On accurate error estimates for the quaternion least squares and weighted least squares problems

ABSTRACT

In this paper, we study error estimates of the quaternion least squares problem and the quaternion weighted least squares problem. For different cases, we define different condition numbers and effective condition numbers. We propose upper bounds of absolute and relative errors of the minimal norm LS solution and the general LS solution of the quaternion least squares problem and the quaternion weighted least squares problem, respectively, by using different condition numbers and effective condition numbers to make error estimates sharper.

KEYWORDS:

• Error estimate
• effective condition number
• quaternion matrix
• least squares problem
• weighted least squares problem

2010 AMS SUBJECT CLASSIFICATIONS:

• 65F20
• 65F35
• 15A12
• 15A18

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Natural Science Foundation of Shandong [grant ZR2012FQ005], Science and Technology Project of Department of Education, Shandong Province [grant J15LI10], and the Science Foundation of Liaocheng University [grants 31805, 318011318].