Drazin inverse based numerical methods for singular linear differential systems

[EN] In this paper, numerical methods for the solution of linear singular differential system are analyzed. The numerical solution of initial value problem by means of a corresponding finite difference approach and a possible implementation of the product Drazin inverse by vector is discussed. Examples of index-1 and index-2 DAEs have been studied numerically.This paper was partially supported by Grant GS1 DGI MTM2010-18228, by Ministry of Education of Argentina (PPUA, grant Resol. 228, SPU, 14-15-222) and by Universidad Nacional de La Pampa, Facultad de Ingenieria (grant Resol. No. 049/11).Coll Aliaga, PDC.; Ginestar Peiro, D.; Sánchez Juan, E.; Thome Coppo, NJ. (2012). Drazin inverse based numerical methods for singular linear differential systems. ADVANCES IN ENGINEERING SOFTWARE. (50):37-43. https://doi.org/10.1016/j.advengsoft.2012.04.001S37435

A Matrix Iteration for Finding Drazin Inverse with Ninth-Order Convergence

The aim of this paper is twofold. First, a matrix iteration for finding approximate inverses of nonsingular square matrices is constructed. Second, how the new method could be applied for computing the Drazin inverse is discussed. It is theoretically proven that the contributed method possesses the convergence rate nine. Numerical studies are brought forward to support the analytical parts

Generalized inverses estimations by means of iterative methods with memory

[EN] A secant-type method is designed for approximating the inverse and some generalized inverses of a complex matrix A. For a nonsingular matrix, the proposed method gives us an approximation of the inverse and, when the matrix is singular, an approximation of the Moore-Penrose inverse and Drazin inverse are obtained. The convergence and the order of convergence is presented in each case. Some numerical tests allowed us to confirm the theoretical results and to compare the performance of our method with other known ones. With these results, the iterative methods with memory appear for the first time for estimating the solution of a nonlinear matrix equations.This research was supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE), Generalitat Valenciana PROMETEO/2016/089, and FONDOCYT 029-2018 Republica Dominicana.Artidiello, S.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vassileva, MP. (2020). Generalized inverses estimations by means of iterative methods with memory. Mathematics. 8(1):1-13. https://doi.org/10.3390/math8010002S11381Li, X., & Wei, Y. (2004). Iterative methods for the Drazin inverse of a matrix with a complex spectrum. Applied Mathematics and Computation, 147(3), 855-862. doi:10.1016/s0096-3003(02)00817-2Li, H.-B., Huang, T.-Z., Zhang, Y., Liu, X.-P., & Gu, T.-X. (2011). Chebyshev-type methods and preconditioning techniques. Applied Mathematics and Computation, 218(2), 260-270. doi:10.1016/j.amc.2011.05.036Soleymani, 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.2012.08.004Soleymani, 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/s12190-014-0759-4Gu, X.-M., Huang, T.-Z., Ji, C.-C., Carpentieri, B., & Alikhanov, A. A. (2017). Fast Iterative Method with a Second-Order Implicit Difference Scheme for Time-Space Fractional Convection–Diffusion Equation. Journal of Scientific Computing, 72(3), 957-985. doi:10.1007/s10915-017-0388-9Li, M., Gu, X.-M., Huang, C., Fei, M., & Zhang, G. (2018). A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations. Journal of Computational Physics, 358, 256-282. doi:10.1016/j.jcp.2017.12.044Schulz, G. (1933). Iterative Berechung der reziproken Matrix. ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik, 13(1), 57-59. doi:10.1002/zamm.19330130111Li, W., & Li, Z. (2010). A family of iterative methods for computing the approximate inverse of a square matrix and inner inverse of a non-square matrix. Applied Mathematics and Computation, 215(9), 3433-3442. doi:10.1016/j.amc.2009.10.038Chen, H., & Wang, Y. (2011). A Family of higher-order convergent iterative methods for computing the Moore–Penrose inverse. Applied Mathematics and Computation, 218(8), 4012-4016. doi:10.1016/j.amc.2011.05.066Monsalve, M., & Raydan, M. (2011). A Secant Method for Nonlinear Matrix Problems. Numerical Linear Algebra in Signals, Systems and Control, 387-402. doi:10.1007/978-94-007-0602-6_18Jay, L. O. (2001). Bit Numerical Mathematics, 41(2), 422-429. doi:10.1023/a:1021902825707Cordero, A., & Torregrosa, J. R. (2007). Variants of Newton’s Method using fifth-order quadrature formulas. Applied Mathematics and Computation, 190(1), 686-698. doi:10.1016/j.amc.2007.01.06

A Higher Order Iterative Method for Computing the Drazin Inverse

A method with high convergence rate for finding approximate inverses of nonsingular matrices is suggested and established analytically. An extension of the introduced computational scheme to general square matrices is defined. The extended method could be used for finding the Drazin inverse. The application of the scheme on large sparse test matrices alongside the use in preconditioning of linear system of equations will be presented to clarify the contribution of the paper

Iterative Methods for the Drazin Inverse of a Matrix With a Complex Spectrum

Iterative methods for the Drazin inverse of a square matrix with a real spectrum have been developed recently. These methods are generalized in the case of matrices with complex spectra. Semiiterative method for the Drazin inverse is also discussed. Numerical examples are given to illustrate the theoretic results