Effective partitioning method for computing weighted Moore-Penrose inverse

We introduce a method and an algorithm for computing the weighted Moore-Penrose inverse of multiple-variable polynomial matrix and the related algorithm which is appropriated for sparse polynomial matrices. These methods and algorithms are generalizations of algorithms developed in [M.B. Tasic, P.S. Stanimirovic, M.D. Petkovic, Symbolic computation of weighted Moore-Penrose inverse using partitioning method, Appl. Math. Comput. 189 (2007) 615-640] to multiple-variable rational and polynomial matrices and improvements of these algorithms on sparse matrices. Also, these methods are generalizations of the partitioning method for computing the Moore-Penrose inverse of rational and polynomial matrices introduced in [P.S. Stanimirovic, M.B. Tasic, Partitioning method for rational and polynomial matrices, Appl. Math. Comput. 155 (2004) 137-163; M.D. Petkovic, P.S. Stanimirovic, Symbolic computation of the Moore-Penrose inverse using partitioning method, Internat. J. Comput. Math. 82 (2005) 355-367] to the case of weighted Moore-Penrose inverse. Algorithms are implemented in the symbolic computational package MATHEMATICA

Representations and symbolic computation of generalized inverses over fields

This paper investigates representations of outer matrix inverses with prescribed range and/or none space in terms of inner inverses. Further, required inner inverses are computed as solutions of appropriate linear matrix equations (LME). In this way, algorithms for computing outer inverses are derived using solutions of appropriately defined LME. Using symbolic solutions to these matrix equations it is possible to derive corresponding algorithms in appropriate computer algebra systems. In addition, we give sufficient conditions to ensure the proper specialization of the presented representations. As a consequence, we derive algorithms to deal with outer inverses with prescribed range and/or none space and with meromorphic functional entries.Agencia Estatal de investigaciónUniversidad de Alcal

Representations and geometrical properties of generalized inverses over fields

In this paper, as a generalization of Urquhart’s formulas, we present a full description of the sets of inner inverses and (B, C)-inverses over an arbitrary field. In addition, identifying the matrix vector space with an affine space, we analyze geometrical properties of the main generalized inverse sets. We prove that the set of inner inverses, and the set of (B, C)-inverses, form affine subspaces and we study their dimensions. Furthermore, under some hypotheses, we prove that the set of outer inverses is not an affine subspace but it is an affine algebraic variety. We also provide lower and upper bounds for the dimension of the outer inverse set.Agencia Estatal de InvestigaciónUniversidad de Alcal

Computation of Moore-Penrose generalized inverses of matrices with meromorphic function entries

J.R. Sendra is member of the Research Group ASYNACS (Ref.CT-CE2019/683)In this paper, given a field with an involutory automorphism, we introduce the notion of Moore-Penrose field by requiring that all matrices over the field have Moore-Penrose inverse. We prove that only characteristic zero fields can be Moore-Penrose, and that the field of rational functions over a Moore-Penrose field is also Moore-Penrose. In addition, for a matrix with rational functions entries with coefficients in a field K, we find sufficient conditions for the elements in K to ensure that the specialization of the Moore-Penrose inverse is the Moore-Penrose inverse of the specialization of the matrix. As a consequence, we provide a symbolic algorithm that, given a matrix whose entries are rational expression over C of finitely many meromeorphic functions being invariant by the involutory automorphism, computes its Moore-Penrose inverve by replacing the functions by new variables, and hence reducing the problem to the case of matrices with complex rational function entries.Ministerio de Economía y CompetitividadEuropean Regional Development Fun

Application of block Cayley-Hamilton theorem to generalized inversion

In this paper we propose two algorithms for computation of the outer inverse with prescribed range and null space and the Drazin inverse of block matrix. The proposed algorithms are based on the extension of the Leverrier-Faddeev algorithm and the block Cayley-Hamilton theorem. These algorithms are implemented using symbolic and functional possibilities of the packages {\it Mathematica} and using numerical possibilities of {\it Matlab}

Symbolic computation of Hankel determinants and matrix generalized inverses

In this thesis, existing methods for symbolic computation of Hankel deteriminants and matrix generalized inverses are modified and new are introducted. There are derived closed-form expressions for Hankel determinants of different classes of sequences. It is constructed the method for rapid computation of generalized inverses whose complexity reaches theoretical lower bound. There are also constructed new methods for computation of generalized inverses of rational and polynomial matrices

Solution strategies for nonlinear conservation laws

Nonlinear conservation laws form the basis for models for a wide range of physical phenomena. Finding an optimal strategy for solving these problems can be challenging, and a good strategy for one problem may fail spectacularly for others. As different problems have different challenging features, exploiting knowledge about the problem structure is a key factor in achieving an efficient solution strategy. Most strategies found in literature for solving nonlinear problems involve a linearization step, usually using Newton's method, which replaces the original nonlinear problem by an iteration process consisting of a series of linear problems. A large effort is then spent on finding a good strategy for solving these linear problems. This involves choosing suitable preconditioners and linear solvers. This approach is in many cases a good choice and a multitude of different methods have been developed. However, the linearization step to some degree involves a loss of information about the original problem. This is not necessarily critical, but in many cases the structure of the nonlinear problem can be exploited to a larger extent than what is possible when working solely on the linearized problem. This may involve knowledge about dominating physical processes and specifically on whether a process is near equilibrium. By using nonlinear preconditioning techniques developed in recent years, certain attractive features such as automatic localization of computations to parts of the problem domain with the highest degree of nonlinearities arise. In the present work, these methods are further refined to obtain a framework for nonlinear preconditioning that also takes into account equilibrium information. This framework is developed mainly in the context of porous media, but in a general manner, allowing for application to a wide range of problems. A scalability study shows that the method is scalable for challenging two-phase flow problems. It is also demonstrated for nonlinear elasticity problems. Some models arising from nonlinear conservation laws are best solved using completely different strategies than the approach outlined above. One such example can be found in the field of surface gravity waves. For special types of nonlinear waves, such as solitary waves and undular bores, the well-known Korteweg-de Vries (KdV) equation has been shown to be a suitable model. This equation has many interesting properties not typical of nonlinear equations which may be exploited in the solver, and strategies usually reserved to linear problems may be applied. In this work includes a comparative study of two discretization methods with highly different properties for this equation

Numerical detection of complex singularities in two and three dimensions

Singularities often occur in solutions to partial differential equations; important examples include the formation of shock fronts in hyperbolic equations and self-focusing type blow up in nonlinear parabolic equations. Information about formation and structure of singularities can have significant role in interfacial fluid dynamics such as Kelvin-Helmholtz instability, Rayleigh-Taylor instability, and Hele-Shaw flow. In this thesis, we present a new method for the numerical analysis of complex singularities in solutions to partial differential equations. In the method, we analyze the decay of Fourier coefficients using a numerical form fit to ascertain the nature of singularities in two and three-dimensional functions. Our results generalize a well known method for the analysis of singularities in one-dimensional functions to higher dimensions. As an example, we apply this method to analyze the complex singularities for the 2D inviscid Burger\u27s equation

An Introduction to the Mechanics of Incompressible Fluids

This open access book allows the reader to grasp the main bulk of fluid flow problems at a brisk pace. Starting with the basic concepts of conservation laws developed using continuum mechanics, the incompressibility of a fluid is explained and modeled, leading to the famous Navier-Stokes equation that governs the dynamics of fluids. Some exact solutions for transient and steady-state cases in Cartesian and axisymmetric coordinates are proposed. A particular set of examples is associated with creeping or Stokes flows, where viscosity is the dominant physical phenomenon. Irrotational flows are treated by introducing complex variables. The use of the conformal mapping and the Joukowski transformation allows the treatment of the flow around an airfoil. The boundary layer theory corrects the earlier approach with the Prandtl equations, their solution for the case of a flat plate, and the von Karman integral equation. The instability of fluid flows is studied for parallel flows using the Orr-Sommerfeld equation. The stability of a circular Couette flow is also described. The book ends with the modeling of turbulence by the Reynolds-averaged Navier-Stokes equations and large-eddy simulations. Each chapter includes useful practice problems and their solutions. The book is useful for engineers, physicists, and scientists interested in the fascinating field of fluid mechanics