Liping Zhang. Linear Convergence of the Collatz Method for Computing the Perron Eigenpair of a Primitive Dual Number Matrix[J]. Communications on Applied Mathematics and Computation, 2026, 8(1): 232-247.

参考文献

[1] Angeles, J.: The dual generalized inverses and their applications in kinematic synthesis. In: Lenarcic, J., Husty, M. (eds) Latest Advances in Robot Kinematics, pp. 1-12. Springer, Dordrecht (2012)
[2] Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia (1994)
[3] Cui, C., Qi, L.: A power method for computing the dominant eigenvalue of a dual quaternion Hermitian matrix. arXiv:2304.04355 (2023)
[4] Gu, Y.L., Luh, L.: Dual-number transformation and its applications to robotics. IEEE J. Robot. Autom. 3, 615-623 (1987)
[5] Ling, C., He, H., Qi, L.: Singular values of dual quaternion matrices and their low-rank approximations. Numer. Funct. Anal. Optim. 43, 1423-1458 (2022)
[6] Ling, C., Qi, L., Yan, H.: Minimax principle for eigenvalues of dual quaternion Hermitian matrices and generalized inverses of dual quaternion matrices. Numer. Funct. Anal. Optim. 44, 1371-1394 (2023)
[7] Pennestri, E., Valentini, P.P.: Linear dual algebra algorithms and their applications to kinematics. In: Bottasso, C.L. (ed) Multibody Dynamics, pp. 207-229. Springer, Dordrecht (2009)
[8] Pennestri, E., Valentini, P.P., De Falco, D., Angeles, J.: Dual Cayley-Klein parameters and Möbius transform: theory and applications. Mech. Mach. Theory 106, 50-67 (2016)
[9] Qi, L.: Standard dual quaternion optimization and its applications in hand-eye calibration and SLAM. Commun. Appl. Math. Comput. 5, 1469-1483 (2023)
[10] Qi, L.: Motion, dual quaternion optimization and motion optimization. Commun. Appl. Math. Comput. (2023). https://doi.org/10.1007/s42967-023-00262-0
[11] Qi, L., Alexander, D.M., Chen, Z., Ling, C., Luo, Z.: Low rank approximation of dual complex matrices. arXiv:2201.12781 (2022)
[12] Qi, L., Cui, C.: Eigenvalues and Jordan forms of dual complex matrices. Commun. Appl. Math. Comput. (2023). https://doi.org/10.1007/s42967-023-00299-1
[13] Qi, L., Cui, C.: Dual Markov chain and dual number matrices with nonnegative standard parts. Commun. Appl. Math. Comput. (2024). https://doi.org/10.1007/s42967-024-00388-9
[14] Qi, L., Ling, C., Yan, H.: Dual quaternions and dual quaternion vectors. Commun. Appl. Math. Comput. 4, 1494-1508 (2022)
[15] Qi, L., Luo, Z.: Eigenvalues and singular value decomposition of dual complex matrices. arXiv:2110.02050 (2021)
[16] Qi, L., Luo, Z.: Eigenvalues and singular value decomposition of dual quaternion matrices. Pac. J. Optim. 19, 257-272 (2023)
[17] Qi, L., Wang, X., Cui, C.: Augmented quaternion and augmented unit quaternion optimization. arXiv:2301.03174v2 (2023)
[18] Qi, L., Wang, X., Luo, Z.: Dual quaternion matrices in multi-agent formation control. Commun. Math. Sci. 21, 1865-1874 (2023)
[19] Varga, R.: Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs (1962)
[20] Wang, H.: Characterization and properties of the MPDGI and DMPGI. Mech. Mach. Theory 158, 104212 (2021)
[21] Wang, H., Cui, C., Wei, Y.: The QLY least-squares and the QLY least squares minimal-norm of linear dual least squares problems. Linear Multilinear Algebra 72, 1985-2002 (2024). https://doi.org/10.1080/03081087.2023.2223348

Linear Convergence of the Collatz Method for Computing the Perron Eigenpair of a Primitive Dual Number Matrix

Linear Convergence of the Collatz Method for Computing the Perron Eigenpair of a Primitive Dual Number Matrix

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