A relaxed gradient based algorithm for solving generalized coupled Sylvester matrix equations. Issue 10 (July 2018)

• Record Type: Journal Article
• Authors: Sheng, Xingping
• Is Part Of: Journal of the Franklin Institute. Volume 355:Issue 10(2018)
• Page Start: 4282
• Page End: 4297
• Publication Date: 2018-07
• Subjects: Science -- Periodicals; Technology -- Periodicals; Patents -- United States -- Periodicals; 505
• DOI: 10.1016/j.jfranklin.2018.04.008
• Languages: English
• ISSNs: 0016-0032

Abstract

Abstract: The present work proposes a relaxed gradient based iterative (RGI) algorithm to find the solutions of coupled Sylvester matrix equations A X + Y B = C, D X + Y E = F . It is proved that the proposed iterative method can obtain the solutions of the coupled Sylvester matrix equations for any initial matrices X 0 and Y 0 . Next the RGI algorithm is extended to the generalized coupled Sylvester matrix equations of the form A i 1 X 1 B i 1 + A i 2 X 2 B i 2 + ⋯ + A i p X p B i p = C i, ( i = 1, 2, …, p ) . Then, we compare their convergence rate and find RGI is faster than GI, which has maximum convergence rate, under an appropriative positive number ω and the same convergence factor µ 1 and µ 2 . Finally, a numerical example is included to demonstrate that the introduced iterative algorithm is more efficient than the gradient based iterative (GI) algorithm of (Ding and Chen 2006) in speed, elapsed time and iterative steps.