The closure property of the Schur complement for Nekrasove matrices and its applications in solving large linear systems
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The closure property of the Schur complement for Nekrasove matrices and its applications in solving large linear systems with Schur-based method Jianzhou Liu1,2 · Yebo Xiong1 · Yun Liu1 Received: 18 July 2020 / Revised: 11 September 2020 / Accepted: 25 September 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, several conditions are presented to keep the Schur complement via a nonleading principle submatrix of some special matrices including Nekrasov matrices being a Nekrasov matrix, which is useful in the Schur-based method for solving large linear equations. And we give some infinity norm bounds for the inverse of Nekrasov matrices and its Schur complement to help measure whether the classical iterative methods are convergent or not. At last, in the applications of solving large linear equations by Schur-based method, some numerical experiments are presented to show the efficiency and superiority of our results. Keywords Nekrasov matrices · Schur complement · Schur-based method · Linear systems · Infinity norm bound Mathematics Subject Classification 15A06 · 93C05 · 15B99
1 Introduction In general, it is hard to obtain the exact solutions to all differential equations. Thus, variational method, finite difference method, and finite element method, etc. are applied to compute their numerical solutions. Then, the problem to solve some differential equations, such as the Euler equation, the elliptic equation and the Dirichlet problem (Elsner and Mehrmann 1991; Hadjidimos 1983), and so on often transforms into a problem of solving linear systems. Luckily, the coefficient matrix is usually a special matrix like diagonal dominant matrix, H-matrix or Nekrasov matrix, and so on (Forsythe 1953; Karlqvist 1952).
Communicated by Zhong-Zhi Bai.
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Jianzhou Liu [email protected]
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School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan, People’s Republic of China
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Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, People’s Republic of China 0123456789().: V,-vol
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For instance, considering the two-point boundary value problem: (Ames 2014; Thomas 2013): du du d p +r Lu = − + qu = f , a < x < b, dx dx dx with u(a) = α, u(b) = β, and p is a first-order continuous differentiable function on the region [a, b], p(x) ≥ pmin > 0, r , q, and f are continuous on the region [a, b], and α and β are constant numbers. The two-point boundary problem is useful in automatic control, design of various electronic devices, calculation of trajectory, study of the stability of aircraft and missile flight, and study of the stability of chemical reaction process (Braun and Golubitsky 1983; Chicone 2006; Nieto and Rodríguez-López 2005). And when the direct differential method with uniform mesh is applied to the problem, we get a linear equations: u i+1 − u i−1 1 + qi u i [ p 1 u i+1 − ( pi+ 1 + pi− 1 )u i + pi− 1 u i−1 ] + ri 2 2 2 h 2 i+ 2 2h i = 1, 2
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