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 general class of arbitrary order iterative methods for computing generalized inverses

[EN] A family of iterative schemes for approximating the inverse and generalized inverse of a complex matrix is designed, having arbitrary order of convergence p. For each p, a class of iterative schemes appears, for which we analyze those elements able to converge with very far initial estimations. This class generalizes many known iterative methods which are obtained for particular values of the parameters. The order of convergence is stated in each case, depending on the first non-zero parameter. For different examples, the accessibility of some schemes, that is, the set of initial estimations leading to convergence, is analyzed in order to select those with wider sets. This wideness is related with the value of the first non-zero value of the parameters defining the method. Later on, some numerical examples (academic and also from signal processing) are provided to confirm the theoretical results and to show the feasibility and effectiveness of the new methods. (C) 2021 The Authors. Published by Elsevier Inc.This research was supported in part by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE) and in part by VIE from Instituto Tecnologico de Costa Rica (Research #1440037)Cordero Barbero, A.; Soto-Quiros, P.; Torregrosa Sánchez, JR. (2021). A general class of arbitrary order iterative methods for computing generalized inverses. Applied Mathematics and Computation. 409:1-18. https://doi.org/10.1016/j.amc.2021.126381S11840

Numerical iterative methods for nonlinear problems.

The primary focus of research in this thesis is to address the construction of iterative methods for nonlinear problems coming from different disciplines. The present manuscript sheds light on the development of iterative schemes for scalar nonlinear equations, for computing the generalized inverse of a matrix, for general classes of systems of nonlinear equations and specific systems of nonlinear equations associated with ordinary and partial differential equations. Our treatment of the considered iterative schemes consists of two parts: in the first called the ’construction part’ we define the solution method; in the second part we establish the proof of local convergence and we derive convergence-order, by using symbolic algebra tools. The quantitative measure in terms of floating-point operations and the quality of the computed solution, when real nonlinear problems are considered, provide the efficiency comparison among the proposed and the existing iterative schemes. In the case of systems of nonlinear equations, the multi-step extensions are formed in such a way that very economical iterative methods are provided, from a computational viewpoint. Especially in the multi-step versions of an iterative method for systems of nonlinear equations, the Jacobians inverses are avoided which make the iterative process computationally very fast. When considering special systems of nonlinear equations associated with ordinary and partial differential equations, we can use higher-order Frechet derivatives thanks to the special type of nonlinearity: from a computational viewpoint such an approach has to be avoided in the case of general systems of nonlinear equations due to the high computational cost. Aside from nonlinear equations, an efficient matrix iteration method is developed and implemented for the calculation of weighted Moore-Penrose inverse. Finally, a variety of nonlinear problems have been numerically tested in order to show the correctness and the computational efficiency of our developed iterative algorithms

Gaussian Belief Propagation Based Multiuser Detection

In this work, we present a novel construction for solving the linear multiuser detection problem using the Gaussian Belief Propagation algorithm. Our algorithm yields an efficient, iterative and distributed implementation of the MMSE detector. We compare our algorithm's performance to a recent result and show an improved memory consumption, reduced computation steps and a reduction in the number of sent messages. We prove that recent work by Montanari et al. is an instance of our general algorithm, providing new convergence results for both algorithms.Comment: 6 pages, 1 figures, appeared in the 2008 IEEE International Symposium on Information Theory, Toronto, July 200

Optimal low-rank approximations of Bayesian linear inverse problems

In the Bayesian approach to inverse problems, data are often informative, relative to the prior, only on a low-dimensional subspace of the parameter space. Significant computational savings can be achieved by using this subspace to characterize and approximate the posterior distribution of the parameters. We first investigate approximation of the posterior covariance matrix as a low-rank update of the prior covariance matrix. We prove optimality of a particular update, based on the leading eigendirections of the matrix pencil defined by the Hessian of the negative log-likelihood and the prior precision, for a broad class of loss functions. This class includes the F\"{o}rstner metric for symmetric positive definite matrices, as well as the Kullback-Leibler divergence and the Hellinger distance between the associated distributions. We also propose two fast approximations of the posterior mean and prove their optimality with respect to a weighted Bayes risk under squared-error loss. These approximations are deployed in an offline-online manner, where a more costly but data-independent offline calculation is followed by fast online evaluations. As a result, these approximations are particularly useful when repeated posterior mean evaluations are required for multiple data sets. We demonstrate our theoretical results with several numerical examples, including high-dimensional X-ray tomography and an inverse heat conduction problem. In both of these examples, the intrinsic low-dimensional structure of the inference problem can be exploited while producing results that are essentially indistinguishable from solutions computed in the full space