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In this study, we obtained results for the computation of eigen-pairs, singular value decomposition, pseudoinverse, and the least squares problem for elliptic quaternion matrices. Moreover, we established algorithms based on these results and provided illustrative numerical experiments to substantiate the accuracy of our conclusions. In the experiments, it was observed that the p-value in the algebra of elliptic quaternions directly affects the performance of the problem under consideration. Selecting the optimal p-value for problem-solving and the elliptic behavior of many physical systems make this number system advantageous in applied sciences. [ABSTRACT FROM AUTHOR]

We introduce the dual-Hilbert space and study the basic properties of a dual operator and its generalized inverse on this space. We provide upper bounds on the perturbation of the dual Moore–Penrose inverse of the dual operator if the dual operator is injective or surjective. If the null space or range space of the perturbed dual operator is invariant, stable perturbations are used to give the perturbation bounds for the dual Moore–Penrose inverse. Additionally, given the aforementioned conditions, perturbation bounds for the least squares solution are provided. The upper bounds on the distance between the solution of a perturbed least squares problem and the set of all of its unperturbed solutions under the dual operator norm are also presented. [ABSTRACT FROM AUTHOR]

This paper mainly studies the least-squares solutions of matrix equation $ AXB = C $ under a semi-tensor product. According to the definition of the semi-tensor product, the equation is transformed into an ordinary matrix equation. Then, the least-squares solutions of matrix-vector and matrix equations respectively investigated by applying the derivation of matrix operations. Finally, the specific form of the least-squares solutions is given.

This paper mainly studies the least-squares solutions of matrix equation A X B = C under a semi-tensor product. According to the definition of the semi-tensor product, the equation is transformed into an ordinary matrix equation. Then, the least-squares solutions of matrix-vector and matrix equations respectively investigated by applying the derivation of matrix operations. Finally, the specific form of the least-squares solutions is given. [ABSTRACT FROM AUTHOR]

This paper considers the Hermitian solutions of a new system of commutative quaternion matrix equations, where we establish both necessary and sufficient conditions for the existence of solutions. Furthermore, we derive an explicit general expression when it is solvable. In addition, we also provide the least squares Hermitian solution in cases where the system of matrix equations is not consistent. To illustrate our main findings, in this paper we present two numerical algorithms and examples. [ABSTRACT FROM AUTHOR]

In this study, we obtained results for the computation of eigen-pairs, singular value decomposition, pseudoinverse, and the least squares problem for elliptic quaternion matrices. Moreover, we established algorithms based on these results and provided illustrative numerical experiments to substantiate the accuracy of our conclusions. In the experiments, it was observed that the p-value in the algebra of elliptic quaternions directly affects the performance of the problem under consideration. Selecting the optimal p-value for problem-solving and the elliptic behavior of many physical systems make this number system advantageous in applied sciences.

A strongly underrelaxed sequential version of the Hildreth's iterative algorithm for norm minimization over linear inequalities is presented. Proofs are given showing that the algorithm converges from any starting point to its projection onto the linear constraint set in the feasible case and to the nearest least squares solution in the general case. [ABSTRACT FROM AUTHOR]

This paper considers the Hermitian solutions of a new system of commutative quaternion matrix equations, where we establish both necessary and sufficient conditions for the existence of solutions. Furthermore, we derive an explicit general expression when it is solvable. In addition, we also provide the least squares Hermitian solution in cases where the system of matrix equations is not consistent. To illustrate our main findings, in this paper we present two numerical algorithms and examples.

In this article, our interest is the quaternion matrix equation (AXB,DXE)=(C,F), and we study its minimal norm centrohermitian least squares solution and skew centrohermitian least squares solution. By applying of the real representation matrices of quaternion matrices and relative properties, we convert the quaternion least squares problems with constrained variables into the corresponding real least squares problems with free variables, and then we obtain the solutions of corresponding problems. The final results can be expressed only by real matrices and vectors, and thus the corresponding algorithms only involves real operations and avoid complex quaternion operations. Therefore, they are portable and convenient. In the end, we give two examples to verify the effectiveness of the purposed algorithms.

In this article, we proposed an enhancement to the convergence rate of Landweber's method by incorporating the concept of momentum acceleration. Landweber's method is commonly used to solve least squares problems of the form min x ‖ A x − b ‖. Our approach is based on Landweber's method, which is acknowledged as a particular case of the methodologies outlined in Ding and Chen (2006). Through optimizing the momentum parameter, we were able to demonstrate the superior performance of the momentum-accelerated Landweber method. Specifically, we established that when A is a nonsquare m × n matrix with Rank (A) = n and σ min (A) ≠ σ max (A) , the momentum-accelerated Landweber method with the optimal parameter consistently outperforms the standard Landweber method. Our numerical experiments have confirmed the theoretical findings, demonstrating a notable improvement in the convergence rate of the Landweber method. [ABSTRACT FROM AUTHOR]

In this paper, we propose an efficient method for some special solutions of the quaternion matrix equation AXB+CYD=E. By integrating real representation of a quaternion matrix with H-representation, we investigate the minimal norm least squares solution of the previous quaternion matrix equation over different constrained matrices and obtain their expressions. In this way, we first apply H-representation to solve quaternion matrix equation with special structure, which not only broadens the application scope of H-representation, but further expands the research idea of solving quaternion matrix equation. The algorithms only include real operations. Consequently, it is very simple and convenient, and it can be applied to all kinds of quaternion matrix equation with similar problems. The numerical example is provided to illustrate the feasibility of our algorithms.

Yuan and Liao [S.F. Yuan, A.P. Liao, Least squares Hermitian solution of the complex matrix equation AXB+CXD=E$$ AXB+ CXD=E $$ with the least norm. J. Frankl. Inst. 351 (2014), 4978‐4997] gave a new product for complex matrices and vectors and obtained the least squares Hermitian solution with the least norm of complex matrix equation AXB+CXD=E$$ AXB+ CXD=E $$. In this paper, we generalize the product of complex matrices and vectors to quaternion tensors and vectors and consider the least squares Hermitian problem of quaternion tensor equation A∗NX=B$$ \mathcal{A}{\ast}_N\mathcal{X}=\mathcal{B} $$, where the symbol ∗N$$ {\ast}_N $$ is the Einstein product of two tensors. Our main work is to derive the explicit expression of least squares Hermitian solution with the least norm and numerical algorithm to obtain the solution. We also provide numerical examples to justify the feasibility of our algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we put forward two efficient methods for solving the commutative quaternion Stein matrix equation X + A X B = C . By combining real representation of a commutative quaternion matrix and H -representation or generalized H -representation of special matrices, we investigate the minimal norm least squares L-structured and generalized L-structured solutions of the previous commutative quaternion matrix equation and derive their expressions. In this way, we first present a theoretical study on extending L-structured real matrices to generalized L-structured matrices, and introduce some generalized L-structured matrices. Based on them, we then discuss their applications in commutative quaternion Stein matrix equation. The algorithms only involve real operations. Consequently, it is very simple and convenient, and it can be used for all kinds of commutative quaternion matrix equation with similar problems. Furthermore, an illustrative example is provided to show the feasibility of the given methods. [ABSTRACT FROM AUTHOR]

In this paper, the idea of partitioning is used to solve quaternion least squares problem, we divide the quaternion Bisymmetric matrix into four blocks and study the relationship between the block matrices. Applying this relation, the real representation of quaternion, and M-P inverse, we obtain the least squares Bisymmetric solution of quaternion matrix equation AXB=C and its compatable conditions. Finally, we verify the effectiveness of the method through numerical examples.

In this paper, we study several constrainted least squares solutions of quaternion Sylvester matrix equation. We first propose a real vector representation of quaternion matrix and study its properties. By using this real vector representation, semi-tensor product of matrices, swap matrix and Moore-Penrose inverse, we derive compatible conditions and the expressions of several constrainted least squares solutions of quaternion Sylvester equation.

In this work, a new algorithm for solving symmetric indefinite systems of linear equations is presented. It factorizes the matrix into the form L D L t using Jacobi rotations in order to increase the pivot´s absolute value. Furthermore, Rook´s pivoting strategy is also adapted and implemented. In determinate compatible systems, the computational cost of the algorithm was similar to the cost of the Bunch-Kaufman method, but the error was approximately 50 % smaller for intermediate and large matrices, regardless of the condition number of the coefficient matrix. Furthermore, unlike Bunch-Kaufman, the new algorithm calculates with little additional cost the fundamental basis of the null space, and obtains the minimal least squares and minimum norm solutions. In minimal least squares with minimum norm problems, the new algorithm was compared with the LAPACK Complete Orthogonal Decomposition algorithm, among others. The obtained error with both algorithms was similar but the computational cost was at least 20 % smaller with the new algorithm, even though the Complete Orthogonal Decomposition is implemented in a blocked form. [ABSTRACT FROM AUTHOR]

In this paper, using the real representation method, we study the reduced biquaternion matrix equation A X = B . Taking advantage of the special structure of real representation of reduced biquaternion, we transform the problem of reduced biquaternion matrix into corresponding problem of real matrix. We propose the expressions of the special minimal norm least squares solution of the reduced biquaternion matrix equation A X = B , and the corresponding algorithms only perform real arithmetic. Numerical examples are provided to illustrate that our algorithm are efficient and easily understood. We also apply the minimal norm pure imaginary least squares reduced biquaternion solution to color image restoration. [ABSTRACT FROM AUTHOR]

In this paper, we study the quaternion matrix equation (1.1) by using the semi-tensor product of matrices. First, we propose a real vector representation of a quaternion matrix and study its properties. Then combining this real vector representation with semi-tensor product of matrices, the explicit expressions of the least squares solution, the least squares J-self-conjugate solution and the least squares anti-J-self-conjugate solution of (1.1) are proposed. We give the equivalence conditions for the compatibility of (1.1) and the expressions of the minimal norm solution with above-mentioned properties. We also propose the corresponding algorithms and give numerical experiments to verify the effectiveness of our methods. [ABSTRACT FROM AUTHOR]

Let η H Q k × k and η A Q k × k represent the sets of all k × k η-Hermitian quaternion matrices and η-anti-Hermitian quaternion matrices, respectively. On the basis of the real representation matrix of a quaternion matrix and its particular structure, we convert the least squares problem of the quaternion matrix equation AXB + CY D = E over X ∈ η H Q n × n , Y ∈ η A Q k × k into the corresponding problem of the real matrix equation over free variables, and then we establish its unique minimal norm least squares solution. Our resulting expressions are expressed only by real matrices, and the algorithm only includes real operations. Consequently, they are very simple and convenient. Compared with the existing method [S.F. Yuan, Q.W. Wang, and X. Zhang, Least-squares problem for the quaternion matrix equation AXB + CYD = E over different constrained matrices, Int. J. Comput. Math. 90 (2013), pp. 565–576], the final two examples show that our method is more efficient and superior. [ABSTRACT FROM AUTHOR]

This paper is concerned with the solution to the least squares problem for the quaternion Sylvester tensor equation. An iterative algorithm based on tensor format is presented to solve this problem. The convergence properties of the proposed iterative method are studied. We also consider the best approximate problem related to quaternion Sylvester tensor equation. Numerical examples are provided to confirm the theoretical results, which demonstrate that the proposed algorithm is effective and feasible for solving the least squares problem of the quaternion Sylvester tensor equation. [ABSTRACT FROM AUTHOR]

The Douglas{Rachford algorithm (DRA) is a powerful optimization method for minimizing the sum of two convex (not necessarily smooth) functions. The vast majority of previous research dealt with the case when the sum has at least one minimizer. In the absence of minimizers, it was recently shown that for the case of two indicator functions, the DRA converges to a best approximation solution. In this paper, we present a new convergence result on the DRA applied to the problem of minimizing a convex function subject to a linear constraint. Indeed, a normal solution may be found even when the domain of the objective function and the linear subspace constraint have no point in common. As an important application, a new parallel splitting result is provided. We also illustrate our results through various examples. [ABSTRACT FROM AUTHOR]

Partial differential equations (PDEs) such as those that describe linear elastic fracture mechanics admit analytic solutions with low regularity at crack tips. Existing numerical methods for such PDEs suffer from the constraint of this low regularity and fail to deliver an optimal high-order rate of convergence. We approach this problem by (i) choosing an artificial interface to enclose the center of low regularity and (ii) representing the solution in the interior of artificial interface as unknown linear combination of known low-regularity modes of the solution. This results in an interface formulation of the original PDE, and the linear combination is represented in the interface conditions. By enforcing the smooth component of the numerical solution in the interior domain to be approximately zero, a least squares problem is obtained for the unknown coefficients. The solution of this least squares problem will provide the approximate interface conditions for the numerical solution of the PDE in the exterior domain. The potential of our interface formulation is favorably demonstrated by numerical experiments on one- and two-dimensional Poisson equations with low-regularity solutions. High-order numerical solutions of unknown coefficients and PDEs are obtained. This proves the potential of the proposed interface formulation as the theoretical basis for solving linear elastic fracture mechanics problems. We address the relations between our interface formulation and domain decomposition methods, as well as a regularization strategy for the Poisson–Boltzmann equation with singular charge density. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of the large scale singular Sylvester equation of the form A X − X D ⊤ = E (or A X B ⊤ − C X D ⊤ = E), where the spectra Λ (A) and Λ (D) (or, Λ (A − λ C) and Λ (D − λ B)) have a nonempty intersection. Using appropriate invariant subspaces, the singular Sylvester equation is rewritten as four Sylvester equations, three of which are nonsingular and one singular. When the invariant subspaces are small, so are three of the equations (including the singular one) which can be solved efficiently. The fourth is large but nonsingular with structures and may be solved using the projection method with Krylov subspaces or techniques involving hierarchical matrices. Some numerical examples for the subspace method are provided. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the best fit solution for the second order differential problem with one initial and other integral conditions. We obtain the representation of that minimizer and present an example.paration.

In this paper, we present an iterative method for finding the least squares solution of a class of generalized Sylvester-transpose matrix equations with the norm inequality constraint. We prove that if the constrained matrix equations are consistent, the solution can be obtained within finite iterative steps in the absence of round-off errors; if constrained matrix equations are inconsistent, the least squares solution can be obtained within finite iterative steps in the absence of round-off errors. Finally, numerical examples are provided to illustrate the efficiency of the proposed method and testify the conclusions suggested in this paper. [ABSTRACT FROM AUTHOR]

The iterative algorithm of a class of generalized coupled Sylvester-transpose matrix equations is presented. We prove that if the system is consistent, a solution can be obtained within finite iterative steps in the absence of round-off errors for any initial matrices; if the system is inconsistent, the least squares solution can be obtained within finite iterative steps in the absence of round-off errors. Furthermore, we provide a method for choosing the initial matrices to obtain the least Frobenius norm least squares solution of the problem. Finally, numerical examples are presented to demonstrate that the algorithm is efficient. [ABSTRACT FROM AUTHOR]

Many real life situations can be described using twice continuously differentiable functions over convex sets with interior points. Such functions have an interesting separation property: At every interior point of the set they separate particular classes of quadratic convex functions from classes of quadratic concave functions. Using this property we introduce new characterizations of the derivative and its zero points. The results are applied to the study of sensitivity of the Cobb-Douglas production function. They are also used to describe the least squares solutions in linear and nonlinear regression.

Due to the lack of efficient approaches to quantify actual loss of effect (LOE) fault diagnosability in dynamic systems, a novel presented is presented in this paper. Discrete-time state space models are modified into dynamic models with time window of certain length, and the similarity of different multivariate distributions, instead of fault diagnosability evaluation, is analyzed. The definitions of detectability and isolability of LOE fault are given. Diagnosability performance is quantified by the method based on minimum Bhattacharyya distance. Least-squares method is utilized to enhance applicability of the novel proposed approach. In addition, a simulation example is employed to show the validity of the proposed approach and to analyze maximum effectiveness coefficient of LOE fault in dynamic systems. [ABSTRACT FROM AUTHOR]

Letandwe first solve the minimum Frobenius norm residual problem (Problem LSP):with unknown Toeplitz matricesXandY. We then consider a best approximation problem: given Toeplitz matricesand, findsuch thatwhereis the solution set of Problem LSP. We show that the best approximation solutionis unique and derive an explicit formula for it. [ABSTRACT FROM AUTHOR]

Let R ∈ C m × m and S ∈ C n × n be nontrivial k -involutions if their minimal polynomials are both x k − 1 for some k ≥ 2 , i.e., R k − 1 = R − 1 ≠ ± I and S k − 1 = S − 1 ≠ ± I . We say that A ∈ C m × n is ( R , S , μ ) -symmetric if R A S − 1 = ζ μ A , and A is ( R , S , α , μ ) -symmetric if R A S − α = ζ μ A with α , μ ∈ { 0 , 1 , … , k − 1 } and α ≠ 0 . Let S be one of the subsets of all ( R , S , μ ) -symmetric and ( R , S , α , μ ) -symmetric matrices. Given X ∈ C n × r , Y ∈ C s × m , B ∈ C m × r and D ∈ C s × n , we characterize the matrices A in S that minimize ‖ A X − B ‖ 2 + ‖ Y A − D ‖ 2 (Frobenius norm) under the assumption that R and S are unitary. Moreover, among the set S ( X , Y , B , D ) ⊂ S of the minimizers of ‖ A X − B ‖ 2 + ‖ Y A − D ‖ 2 = min , we find the optimal approximate matrix A ∈ S ( X , Y , B , D ) that minimizes ‖ A − G ‖ to a given unstructural matrix G ∈ C m × n . We also present the necessary and sufficient conditions such that A X = B , Y A = D is consistent in S . If the conditions are satisfied, we characterize the consistent solution set of all such A . Finally, a numerical algorithm and some numerical examples are given to illustrate the proposed results. [ABSTRACT FROM AUTHOR]

In this paper, we consider mth-order linear discrete problems with m nonlocal conditions. We investigate a generalized Green's function, describing the minimum norm least squares solution, and present its properties, which resemble properties of an ordinary Green's function. [ABSTRACT FROM AUTHOR]

In this paper, by using the real representations of quaternion matrices, the particular structure of the real representations of quaternion matrices, the Kronecker product of matrices and the Moore–Penrose generalized inverse, we obtain the expressions of the minimal norm least squares solution, the pure imaginary least squares solution, and the real least squares solution for the quaternion matrix equation A X B + C X D = E , respectively. Our resulting formulas only involve real matrices, and therefore are simpler than those reported in Yuan (2014). The corresponding algorithms only perform real arithmetic which also consider the particular structure of the real representations of quaternion matrices, and therefore are very efficient and portable. Numerical examples are provided to illustrate the efficiency of our algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we first establish the necessary and sufficient conditions for the existence and the explicit expressions of the Hermitian { P , k + 1 } -(anti-)reflexive solutions of the matrix equation A X = B , and meanwhile the best approximation solution is considered. Then, if the solvability conditions are not satisfied, the least squares Hermitian { P , k + 1 } -(anti-)reflexive solutions and the least squares Hermitian { P , k + 1 } -(anti-)reflexive solutions with the minimum norm of the above matrix equation are respectively derived. In addition, two algorithms are shown to compute the least squares Hermitian { P , k + 1 } -(anti-)reflexive solutions, and the corresponding numerical examples are also given to illustrate the feasibility of the algorithms. [ABSTRACT FROM AUTHOR]

This paper focuses on L-structured quaternion matrices. L-structured real matrices, conditions for the existence of solutions and the general solution of linear matrix equations were studied in the paper [Magnus JR. L-structured matrices and linear matrix equations, Linear Multilinear Algebra 1983;14:67–88]. In this paper, we present a theoretical study extending L-structured real matrices to L-structured quaternion matrices, and introduce some L-structured quaternion matrices. Based on them, we then discuss their applications in quaternion matrix equations. [ABSTRACT FROM AUTHOR]

We consider estimation of parameters in models defined by systems of ordinary differential equations (ODEs). This problem is important because many processes in different fields of science are modelled by systems of ODEs. Various estimation methods based on smoothing have been suggested to bypass numerical integration of the ODE system. In this paper, we do not propose another method based on smoothing but show how some of the existing ones can be brought together under one unifying framework. The framework is based on generalized Tikhonov regularization and extremum estimation. We define an approximation of the ODE solution by viewing the system of ODEs as an operator equation and exploiting the connection with regularization theory. Combining the introduced regularized solution with an extremum criterion function provides a general framework for estimating parameters in ODEs, which can handle partially observed systems. If the extremum criterion function is the negative log-likelihood, then suitable regularized solutions yield estimators that are consistent and asymptotically efficient. The well-known generalized profiling procedure fits into the proposed framework. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Single-hidden-layer feedforward networks with randomly generated additive or radial basis function hidden nodes have been theoretically proved that they can approximate any continuous function. Meanwhile, an incremental algorithm referred to as incremental extreme learning machine (I-ELM) was proposed which outperforms many popular learning algorithms. However, I-ELM may produce redundant nodes which increase the network architecture complexity and reduce the convergence rate of I-ELM. Moreover, the output weight vector obtained by I-ELM is not the least squares solution of equation Hβ = T. In order to settle these problems, this paper proposes an orthogonal incremental extreme learning machine (OI-ELM) and gives the rigorous proofs in theory. OI-ELM avoids redundant nodes and obtains the least squares solution of equation Hβ = T through incorporating the Gram-Schmidt orthogonalization method into I-ELM. Simulation results on nonlinear dynamic system identification and some benchmark real-world problems verify that OI-ELM learns much faster and obtains much more compact neural networks than ELM, I-ELM, convex I-ELM and enhanced I-ELM while keeping competitive performance. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the properties of a generalized Green's function describing the minimum norm least squares solution for a second order discrete problem with two nonlocal conditions. The properties obtained of a generalized Green's function resemble analogous properties of an ordinary Green's function that describes the unique exact solution if it exists. Several features are illustrated by examples. [ABSTRACT FROM AUTHOR]