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Volume 47 Number 5 Volume 47 Number 6 Volume 48 Number 1

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Research articles

ScienceAsia 47 (2021): 785-792 |doi: 10.2306/scienceasia1513-1874.2021.104

Iterative methods for the quadratic bilinear equation arising from a class of quadratic dynamic systems

Bo Yu, Ning Dong*, Qiong Tang

ABSTRACT:     To solve the quadratic bilinear equation arising from the dynamical system, a fixed-point iterative method and Newton’s method are considered in this paper. The iteration sequence generated by two methods, starting from the zero matrix is proved to be monotonically increasing and convergent to the minimal positive (semi-)definite solution. Besides, a double Newton step is given to accelerate the current Newton’s iteration when the equation is near or in the semi-stable case. Numerical experiments demonstrate the effectiveness of the fixed-point iteration and Newton’s method with the ADI preconditioning. In particular, the adapted double Newton step can efficiently decrease iterative steps of Newton’s method when the equation is semi-stable.

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a School of Science, Hunan University of Technology, Zhuzhou 412008 China

* Corresponding author, E-mail: dongning_158@sina.com

Received 25 Feb 2021, Accepted 3 Sep 2021