Fourier operational matrices of differentiation . . .

This paper introduces Fourier operational matrices of differentiation and transmission for solving high-order linear differential and difference equations with constant coefficients. Moreover, we extend our methods for generalized Pantograph equations with variable coefficients by using Legendre Gauss collocation nodes. In the case of numerical solution of Pantograph equation, an error problem is constructed by means of the residual function and this error problem is solved by using the mentioned collocation scheme. When the exact solution of the problem is not known, the absolute errors can be computed approximately by the numerical solution of the error problem. The reliability and efficiency of the presented approaches are demonstrated by several numerical examples, and also the results are compared with different methods

LSMR Iterative Method for General Coupled Matrix Equations

By extending the idea of LSMR method, we present an iterative method to solve the general coupled matrix equations ∑k=1qAikXkBik=Ci, i=1,2,…,p, (including the generalized (coupled) Lyapunov and Sylvester matrix equations as special cases) over some constrained matrix groups (X1,X2,…,Xq), such as symmetric, generalized bisymmetric, and (R,S)-symmetric matrix groups. By this iterative method, for any initial matrix group (X1(0),X2(0),…,Xq(0)), a solution group (X1*,X2*,…,Xq*) can be obtained within finite iteration steps in absence of round-off errors, and the minimum Frobenius norm solution or the minimum Frobenius norm least-squares solution group can be derived when an appropriate initial iterative matrix group is chosen. In addition, the optimal approximation solution group to a given matrix group (X¯1,X¯2,…,X¯q) in the Frobenius norm can be obtained by finding the least Frobenius norm solution group of new general coupled matrix equations. Finally, numerical examples are given to illustrate the effectiveness of the presented method

New breakdown-free variant of AINV method for nonsymmetric positive definite matrices,

Abstract This paper proposes a new breakdown-free preconditioning technique, called SAINV-NS, of the AINV method of Benzi and Tuma for nonsymmetric positive definite matrices. The resulting preconditioner which is an incomplete factorization of the inverse of a nonsymmetric matrix will be used as an explicit right preconditioner for QMR, BiCGSTAB and GMRES(m) methods. The preconditoner is reliable (pivot breakdown can not occur) and effective at reducing the number of iterations. Some numerical experiments on test matrices are presented to show the efficiency of the new method and comparing to the AINV-A algorithm

Accelerated Circulant and Skew Circulant Splitting Methods for Hermitian Positive Definite Toeplitz Systems

We study the CSCS method for large Hermitian positive definite Toeplitz linear systems, which first appears in Ng's paper published in (Ng, 2003), and CSCS stands for circulant and skew circulant splitting of the coefficient matrix . In this paper, we present a new iteration method for the numerical solution of Hermitian positive definite Toeplitz systems of linear equations. The method is a two-parameter generation of the CSCS method such that when the two parameters involved are equal, it coincides with the CSCS method. We discuss the convergence property and optimal parameters of this method. Finally, we extend our method to BTTB matrices. Numerical experiments are presented to show the effectiveness of our new method

Approximating the inverse and the Moore‐Penrose inverse of complex matrices

[EN] A parametric family of fourth-order schemes for computing the inverse and the Moore-Penrose inverse of a complex matrix is designed. A particular value of the parameter allows us to obtain a fifth-order method. Convergence analysis of the different methods is studied. Every iteration of the proposed schemes involves four matrix multiplications. A numerical comparison with other known methods, in terms of the average number of matrix multiplications and the mean of CPU time, is presented.This research was partially supported by Ministerio de Ciencia, Innovación y Universidades PGC2018-095896-B-C22, Generalitat Valenciana PROMETEO/2016/089, and Schlumberger Foundation-Faculty for Future Program. On the other hand, the authors would like to thank the anonymous referees for their comments and suggestions that have improved the final version of this manuscript.Cordero Barbero, A.; Torregrosa Sánchez, JR.; Zafar, F. (2019). Approximating the inverse and the Moore-Penrose inverse of complex matrices. Mathematical Methods in the Applied Sciences. 42(17):5920-5928. https://doi.org/10.1002/mma.5879592059284217Soleymani, F., & Stanimirović, P. S. (2013). A Higher Order Iterative Method for Computing the Drazin Inverse. The Scientific World Journal, 2013, 1-11. doi:10.1155/2013/708647Weiguo, L., Juan, L., & Tiantian, Q. (2013). A family of iterative methods for computing Moore–Penrose inverse of a matrix. Linear Algebra and its Applications, 438(1), 47-56. doi:10.1016/j.laa.2012.08.004Higham, N. J. (2008). Functions of Matrices. doi:10.1137/1.9780898717778Schulz, G. (1933). Iterative Berechung der reziproken Matrix. ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik, 13(1), 57-59. doi:10.1002/zamm.19330130111Soleymani, F., Salmani, H., & Rasouli, M. (2014). Finding the Moore–Penrose inverse by a new matrix iteration. Journal of Applied Mathematics and Computing, 47(1-2), 33-48. doi:10.1007/s12190-014-0759-4Jay, L. O. (2001). Bit Numerical Mathematics, 41(2), 422-429. doi:10.1023/a:102190282570