Comp. Appl. Math. https://doi.org/10.1007/s40314-018-0654-y A globally convergent variant of mid-point method for finding the matrix sign Nahid Zainali 1 · Taher Lotfi 1 Received: 11 March 2018 / Revised: 3 May 2018 / Accepted: 19 May 2018 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018 Abstract In this research, an efficient variant of mid-point iterative method is given for computing the sign of a square complex matrix having no pure imaginary eigenvalues. It is proven that the method is new and has global convergence with high order of convergence seven. To justify the effectiveness of the new scheme, several comparisons for matrices of different sizes are worked out to show that the new method is efficient. Keywords Matrix sign · Mid-point method · Global convergence · Iterative methods · High order Mathematics Subject Classification 65F60 1 Introductory notes The sign function for the scalar case is defined by sign(z ) =  1, Re(z )> 0, −1, Re(z )< 0, (1) wherein z ∈ C is not located on the imaginary axis. Roberts in 1980 for the first time extended this definition for matrices, which has several important applications in scientific computing, (see e.g., Benner and Quintana-Ortí 1999; Howland 1983; Kenney and Laub 1995) and the references therein. For example, the off-diagonal decay of the matrix sign function is also a well-developed area of study in statistics and statistical physics Hardin et al. (2013). An Communicated by Jinyun Yuan. B Taher Lotfi lotfi@iauh.ac.ir; lotfitaher@yahoo.com Nahid Zainali nahid.zainali@yahoo.com 1 Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran 123