An efficient class of iterative methods for computing generalized outer inverse $${M_{T,S}^{(2)}}$$ M T , S ( 2 )
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An efficient class of iterative methods for computing generalized outer inverse MT(2) ,S Manpreet Kaur1 · Munish Kansal1 Received: 3 March 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract In this paper, we propose a new matrix iteration scheme for computing the generalized outer inverse for a given complex matrix. The convergence analysis of the proposed scheme is established under certain necessary conditions, which indicates that the methods possess at least fourth-order convergence. The theoretical discussions show that the convergence order improves from 4 to 5 for a particular parameter choice. We prove that the sequence of approximations generated by the family satisfies the commutative property of matrices, provided the initial matrix commutes with the matrix under consideration. Some real-world and academic problems are chosen to validate our methods for solving the linear systems arising from statically determinate truss problems, steady-state analysis of a system of reactors, and elliptic partial differential equations. Moreover, we include a wide variety of large sparse test matrices obtained from the matrix market library. The performance measures used are the number of iterations, computational order of convergence, residual norm, efficiency index, and the computational time. The numerical results obtained are compared with some of the existing robust methods. It is demonstrated that our method gives improved results in terms of computational speed and efficiency. Keywords Generalized outer inverse · Rank-deficient matrices · Computational efficiency · Convergence analysis · Schulz method Mathematics Subject Classification 15A09 · 65F30
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Munish Kansal [email protected] Manpreet Kaur [email protected]
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School of Mathematics, Thapar Institute of Engineering and Technology, Patiala, Punjab 147004, India
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M. Kaur, M. Kansal
1 Introduction The mathematical formulation of most scientific and engineering problems involve a system of simultaneous linear equations, whose coefficient matrix is either singular or nonsingular. Forsythe et al. [10] state that it is not advisable to evaluate the inverse of a nonsingular matrix in extensive practical computational problems. Moreover, the computation of the inverse of large sparse matrices is a more difficult and timeconsuming task in various fields of science and engineering. For instance, in the truss problem, which is related to the design of bridges and buildings that demands large spans and in the steady-state analysis of a system of reactors, we obtain the singular as well as large sparse matrices. To solve these problems, the traditional direct methods, such as singular value decomposition (SVD), LU factorization, are time-consuming and require the large memory for saving the data and processing. Moreover, the direct method as Gaussian elimination with partial pivoting (GEPP) cannot be used in parallel computers. Therefore, one has to use the iterative method for finding the generalized inverse of
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