a b s t r a c tIn this paper, we propose a new algorithm to solve the Sylvester matrix equation XA + BX = C . The technique consists of orthogonal reduction of the matrix A to a block upper Hessenberg form P T AP = H and then solving the reduced equation, YH + BY = C for Y through recurrence relation, where Y = XP, and C = CP. We then recover the solution of the original problem via the relation X = YP T . The numerical results show the accuracy and the efficiency of the proposed algorithm. In addition, how the technique described can be applied to other matrix equations was shown.

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The plasmoid instability has revolutionized our understanding of magnetic reconnection in astrophysical environments. By preventing the formation of highly elongated reconnection layers, it is crucial in enabling the rapid energy conversion rates that are characteristic of many astrophysical phenomena. Most of the previous studies have focused on Sweet-Parker current sheets, which, however, are unattainable in typical astrophysical systems. Here, we derive a general set of scaling laws for the plasmoid instability in resistive and visco-resistive current sheets that evolve over time. Our method relies on a principle of least time that enables us to determine the properties of the reconnecting current sheet (aspect ratio and elapsed time) and the plasmoid instability (growth rate, wavenumber, inner layer width) at the end of the linear phase. After this phase the reconnecting current sheet is disrupted and fast reconnection can occur. The scaling laws of the plasmoid instability are not simple power laws, and depend on the Lundquist number (S), the magnetic Prandtl number (P m), the noise of the system (ψ 0), the characteristic rate of current sheet evolution (1/τ), as well as the thinning process. We also demonstrate that previous scalings are inapplicable to the vast majority of the astrophysical systems. We explore the implications of the new scaling relations in astrophysical systems such as the solar corona and the interstellar medium. In both these systems, we show that our scaling laws yield values for the growth rate, wavenumber, and aspect ratio that are much smaller than the Sweet-Parker based scalings.

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