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

Safety assessment of the commensal strain Bacteroides xylanisolvens DSM 23964

AbstractWe recently isolated and characterized the new strain Bacteroides xylanisolvens DSM 23964 and presented it as potential candidate for the first natural probiotic strain of the genus Bacteroides. In order to evaluate the safety of this strain for use in food, the following standard toxicity assays were conducted with this strain in both viable and pasteurized form: in vitro bacterial reverse mutation assay, in vitro chromosomal aberration assay, and 90day subchronic repeated oral toxicity studies in mice. No mutagenic, clastogenic, or toxic effects were detected even at extremely high doses. In addition, no clinical, hematological, ophthalmological, or histopathological abnormality could be observed after necropsy at any of the doses tested. Hence, the NOAEL could be estimated to be greater than 2.3×1011 CFUs, and 2.3×1014 for pasteurized bacteria calculated as equivalent for an average 70kg human being. In addition, the absence of any in vivo pathogenic properties of viable B. xylanisolvens DSM 23964 cells was confirmed by means of an intraperitoneal abscess formation model in mice which also demonstrated that the bacteria are easily eradicated by the host’s immune system. The obtained results support the assumed safety of B. xylanisolvens DSM 23964 for use in food

A collocation method based on the Bernoulli operational matrix for solving highorder linear complex differential equations in a rectangular domain,”

This paper contributes a new matrix method for the solution of high-order linear complex differential equations with variable coefficients in rectangular domains under the considered initial conditions. On the basis of the presented approach, the matrix forms of the Bernoulli polynomials and their derivatives are constructed, and then by substituting the collocation points into the matrix forms, the fundamental matrix equation is formed. This matrix equation corresponds to a system of linear algebraic equations. By solving this system, the unknown Bernoulli coefficients are determined and thus the approximate solutions are obtained. Also, an error analysis based on the use of the Bernoulli polynomials is provided under several mild conditions. To illustrate the efficiency of our method, some numerical examples are given

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

A Sparse-Sparse Iteration for Computing a Sparse Incomplete Factorization of the Inverse of an SPD Matrix

In this paper, a method via sparse-sparse iteration for computing a sparse incomplete factorization of the inverse of a symmetric positive definite matrix is proposed. The resulting factorized sparse approximate inverse is used as a preconditioner for solving symmetric positive definite linear systems of equations by using the preconditioned conjugate gradient algorithm. Some numerical experiments on test matrices from the Harwell-Boeing collection for comparing the numerical performance of the presented method with one available well-known algorithm are also given.Comment: 15 pages, 1 figur

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

Molecular biological Fingerprinting of Yersinia spp. - Strains from Broilers

Im Rahmen der vorliegenden Arbeit wurden aus einem experimentellen Infektionsversuch von Mastgeflügel insgesamt 747 verdächtige Yersinia spp.- Stämme aus 15 Durchgänge (A – O) nicht nur biochemisch sondern auch mittels PCR- Verfahren spezifiziert. Hierbei wurde das Genus (Yersinien), die Spezies (Yersinia enterocolitica) und die Pathogenität (yopT- Gen) der Feldstämme mittels Multiplex- PCR- Verfahren ermittelt. Die PCR- Ergebnisse zeigten, dass 596 (79,8%) Isolate als Yersinia- negativ, 130 (17,4%) Isolate als Y. enterocolitica (yopT+) und 21 (2,8%) Isolate als Yersinia enterocolitica (yopT-) anzusehen waren. Die meiste Yersinia enterocolitica- positive Stämme wurden 2. bis 10. Tag post infectionem gefunden. Ab dem 12. Tag post infectionem wurden keine Yersinia spp.- Stämme identifiziert. Insgesamt wurden 130 Yersinia spp.- Isolate (yopT+) und 21 Isolate (yopT-) für die Feintypisierung mittels PFGE- und AFLP- Methode ausgewählt. Als Referenzstamm wurden wie bei der PCR- Methode die Yersinia enterocolitica DSM 13030 (4/O:3) benutzt. Zusätzlich wurden drei Yersinia enterocolitica- Stämme unterschiedlicher Bioserotypen (4/O:3 yopT- negativ; O:5,27; 1A/O:7,8) und Yersinia pseudotuberculosis (ATCC 29833) mitgeführt um die genotypische Unterschiede zwischen den Referenzstämmen und innerhalb der Spezies Yersinia enterocolitica zu untersuchen. Die Ergebnisse der PFGE- und AFLP- Typisierung von Referenzstämmen zeigten, dass es mittels beider Methoden möglich war, innerhalb der Spezies Yersinia enterocolitica die Bioserotypen voneinander zu unterscheiden. Hierbei konnte festgestellt werden, dass das Profil der Bandenmuster von Yersinia enterocolitica- Stämmen des Bioserovars 4/O:3 sich zu 83,3 bis 87% ähnelte. Alle Yersinia spp.- Stämme, die in verschiedenen Versuchstagen nach der Infektion isoliert wurden, wiesen eine enge genetische Verwandtschaft zum Yersinia enterocolitica (DSM 13030)- Stamm auf und wurden aus diesem Grund als ein Genotyp bezeichnet. Die Bandenmuster- Ähnlichkeit der Yersinia spp.- Stämme betrug bei der PFGE- Methode (91,7 – 100%) und bei AFLP- Methode (90,1 – 99,8%). Die PFGE- und AFLP- Untersuchungen zeigten, dass der im Tierversuch eingesetzte Yersinia enterocolitica DSM 13030- Stamm in allen Isolaten, die mittels PCR bestätigt wurden, wieder zu finden war. Mit den beiden Methoden wurden genetische Fingerabdrücke der Isolate hergestellt, die sich in ihrer diskriminierenden Aussagekraft unterschieden. Beide Typisierungsmethoden basierten auf die Darstellung von Teilen (Fragmenten) des gesamten Genoms. Für den Vergleich der Bandenmuster wurden bei beiden Methoden die cut- off- Werte der Auswertungs- Software bei 90% festgelegt. Alle Isolate die eine Ähnlichkeit von mehr als 90% zeigten, wurden als genetisch eng miteinander verwandt bezeichnet (klonal). Der Vergleich der beiden Methoden miteinander zeigte, dass die AFLP- Methode aufgrund höherer Diskriminierungsstärke, ihrem geringen zeitlichen Aufwand, ihrer sehr guten Reproduzierbarkeit und besseren Handhabung eine hervorragende Methode für die Genotypisierung von Yersinia spp.- Stämmen darstellt.In the context of the presented work from an experimental infection attempt of broiler chickens altogether 747 suspicious Yersinia spp. strains from 15 passages (A to O) were specified not only biochemically but also by PCR procedure. The Genus (Yersinia), the species (Yersinia enterocolitica) and the pathogenicity (yopT gene) of the field strains were identified using multiplex PCR procedure. The PCR results showed that 596 (79.8%) isolates were Yersinia- negative, 130 (17.4%) isolates were Y. enterocolitica (yopT+) and 21 (2.8%) were Yersinia enterocolitica (yopT-). Most of the Y. enterocolitica positive strains were found in the 2nd to 10th day post infection. Starting from 12th day post infection no Yersinia spp. - strains were identified. Altogether 130 Yersinia enterocolitica isolates (yopT+) and 21 Yersinia enterocolitica isolates (yopT-) were selected for genotyping using PFGE and AFLP methods. As reference strains (as with the PCR method) Yersinia enterocolitica DSM 13030 (4/O: 3) were used. In addition, three Yersinia enterocolitica strains of different bioserotyps (4/O:3 yopT negative; O:5,27; 1A/O:7,8) and Yersinia pseudotuberculosis (ATCC 29833) were included to examined the genotype differences between the reference strains and the species Yersinia enterocolitica. The results of the PFGE and AFLP typing of reference strains showed that it was possible using both methods to differentiate bioserotyps within the species Yersinia enterocolitica from each other. On this occasion it could be registered that the profiles of band patterns of Yersinia enterocolitica bioserotyp 4/O: 3 demonstrated 83.3% to 87% similarity. All Yersinia spp. strains, which were isolated at different days after the infection, revealed a close genetic relationship to Yersinia enterocolitica DSM 13030 and for this reason were characterized as one genotype. The band pattern similarity of the Yersinia spp. strains amounted to 91.7 - 100% using the PFGE method and 90.1 - 99.8% using the AFLP method. The PFGE and AFLP investigations showed that the Yersinia enterocolitica DSM 13030 - strain used in the bioassays was regainable. That was confirmed by using PCR. The two genotyping methods used differed in their discriminating power. Both genotyping methods were based on the detection of parts (fragments) of the entire gene. During the comparison of the band patterns with both methods the cut-off value of the analysis software was specified with 90%. All isolates which showed a similarity of >90%, were described as genetically closely related (clone). The comparison of the two methods showed that the AFLP method, due to higher discrimination strength, their small temporal expenditure, their very good reproducibility and better handling proofed to be a very good method for the genotyping of Yersinia spp. strains

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