Improving the condition number of a simple eigenvalue by a rank one matrix

In this work a technique to improve the condition number si of a simple eigenvalue lambda(i) of a matrix A is an element of C-nxn is given. This technique obtains a rank one updated matrix that is similar to A with the eigenvalue condition number of lambda(i) equal to one. More precisely, the similar updated matrix A + v(i)q*, where Av(i) = lambda(i)v(i) and q is a fixed vector, has s(i) = 1 and the remaining condition numbers are at most equal to the corresponding initial condition numbers. Moreover an expression to compute the vector q, using only the eigenvalue lambda(i) and its eigenvector v(i), is given. (C) 2016 Elsevier Ltd. All rights reserved.Supported by the Spanish DGI grant MTM2013-43678-P.Bru García, R.; Cantó Colomina, R.; Urbano Salvador, AM. (2016). Improving the condition number of a simple eigenvalue by a rank one matrix. Applied Mathematics Letters. 58:7-12. https://doi.org/10.1016/j.aml.2016.01.010S7125

On the intersection of the classes of doubly diagonally dominant matrices and S-strictly diagonally dominant matrices

We denote by H0 the subclass of H-matrices consisting of all the matrices that lay simultaneously on the classes of doubly diagonally dominant (DDD) matrices (A = [aij ] ∈ Cn×n : |aii||ajj | ≥ k =i |aik| k =j |ajk|, i = j) and S-strictly diagonally dominant (S-SDD) matrices. Notice that strictly doubly diagonally dominant matrices (also called Ostrowsky matrices) are a subclass of H0. Strictly diagonally dominant matrices (SDD) are also a subclass of H0. In this paper we analyze some properties of the class H0 = DDD ∩ S-SDD

Determinación de H-matrices

Cuando la matriz de comparación de una matriz A, M(A), es M-matriz, se dice que A es H-matriz. Este tipo de matrices aparece, por ejemplo, en la discretización por elementos finitos de ciertas ecuaciones parabólicas no lineales. Además, esta característica garantiza la existencia de precondicionadores para la resolución de sistemas lineales por métodos iterativos de tipo Krylov. Aunque son muchas las caracterizaciones de H-matriz que provienen de las de M-matriz invertible, una H-matriz invertible puede tener una matriz de comparación singular. En este trabajo utilizamos una caracterización de H-matriz con elementos diagonales no nulos, tanto si M(A) es invertible como singular, basada en que ρ ≤ 1, siendo ρ el radio espectral de la matriz de Jacobi de M(A). Proponemos entonces algoritmos para acotar el valor de ρ y concluir si la matriz A es H-matriz o no. Se proponen algoritmos para aproximar el valor de ρ y el vector de Perron asociado y se demuestra su convergencia para matrices irreducibles

Improved balanced incomplete factorization

[EN] . In this paper we improve the BIF algorithm which computes simultaneously the LU factors (direct factors) of a given matrix and their inverses (inverse factors). This algorithm was introduced in [R. Bru, J. Mar´ın, J. Mas, and M. T˚uma, SIAM J. Sci. Comput., 30 (2008), pp. 2302– 2318]. The improvements are based on a deeper understanding of the inverse Sherman–Morrison (ISM) decomposition, and they provide a new insight into the BIF decomposition. In particular, it is shown that a slight algorithmic reformulation of the basic algorithm implies that the direct and inverse factors numerically influence each other even without any dropping for incompleteness. Algorithmically, the nonsymmetric version of the improved BIF algorithm is formulated. Numerical experiments show very high robustness of the incomplete implementation of the algorithm used for preconditioning nonsymmetric linear systemsReceived by the editors January 26, 2009; accepted for publication (in revised form) by V. Simoncini June 1, 2010; published electronically August 12, 2010. This work was supported by Spanish grant MTM 2007-64477, by project IAA100300802 of the Grant Agency of the Academy of Sciences of the Czech Republic, and partially also by the International Collaboration Support M100300902 of AS CR.Bru García, R.; Marín Mateos-Aparicio, J.; Mas Marí, J.; Tuma, M. (2010). Improved balanced incomplete factorization. SIAM Journal on Matrix Analysis and Applications. 31(5):2431-2452. https://doi.org/10.1137/090747804S2431245231

Combined matrices and conditioning

[EN] In this work, we study a lower bound of the condition number of a matrix by its combined matrix. In particular, we construct a special combined matrix in such a way that the sums of its columns are lower bounds of the condition number of the matrix. Cases for special matrices as unitary matrices are considered.We thank the anonymous referees for their valuable reports that have improved this work. This work was partially sup-ported by Spanish grants MTM2017-85669-P and MTM2017-90682-REDT from "Ministerio de Economa y Competitividad", and by the Dominican Republic FONDOCYT grant number 2016-2017-057.Bru García, R.; Gasso Matoses, MT.; Santana-De Asis, MDJ. (2022). Combined matrices and conditioning. Applied Mathematics and Computation. 412:1-8. https://doi.org/10.1016/j.amc.2021.126549S1841

Low-rank update of preconditioners for the nonlinear Richard's equation

Preconditioners for the Conjugate Gradient method are studied to solve the Newton system with symmetric positive definite (SPD) Jacobian. Following the theoretical work in Bergamaschi et al. (2011) [4] we start from a given approximation of the inverse of the initial Jacobian, and we construct a sequence of preconditioners by means of a low rank update, for the linearized systems arising in the Picard Newton solution of the nonlinear discretized Richards equation. Numerical results onto a very large and realistic test case show that the proposed approach is more efficient, in terms of iteration number and CPU time, as compared to computing the preconditioner of choice at every nonlinear iteration.The support of the CARIPARO Foundation (Grant NPDE: Non-linear Partial Differential Equations: models, analysis, and control - theoretic problems), and of the Spanish DGI grant MTM2010-18674 is acknowledged.Bergamaschi, L.; Bru García, R.; Martínez Calomardo, Á.; Mas Marí, J.; Putti, M. (2013). Low-rank update of preconditioners for the nonlinear Richard's equation. Mathematical and Computer Modelling. 57(7):1933-1941. https://doi.org/10.1016/j.mcm.2012.01.013S1933194157

Combined matrices of sign regular matrices

This is the author’s version of a work that was accepted for publication in Linear Algebra and its Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Linear Algebra and its Applications, VOL 498, (2016). DOI 10.1016/j.laa.2014.12.010.The combined matrix of a nonsingular matrix A is the Hadamard (entry wise) product C(A) = A ◦ (A−1)T . Since each row and column sum of C(A) is equal to one, the combined matrix is doubly stochastic when it is nonnegative. In this work, we study the nonnegativity of the combined matrix of sign regular matrices, based upon their signature. In particular, a few coordinates of the signature ε of A play a crucial role in determining whether or not C(A) is nonnegative. © 2014 Elsevier Inc. All rights reserved.Research supported by Spanish DGI grant number MTM2010-18674.Bru García, R.; Gasso Matoses, MT.; Gimenez Manglano, MI.; Santana-De Asis, MDJ. (2016). Combined matrices of sign regular matrices. Linear Algebra and its Applications. 498:88-98. https://doi.org/10.1016/j.laa.2014.12.010S889849

Diagonal entries of the combined matrix of a totally negative matrix

[EN] The combined matrix of a nonsingular matrix A is the Hadamard (entrywise) product . This paper deals with the characterization of the diagonal entries of a combined matrix C(A) of a given nonsingular real matrix A. A partial answer describing the diagonal entries of C(A) in the positive definite case was given by Fiedler in 1964. Recently in 2011, Fiedler and Markham characterized the sequence of diagonal entries of the combined matrix C(A) for any totally positive matrix A when the size is 3. For this case, we characterize totally negative matrices and we find necessary and sufficient conditions for the sequence of diagonal entries of C(A), in both cases, symmetric and nonsymmetric.This work was supported by Spanish DGI [grant number MTM-2014-58159-P]; Dominican Republic FONDOCYT [grant number 2015-1D2-166].Bru García, R.; Gasso Matoses, MT.; Gimenez Manglano, MI.; Santana-De Asis, MDJ. (2017). Diagonal entries of the combined matrix of a totally negative matrix. Linear and Multilinear Algebra. 65(10):1971-1984. https://doi.org/10.1080/03081087.2016.1261079S19711984651

Differential expression of miR-1249-3p and miR-34b-5p between vulnerable and resilient phenotypes of cocaine addiction

Cocaine addiction is a complex brain disorder involving long-term alterations that leadto loss of control over drug seeking. The transition from recreational use to pathologi-cal consumption is different in each individual, depending on the interaction betweenenvironmental and genetic factors. Epigenetic mechanisms are ideal candidates tostudy psychiatric disorders triggered by these interactions, maintaining persistentmalfunctions in specific brain regions. Here we aim to study brain-region-specific epi-genetic signatures following exposure to cocaine in a mouse model of addiction tothis drug. Extreme subpopulations of vulnerable and resilient phenotypes wereselected to identify miRNA signatures for differential vulnerability to cocaine addic-tion. We used an operant model of intravenous cocaine self-administration to evalu-ate addictive-like behaviour in rodents based on the Diagnostic and StatisticalManual of Mental Disorders Fifth Edition criteria to diagnose substance use disor-ders. After cocaine self-administration, we performed miRNA profiling to comparetwo extreme subpopulations of mice classified as resilient and vulnerable to cocaineaddiction. We found that mmu-miR-34b-5p was downregulated in the nucleusaccumbens of vulnerable mice with high motivation for cocaine. On the other hand,mmu-miR-1249-3p was downregulated on vulnerable mice with high levels of motordisinhibition. The elucidation of the epigenetic profile related to vulnerability to cocaine addiction is expected to help find novel biomarkers that could facilitate theinterventions to battle this devastating disorder

Nonnegative combined matrices

The combined matrix of a nonsingular real matrix is the Hadamard (entrywise) product ∘ (−1) . It is well known that row (column) sums of combined matrices are constant and equal to one. Recently, some results on combined matrices of different classes of matrices have been done. In this work, we study some classes of matrices such that their combined matrices are nonnegative and obtain the relation with the sign pattern of . In this case the combined matrix is doubly stochastic.The authors would like to thank the referees for their suggestions that have improved the reading of this paper. This research is supported by Spanish DGI (Grant no. MTM2010-18674).Bru García, R.; Gasso Matoses, MT.; Gimenez Manglano, MI.; Santana, M. (2014). Nonnegative combined matrices. Journal of Applied Mathematics. 2014. https://doi.org/10.1155/2014/182354S2014Fiedler, M., & Markham, T. L. (2011). Combined matrices in special classes of matrices. Linear Algebra and its Applications, 435(8), 1945-1955. doi:10.1016/j.laa.2011.03.054Horn, R. A., & Johnson, C. R. (1991). Topics in Matrix Analysis. doi:10.1017/cbo9780511840371Brualdi, R. A. (1988). Some applications of doubly stochastic matrices. Linear Algebra and its Applications, 107, 77-100. doi:10.1016/0024-3795(88)90239-xMourad, B. (2013). Generalization of some results concerning eigenvalues of a certain class of matrices and some applications. Linear and Multilinear Algebra, 61(9), 1234-1243. doi:10.1080/03081087.2012.746330Bru, R., Corral, C., Giménez, I., & Mas, J. (2008). Classes of general H-matrices. Linear Algebra and its Applications, 429(10), 2358-2366. doi:10.1016/j.laa.2007.10.030Fiedler, M., & Hall, F. J. (2012). G-matrices. Linear Algebra and its Applications, 436(3), 731-741. doi:10.1016/j.laa.2011.08.001Ando, T. (1987). Totally positive matrices. Linear Algebra and its Applications, 90, 165-219. doi:10.1016/0024-3795(87)90313-2Peña, J. M. (2003). On nonsingular sign regular matrices. Linear Algebra and its Applications, 359(1-3), 91-100. doi:10.1016/s0024-3795(02)00437-