Algorithms for generalized inverses

Every rectangular matrix has a Moore-Penrose generalized inverse. The purpose of this paper is to present several computational approaches to the determination of this generalized inverse, which plays a role in least-squares problems.     

Differentiation of generalized inverses for rational and polynomial matrices

Our basic motivation is a direct method for computing the gradient of the pseudo-inverse of well-conditioned system with respect to a scalar, proposed in [13] by Layton. In the present paper we combine the Layton’s method together with the representation of the Moore-Penrose inverse of one-variable polynomial matrix from [24] and developed an algorithm for computing the gradient of the Moore-Penrose inverse for one-variable polynomial matrix. Moreover, using the representation of various types of pseudo-inverses from [26], based on the Grevile’s partitioning method, we derive more general algorithms for computing {1}, {1, 3} and {1, 4} inverses of one-variable rational and polynomial matrices. Introduced algorithms are implemented in the programming language MATHEMATICA. Illustrative examples on analytical matrices are presented.     

A review of infinite matrices and their applications

Infinite matrices, the forerunner and a main constituent of many branches of classical mathematics (infinite quadratic forms, integral equations, differential equations, etc.) and of the modern operator theory, is revisited to demonstrate its deep influence on the development of many branches of mathematics, classical and modern, replete with applications. This review does not claim to be exhaustive, but attempts to present research by the authors in a variety of applications. These include the theory of infinite and related finite matrices, such as sections or truncations and their relationship to the linear operator theory on separable and sequence spaces. Matrices considered here have special structures like diagonal dominance, tridiagonal, sign distributions, etc. and are frequently nonsingular. Moreover, diagonally dominant finite and infinite matrices occur largely in numerical solutions of elliptic partial differential equations.The main focus is the theoretical and computational aspects concerning infinite linear algebraic and differential systems, using techniques like conformal mapping, iterations, truncations etc. to derive estimates based solutions. Particular attention is paid to computable precise error estimates, and explicit lower and upper bounds. Topics include Bessel’s, Mathieu equations, viscous fluid flow, simply and doubly connected regions, digital dynamics, eigenvalues of the Laplacian, etc. Also presented are results in generalized inverses and semi-infinite linear programming.     

A note on the stability of a pth order iteration for finding generalized inverses

A number of papers intended for the numerical computation of generalized inverses by means of higher-order iterative methods have been published recently. This note investigates the numerical stability of a general family of iterative methods in order to complete the previous studies in this trend of research.     

On constrained generalized inverses of matrices and their properties

A matrix G is called a generalized inverse (g-invserse) of matrix A if AGA = A and is denoted by G = A ⁻. Constrained g-inverses of A are defined through some matrix expressions like E(AE)⁻, (FA)⁻ F and E(FAE)⁻ F. In this paper, we derive a variety of properties of these constrained g-inverses by making use of the matrix rank method. As applications, we give some results on g-inverses of block matrices, and weighted least-squares estimators for the general linear model.     

About the generalized LM-inverse and the weighted Moore-Penrose inverse

The recursive method for computing the generalized LM-inverse of a constant rectangular matrix augmented by a column vector is proposed in Udwadia and Phohomsiri (2007) [16] and [17]. The corresponding algorithm for the sequential determination of the generalized LM-inverse is established in the present paper. We prove that the introduced algorithm for computing the generalized LM-inverse and the algorithm for the computation of the weighted Moore-Penrose inverse developed by Wang and Chen (1986) in [23] are equivalent algorithms. Both of the algorithms are implemented in the present paper using the package MATHEMATICA. Several rational test matrices and randomly generated constant matrices are tested and the CPU time is compared and discussed.     

A probing method for computing the diagonal of a matrix inverse

The computation of some entries of a matrix inverse arises in several important applications in practice. This paper presents a probing method for determining the diagonal of the inverse of a sparse matrix in the common situation when its inverse exhibits a decay property, i.e. when many of the entries of the inverse are small. A few simple properties of the inverse suggest a way to determine effective probing vectors based on standard graph theory results. An iterative method is then applied to solve the resulting sequence of linear systems, from which the diagonal of the matrix inverse is extracted. The results of numerical experiments are provided to demonstrate the effectiveness of the probing method. Copyright © 2011 John Wiley & Sons, Ltd.     

New exact solutions for a generalized variable-coefficient KdV equation

New exact solutions for a generalized variable-coefficient KdV equation were obtained using the generalized expansion method [R. Sabry, M.A. Zahran, E.G. Fan, Phys. Lett. A 326 (2004) 93]. The obtained solutions include solitary wave solutions besides Jacobi and Weierstrass doubly periodic wave solutions.     

A residual method for solving nonlinear operator equations and their application to nonlinear integral equations using symbolic computation

A derivative-free residual method for solving nonlinear operator equations in real Hilbert spaces is discussed. This method uses in a systematic way the residual as search direction, but it does not use first order information. Furthermore a convergence analysis and numerical results of the new method applied to nonlinear integral equations using symbolic computation are presented.     

A new rational auxiliary equation method and exact solutions of a generalized Zakharov system

A new rational auxiliary equation method for obtaining exact traveling wave solutions of constant coefficient nonlinear partial differential equations of evolution is proposed. Its effectiveness is evinced by obtaining exact solutions of a generalized Zakharov system, some of which are new. It is shown that the G^′/G and the generalized projective Ricatti expansion methods are special cases of the auxiliary equation method. Further, due the solutions obtained, four other new and practicable rational methods are deduced.     

Conjugate-gradient method for computing the Moore-Penrose inverse and rank of a matrix

A conjugate-gradient method is developed for computing the Moore-Penrose generalized inverseA of a matrix and the associated projectors, by using the least-square characteristics of both the method and the inverseA . Two dual algorithms are introduced for computing the least-square and the minimum-norm generalized inverses, as well asA . It is shown that (i) these algorithms converge for any starting approximation; (ii) if they are started from the zero matrix, they converge toA ; and (iii) the trace of a sequence of approximations multiplied byA is a monotone increasing function converging to the rank ofA. A practical way of compensating the self-correcting feature in the computation ofA is devised by using the duality of the algorithms. Comparison with Ben-Israel's method is made through numerical examples. The conjugate-gradient method has an advantage over Ben-Israel's method.After having completed the present paper, the author received from Professor M. R. Hestenes his paper entitledPseudo Inverses and Conjugate Gradients. This paper treated the same subject and appeared in Communications of the ACM, Vol. 18, pp. 40–43, 1975.     

A sub-ODE method for generalized Gardner and BBM equation with nonlinear terms of any order

In this paper, a method with the aid of a sub-ODE and its solutions is used for constructing new periodic wave solutions for nonlinear Gardner equation and BBM equation with nonlinear terms of any order arising in mathematical physics. As a result, many exact traveling wave solutions are successfully obtained. The method in the paper is very direct and it can also be applied to other nonlinear evolution equations.     

Classification of the centers, their cyclicity and isochronicity for the generalized quadratic polynomial differential systems

In this paper we classify the centers, the cyclicity of their Hopf bifurcation and the isochronicity of the polynomial differential systems in R² of degree d that in complex notation z=x+iy can be written as

     

Edelstein's method and fixed point theorems for some generalized nonexpansive mappings

A new condition for mappings, called condition (C), which is more general than nonexpansiveness, was recently introduced by Suzuki [T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008) 1088-1095]. Following the idea of Kirk and Massa Theorem in [W.A. Kirk, S. Massa, Remarks on asymptotic and Chebyshev centers, Houston J. Math. 16 (1990) 364-375], we prove a fixed point theorem for mappings with condition (C) on a Banach space such that its asymptotic center in a bounded closed and convex subset of each bounded sequence is nonempty and compact. This covers a result obtained by Suzuki [T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008) 1088-1095]. We also present fixed point theorems for this class of mappings defined on weakly compact convex subsets of Banach spaces satisfying property (D). Consequently, we extend the results in [T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008) 1088-1095] to many other Banach spaces.     

Numerical simulation and analysis of generalized difference method on triangular networks for electrical impedance tomography

Electrical impedance tomography is an inverse problem of elliptic differential equations. Numerical methods based on combining generalized difference method and Levenberg–Marquardt iteration on a planar domain are proposed. Positive semi-definiteness and existence of solution of the generalized difference scheme are proved. Element geometry matrix is introduced to shortcut calculation and standardize computer program. A series of numerical experiments verify the reliability of its mathematical model and the feasibility of the algorithm. A class of electrical current patterns is proposed to minimize the number of direct problems to be solved in each iteration. These methods have been applied successfully in practical simulation of electrical impedance tomography.     

Soliton solutions and a Bäcklund transformation for a generalized nonlinear Schrödinger equation with variable coefficients from optical fiber communications

Under investigation in this paper is a generalized nonlinear Schrödinger model with variable dispersion, nonlinearity and gain/loss, which could describe the propagation of optical pulse in inhomogeneous fiber systems. By employing the Hirota method, one- and two-soliton solutions are obtained with the aid of symbolic computation. Furthermore, a general formula which denotes multi-soliton solutions is given. Some main properties of the solutions are discussed simultaneously. As one important property of nonlinear evolution equation, the Bäcklund transformation in bilinear form is also constructed, which is helpful on future research and as far as we know is firstly proposed in this paper.     

A GENERALIZED GRADIENT PROJECTION METHOD FOR OPTIMIZATION PROBLEMS WITH EQUALITY AND INEQUALITY CONSTRAINTS ABOUT ARBITRARY INITIAL POINT

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