On inexact alternating direction implicit iteration for continuous Sylvester equations

In this paper, we study the alternating direction implicit (ADI) iteration for solving the continuous Sylvester equation AX +XB=C, where the coefficient matrices A and B are assumed to be positive semi-definite matrices (not necessarily Hermitian), and at least one of them to be positive definite. We first analyze the convergence of the ADI iteration for solving such a class of Sylvester equations, then derive an upper bound for the contraction factor of this ADI iteration. To reduce its computational complexity, we further propose an inexact variant of the ADI iteration, which employs some Krylov subspace methods as its inner iteration processes at each step of the outer ADI iteration. The convergence is also analyzed in detail. The numerical experiments are given to illustrate the effectiveness of both ADI and inexact ADI iterations.The authors are grateful to the anonymous referees for their valuable comments and suggestions which improved the quality of this paper. Also, The authors would like to thank the supports of the National Natural Science Foundation of China under Grant No. 11371075, the Hunan Key Laboratory of mathematical modeling and analysis in engineering, the Portuguese Funds through FCT-Fundacao para a Ciencia, within the Project UIDB/00013/2020 and UIDP/00013/2020. This work does not have any conflicts of interest

Climatology of the semidiurnal tide at 52-56°N from ground-based radar wind measurements 1985-1995

Long-term wind measurements carried out at 6 northern midlatitude sites (Saskatoon, Sheffield, Juliusruh, Collm, Obninsk, Kazan) are investigated to establish a climatology of the semidiurnal tide in the mesopause region for the narrow latitudinal range between 52°N and 56°N. Comparison of zonal and meridional components shows that in general the horizontal components are circularly polarized. Intercomparison of amplitudes and phases generally shows good agreement between the results from the different measuring systems. The results are compared with an empirical model of the semidiurnal tide. The longitudinal variation of the semidiurnal tide is small in summer, but the tidal amplitudes in winter are larger at Saskatoon and Kazan, compared with the results from the other sites. The possible influence of wave-tidal interaction in the stratosphere on the interannual variability of this difference is discussed