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In this paper, several Kaczmarz-type numerical methods for solving the matrix equations AX = B and XA = C are proposed, where the coefficient matrix A may be full rank or deficient rank. These methods are iterative methods without matrix multiplication. Theoretically, the convergence of these methods is proved. The numerical results show that these methods are more efficient than iterative methods involving matrix multiplication for high-dimensional matrices. [ABSTRACT FROM AUTHOR]

In this paper, we establish some efficient iteration methods for solving the general coupled discrete-time periodic matrix equation. More concretely, we propose some structure preserving matrix versions of subspace methods, such as bi-conjugate gradient, bi-conjugate residual, conjugate gradient squared, bi-conjugate gradient stabilized, generalized minimal residual and restarted generalized minimal residual methods. We demonstrate experimentally that the proposed iteration methods are feasible and efficient for the periodic matrix equation. [ABSTRACT FROM AUTHOR]

Finite tight frames are used widely for many applications. An important problem is to construct finite frames containing specified elements as needed in real world applications. In this paper we provide a stable method for that purpose. We establish an identity for constructing equi-norm tight frames from the specified elements and present a method for generating equi-norm tight frames explicitly from the outcome of the singular value decomposition of the matrix of the given elements. Under certain conditions, the number of columns of the unit-norm tight frames generated by our method reaches the minimum bound, improving a result in the literature. Finally, we illustrate our method through a numerical example. [ABSTRACT FROM AUTHOR]

Suppose we have an observed data matrix D which can be decomposed as the sum of a sparse matrix E and a low-rank matrix A. The purpose of this paper is to recover the sparse component and low-rank component individually from a given observation matrix. In this paper, a novel method proposed here deviates from the other approaches listed is the splitting of the constraint D = A + E into two constraints, A = A 0 and E = E 0 , where A 0 and E 0 are the real low-rank and sparse part of data matrix D. This separation strategy is referred to as the separable surrogate function (SSF). In such case, two iterative methods are designed to solve this problem. Correspondingly, the convergence analysis of these two iterative methods is given respectively. Simulations about real-data examples and applications on images decomposition show the feasibility and effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

As is known, Alternating-Directional Doubling Algorithm (ADDA) is quadratically convergent for computing the minimal nonnegative solution of an irreducible singular M-matrix algebraic Riccati equation (MARE) in the noncritical case or a nonsingular MARE, but ADDA is only linearly convergent in the critical case. The drawback can be overcome by deflating techniques for an irreducible singular MARE so that the speed of quadratic convergence is still preserved in the critical case and accelerated in the noncritical case. In this paper, we proposed an improved deflating technique to accelerate further the convergence speed – the double deflating technique for an irreducible singular MARE in the critical case. We proved that ADDA is quadratically convergent instead of linearly when it is applied to the deflated algebraic Riccati equation (ARE) obtained by a double deflating technique. We also showed that the double deflating technique is better than the deflating technique from the perspective of dimension of the deflated ARE. Numerical experiments are provided to illustrate that our double deflating technique is effective. [ABSTRACT FROM AUTHOR]

In this paper we show how to improve the approximate solution of the large Sylvester equation obtained by an arbitrary method. Such problems appear in many areas of control theory such as the computation of Hankel singular values, model reduction algorithms and others. Moreover, we propose a new method based on refinement process and weighted block Arnoldi algorithm for solving large Sylvester matrix equation. The numerical tests report the effectiveness of these methods. [ABSTRACT FROM AUTHOR]

Letbe the set of allmatrices over the real quaternion algebra.,,, whereis the conjugate of the quaternion. We call thatis-Hermitian, if,;is-bihermitian, if. We in this paper, present the solvability conditions and the general-Hermitian solution to a system of linear real quaternion matrix equations. As an application, we give the necessary and sufficient conditions for the systemto have an-bihermitian solution. We establish an expression of the-bihermitian to the system when it is solvable. We also obtain a criterion for a quaternion matrix to be-bihermitian. Moreover, we provide an algorithm and a numerical example to illustrate the theory developed in this paper. [ABSTRACT FROM PUBLISHER]

This paper introduces an alternating direction method of multipliers (ADMM) for finding solutions to a class of Sylvester matrix equationAXB = Esubject to a linear matrix inequality constraintCXD≥G. Preliminary convergence properties of ADMM are presented. Numerical experiments are performed to illustrate the feasibility and effectiveness of ADMM. In addition, some numerical comparisons with a recent algorithm are also given. [ABSTRACT FROM PUBLISHER]

In this paper, we propose and discuss a new class of complex nonsymmetric algebraic Riccati equations (NAREs) whose four coefficient matrices form a matrix with its ω-comparison matrix being an irreducible singular M-matrix. We also prove that the extremal solutions of the NAREs exist uniquely in the noncritical case and exist in the critical case. Some good properties of the solutions are also shown. Besides, some classical numerical methods, including the Schur methods, Newton's method, the fixed-point iterative methods and the doubling algorithms, are also applied to solve the NAREs, and the convergence analysis of these methods are given in details. For the doubling algorithms, we also give out the concrete parameter selection strategies. The numerical results show that our methods are efficient for solving the NAREs. [ABSTRACT FROM AUTHOR]

In this paper, we present an efficient method for nonnegative matrix factorization (NMF) based on the alternating nonnegative least-squares framework. To solve the nonnegativity constrained least-squares problems efficiently, we propose an extrapolated quadratic regularization projected Barzilai–Borwein (EQRPBB) method utilizing the extrapolation technique and a modified nonmonotone line search. The efficiency of the proposed method is demonstrated through experiments on synthetic and image datasets. We observe that our method significantly outperform existing ones in terms of computational speed. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new Schulz-type method to find the pseudo-inverse (also known as the Moore–Penrose inverse) of a singular or rectangular real (or complex) matrix. It is proved that the method converges quadratically. A wide set of numerical comparisons of our method with nine higher order methods shows that the average number of matrix–matrix multiplications and the average CPU time of proposed method are considerably less than those of other methods. So, our new method can be considered as a fast method. For each of sizesand,n=100,200,300,400, ten random matrices were chosen to make these comparisons. So, overall 800 problems were solved. [ABSTRACT FROM AUTHOR]

For given skew-Hermitian unitary matrixJ, i.e.,, a matrixis termed generalized Hamiltonian matrix if. In this paper, an iterative method is constructed to solve the generalized Hamiltonian solutions of the coupled Sylvester matrix equations with conjugate transpose. It is proved that the iterative method is unconditionally convergent for any initial generalized Hamiltonian matrices. With it, the generalized Hamiltonian solutions can be obtained within finite iteration steps in the absence of roundoff errors. Finally, numerical examples are presented to illustrate the efficiency and applicability of the method. [ABSTRACT FROM PUBLISHER]

In the current paper, a new serial algorithm for solving nearly penta-diagonal linear systems is presented. The computational cost of the algorithm is less than or almost equal to those of recent successful algorithms [J. Jia, Q. Kong, and T. Sogabe,A fast numerical algorithm for solving nearly penta-diagonal linear systems, Int. J. Comput. Math. 89 (2012), pp. 851–860; X.G. Lv and J. Le,A note on solving nearly penta-diagonal linear systems, Appl. Math. Comput. 204 (2008), pp. 707–712; S.N. Neossi Nguetchue and S. Abelman,A computational algorithm for solving nearly penta-diagonal linear systems, Appl. Math. Comput. 203 (2008), pp. 629–634]. Moreover, it is suitable for developing its parallel algorithms. One of the parallel algorithms is given and is shown to be promising. The implementation of the algorithms using Computer Algebra Systems such as MATLAB and MAPLE is straightforward. Two numerical examples are given in order to illustrate the validity and efficiency of our algorithms. [ABSTRACT FROM PUBLISHER]

This paper describes iterative methods for solving the general linear matrix equation including the well-known Lyapunov matrix equation, Sylvester matrix equation and some related matrix equations encountered in control system theory, as special cases. We develop the methods from the optimization point of view in the sense that the iterative algorithms are constructed to solve some optimization problems whose solutions are closely related to the unique solution to the linear matrix equation. Actually, two optimization problems are considered and, therefore, two iterative algorithms are proposed to solve the linear matrix equation. To solve the two optimization problems, the steepest descent method is adopted. By means of the so-called weighted inner product that is defined and studied in this paper, the convergence properties of the algorithms are analysed. It is shown that the algorithms converge at least linearly for arbitrary initial conditions. The proposed approaches are expected to be numerically reliable as only matrix manipulation is required. Numerical examples show the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we present a fast numerical algorithm for solving nearly penta-diagonal linear systems and show that the computational cost is less than those of three algorithms in El-Mikkawy and Rahmo, [Symbolic algorithm for inverting cyclic penta-diagonal matrices recursively–Derivation and implementation, Comput. Math. Appl. 59 (2010), pp. 1386–1396], Lv and Le [A note on solving nearly penta-diagonal linear systems, Appl. Math. Comput. 204 (2008), pp. 707–712] and Neossi Nguetchue and Abelman [A computational algorithm for solving nearly penta-diagonal linear systems, Appl. Math. Comput. 203 (2008), pp. 629–634.]. In addition, an efficient way of evaluating the determinant of a nearly penta-diagonal matrix is also discussed. The algorithm is suited for implementation using computer algebra systems (CAS) such as MATLAB, MACSYMA and MAPLE. Some numerical examples are given in order to illustrate the efficiency of our algorithm. [ABSTRACT FROM AUTHOR]

This paper is a matrix iterative method presented to compute the solutions of the matrix equation, AXB=C, with unknown matrix X∈S, where S is the constrained matrices set like symmetric, symmetric-R-symmetric and (R, S)-symmetric. By this iterative method, for any initial matrix X0∈S, a solution X* can be obtained within finite iteration steps if exact arithmetics were used, and the solution X* with the minimum Frobenius norm can be obtained by choosing a special kind of initial matrix. The solution [image omitted] , which is nearest to a given matrix ~X in Frobenius norm, can be obtained by first finding the minimum Frobenius norm solution of a new compatible matrix equation. The numerical examples given here show that the iterative method proposed in this paper has faster convergence and higher accuracy than the iterative methods proposed in [G.-X. Huang, F. Yin, and K. Guo, An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation AXB=C, J. Comput. Appl. Math. 212 (2008), pp. 231-244; Y. Lei and A.-P. Liao, A minimal residual algorithm for the inconsistent matrix equation AXB=C over symmetric matrices, Appl. Math. Comput. 188 (2007), pp. 499-513; Z.-Y. Peng, An iterative method for the least squares symmetric solution of the linear matrix equation AXB=C, Appl. Math. Comput. 170 (2005), pp. 711-723; Y.-X. Peng, X.-Y. Hu, and L. Zhang, An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB=C, Appl. Math. Comput. 160 (2005), pp. 763-777]. [ABSTRACT FROM AUTHOR]

In this paper, a new and different neural network, called Zhang Neural Network (ZNN) is appropriated from discrete time-varying matrix problems and applied to the angle parameter-varying matrix field of values (FoV) problem. This problem acts as a test bed for newly discovered convergent 1-step ahead finite difference formulas of high truncation error orders. The ZNN method that uses a look-ahead finite difference scheme of error order 6 gives us 15+ accurate digits of the FoV boundary points in record time when applied to hermitian matrix flows A (t). [ABSTRACT FROM AUTHOR]

In this paper, we establish results fully addressing two open problems proposed recently by I. Ivanov, see Nonlinear Analysis 69 (2008) 4012–4024, with respect to the convergence of the accelerated Riccati iteration method for solving the continuous coupled algebraic Riccati equation, or CCARE for short. These results confirm several desirable features of that method, including the monotonicity and boundedness of the sequences it produces, its capability of determining whether the CCARE has a solution, the extremal solutions it computes under certain circumstances, and its faster convergence than the regular Riccati iteration method. [ABSTRACT FROM AUTHOR]

Canonical correlation analysis (CCA) is a cornerstone of linear dimensionality reduction techniques that jointly maps two datasets to achieve maximal correlation. CCA has been widely used in applications for capturing data features of interest. In this paper, we establish a range constrained orthogonal CCA (OCCA) model and its variant and apply them for three data analysis tasks of datasets in real-life applications, namely unsupervised feature fusion, multi-target regression and multi-label classification. Numerical experiments show that the OCCA and its variant produce superior accuracy compared to the traditional CCA. [ABSTRACT FROM AUTHOR]

In this paper, Newton's iterative method to solve the nonlinear matrix equation is studied. For the given initial matrix Q, the main results that the matrix sequence generated by the iterative method is contained in a fixed open ball, and that the matrix sequence generated by the iterative method converges to the only solution of the nonlinear matrix equation in a fixed closed ball are proved. In addition, the error estimate of the approximate solution in the closed ball and a numerical example to illustrate the convergence results are given. [ABSTRACT FROM AUTHOR]

Metabolic P (MP) grammars are a particular class of multiset rewriting grammars introduced in the MP theory for modelling metabolic processes. In this paper, a new algebraic formulation of inverse problems, based on MP grammars and Kronecker product, is given, for further motivating the correctness of the LGSS (Log-Gain Stoichiometric Stepwise) algorithm, introduced in 2010s for solving inverse problems in the MP framework. At the end of the paper, a section is included that introduces the problem of multicollinearity, which could arise during the execution of LGSS, and that defines an algorithm, based on a hierarchical clustering technique, that solves it in a suitable way. [ABSTRACT FROM PUBLISHER]

In this paper, we present an iterative method to compute the symmetric solutions of the matrix inequalityAX≥Bin least-squares sense. The presented iteration method can be also used to compute the symmetric solutions of the matrix equationAX=Band the consistent matrix inequalityAX≥B. Numerical experiments are given to illustrate the usefulness of the proposed approach. [ABSTRACT FROM PUBLISHER]

In this paper, an iterative method is presented for finding the bisymmetric solutions of a pair of consistent matrix equations A1XB1=C1, A2XB2=C2, by which a bisymmetric solution can be obtained in finite iteration steps in the absence of round-off errors. Moreover, the solution with least Frobenius norm can be obtained by choosing a special kind of initial matrix. In the solution set of the matrix equations, the optimal approximation bisymmetric solution to a given matrix can also be derived by this iterative method. The efficiency of the proposed algorithm is shown by some numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, the characteristic polynomial of general heptadiagonal matrices is derived as well as eigenvectors associated to a prescribed eigenvalue. A symbolic algorithm to compute the determinant of heptadiagonal matrices is also presented allowing a suite implementation through computational software programs. [ABSTRACT FROM AUTHOR]

The Bethe–Salpeter equation (BSE) is a reliable model for estimating the absorption spectra in molecules and solids on the basis of accurate calculation of the excited states from first principles. Direct diagonalisation of the BSE matrix is practically intractable due toO(N6) complexity scaling in the size of the atomic orbital basis set,N. In this paper, we introduce and analyse a reduced basis approach to computation of the Bethe–Salpeter excitation energies which can lead to a relaxation of the numerical costs down toO(N3). The BSE operator is specified in terms of the two-electron integrals in the Hartree–Fock molecular orbital basis and the respective energies, calculated by the tensor-based solver described in previous works. The reduced basis method includes two steps. First, the diagonal plus low-rank approximation to fully populated blocks in the BSE matrix is calculated, enabling an easier partial eigenvalue solver for a large auxiliary system relying only on matrix–vector multiplications with rank-structured matrices. Second, a small subset of eigenvectors from the auxiliary eigenvalue problem is selected to build a projection of the exact BSE system onto this reduced basis set. Numerical tests demonstrate the ϵ-rank bounds for the blocks of the BSE matrix on examples of some compact molecules. The accuracy of the reduced basis approach vs. the effective matrix rank is illustrated. [ABSTRACT FROM PUBLISHER]

Letandbe two real symmetric orthogonal matrices. Anreal matrixis said to be an-symmetric matrix if. The-symmetric matrices have been widely used in engineering and scientific computing. In this paper, we construct an algorithm to solve the general coupled matrix equations, whereis a-symmetric matrix. The algorithm produces suitablesuch thatis minimized within a finite number of iteration steps in the absence of round-off errors. We show that the algorithm is stable any case. The algorithm requires little storage capacity. Numerical examples are given to show that the algorithm is efficient. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following minimization problem:where,,,andare given. An efficient inequality relaxation technique is presented to relax the matrix inequality constraint so that there is an optimal solution which is(R,S)-symmetric that minimize, and also satisfies the corrected matrix inequality constraint. A hybrid algorithm with convergence analysis is given to solve this problem. Numerical examples show that the algorithm requires less CPU times when compared with some other methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new method for polynomial optimization with real or complex decision variables. The main ingredient of the approach is to apply the classical alternating direction method of multipliers based on the augmented Lagrangian function. In this particular case, this allows us to fully exploit the multi-block structure of the polynomial functions, even though the optimization model encountered is highly non-linear and non-convex. The new method is shown to be convergent under some conditions, and the numerical results show that the algorithm returns high quality solutions and runs much faster than the two other competing algorithms. [ABSTRACT FROM AUTHOR]

The online social networking phenomenon is growing rapidly all around the world. As a consequence, in recent years, several studies have been devoted to the analysis of social network sites. A specific issue that has been addressed is the identification of leaders of the social network based on well-known algorithms such as PageRank. In this paper, we propose a novel algorithm based on the PageRank method to determine the minimum number of new friends required by a user of a social network to become the user with the highest PageRank in the network. We provide formal mathematical definitions and validate our proposal with some experiments based on artificial and real data. [ABSTRACT FROM PUBLISHER]

Detection of copositivity plays an important role in combinatorial and quadratic optimization. Recently, an algorithm for copositivity detection by simplicial partition has been proposed. In this paper, we develop an improved depth-first simplicial partition algorithm which reduces memory requirements significantly and therefore enables copositivity checks of much larger matrices - of size up to a few thousands instead of a few hundreds. The algorithm has been investigated experimentally on a number of MaxClique problems as well as on generated random problems. We present numerical results showing that the algorithm is much faster than a recently published linear algebraic algorithm for copositivity detection based on traditional ideas - checking properties of principal sub-matrices. We also show that the algorithm works very well for solving MaxClique problems through copositivity checks. [ABSTRACT FROM AUTHOR]

The present paper develops an “Erlangization” method for fluid queues. It then applies the results we derive to a forestry problem: the evolution of an uncontrolled fire perimeter over time. Specifically, we focus on the probability of containing a fire prior to reaching a randomly distributed, finite time horizon. Transitions to lower non-zero levels are also investigated. A preliminary model is introduced to demonstrate the potential of the application, and numerical results are given for illustrative purposes. [ABSTRACT FROM AUTHOR]

We perform extensive theoretical analysis on an inversion-free algorithm to obtain a maximal positive definite solution (PDS) of a general nonlinear matrix equation (NME) with suitable square matrices. We illustrate some estimations for obtaining the solution, rate of convergence and convergence analysis of the presented algorithm. To ensure its validity and usefulness, several different types of test problems, including randomly generated ones, are given. Comparisons with the existing iterative methods are done through error and computational time. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the Stirling numbers of both kinds can be written as sums over integer partitions. As corollaries, we rewrite some identities with Stirling numbers of both kinds without Stirling numbers. [ABSTRACT FROM PUBLISHER]

□ We present a new approach to carry out transient analysis for Markov modulated fluid flows. The system of partial differential equations for the time-dependent distributions is transformed into a system of ordinary differential equations by Laplace transform. This system of ordinary differential equations is solved by means of decomposing its coefficient matrix into two parts, one with eigenvalues with negative real parts and the other with positive real parts. The results we obtained are equivalent, but in simpler form, to those of Ahn and Ramaswami, which are derived by probabilistic arguments. [ABSTRACT FROM PUBLISHER]

We present a new method for computing zeros of polynomial systems using the algebraic solver and the QR-method. It is based on the theory of algebraic solvers. The unstable calculation of the determinant of the large matrix is replaced by a stable technique using the QR-method. Algorithms and numerical results are presented. [ABSTRACT FROM AUTHOR]

This work discusses the iterative refinement for finding the numerical inverse of a given non-singular matrix. The algorithm and convergence analysis of the iterative procedures are presented. Numerical examples are given to show the effectiveness of the iterative refinement. [ABSTRACT FROM AUTHOR]

The explicit expressions of normwise, mixed and componentwise condition numbers for the K-weighted pseudoinverse L K † are first presented. With the intermediate result, i.e. the derivative of L K † , we can recover the explicit expressions of condition numbers for solution of least squares problem with equality constraint. Then, we investigate the statistical estimation of condition numbers of L K † using the probabilistic spectral norm estimator and the small-sample statistical condition estimation method, and devise three algorithms. Finally, numerical examples are provided to illustrate these results. [ABSTRACT FROM AUTHOR]

We present an iterative method based on the Hermitian and skew-Hermitian splitting (HSS) for solving the continuous Sylvester equation. By using the HSS of the coefficient matricesAandB, we establish a method which is practically inner/outer iterations, by employing a conjugate gradient on the normal equations (CGNR)-like method as inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergent splitting of the coefficient matrices. Via this method, a Sylvester equation with coefficient matricesand(which are the skew-Hermitian part ofAandB, respectively) is solved iteratively by a CGNR-like method. Convergence conditions of this method are studied and numerical examples show the efficiency of this method. In addition, we show that the quasi-Hermitian splitting can induce accurate, robust and effective preconditioned Krylov subspace methods. [ABSTRACT FROM PUBLISHER]

The saddle-point system is considered, where A is an m × n matrix with full column rank, B is an m × m symmetric and positive definite matrix, and b is an mvector. When the perturbed systems are still saddle-point systems, perturbation theory is presented under three cases, such as the case of A and b perturbed, the case of B and b perturbed, and the case of A, B and b perturbed.We find that the bounds for the errors of the solution depend on A and B. To remove their influence, some conditions are disposed. To improve the accuracy of the solution, a scaling is given. When the perturbed systems are generalized saddle-point systems, the block LDLT factorization is applied. Thus the sensitivity of the block LDLT factorization should be discussed, then perturbation theorems for this case are presented. [ABSTRACT FROM AUTHOR]

Many studies of mechanical systems in engineering are based on second-order matrix models. This work discusses the second-order generalization of previous research on matrix differential equations dealing with the construction of approximate solutions for non-stiff initial problemsusing higher degree matrix splines without any dimensional increase. An estimation of the approximation error for some illustrative examples are presented by using Mathematica. Several MatLabfunctions have also been developed, comparing, under equal conditions, accuracy and execution times with built-in MatLabfunctions. Experimental results show the advantages of solving the above initial problem by using the implemented MatLabfunctions. [ABSTRACT FROM AUTHOR]

We revisit an earlier idea of Marcus and Pesce from 1987 for generating fields of values ofbymatricesvia 2 by 2 matrix compressions and their easily constructed elliptical field of values. This approach is used to reduce the cost of finding the FOV boundary curve of a matrixand makes it more accurate at the same time. The new algorithm succeeds by using fewer eigenanalyses and constructing 2 by 2 matrix compression ellipses offor approximatinginstead of only eigenanalyses. An application to verify a counterexample to theZemanek Conjecture for companion matrices of monic polynomials and their normalized derivatives inand, respectively, in dimension 4 is given. [ABSTRACT FROM PUBLISHER]

Based on a hypersingular integral method, we analyze the use of a coupled Nyström and finite-element procedure for approximating the Helmholtz equation in an exterior domain. The coupling technique uses a smooth artificial boundary and hypersingular integral equation and finite-element computations. We prove convergence of the method. One highlight of this analysis is that we prove Sobolev space error estimates for the Nyström scheme (which is usually analyzed in Hölder spaces). Finally, we discuss some details of implementing the coupled method and prove the discrete Dirichlet–Neumann alternation iterative method solving the problem more rapidly. [ABSTRACT FROM PUBLISHER]

This work gives a new formula for the forward relative error of matrix exponential Taylor approximation and proposes new bounds for it depending on the matrix size and the Taylor approximation order, providing a new efficient scaling and squaring Taylor algorithm for the matrix exponential. A Matlab version of the new algorithm is provided and compared with Padé state-of-the-art algorithms obtaining higher accuracy in the majority of tests at similar or even lower cost. [ABSTRACT FROM PUBLISHER]

Quaternionic least squares (QLS) is an efficient method for solving approximate problems in quaternionic quantum theory. Based on Paige's algorithms LSQR and residual-reducing version of LSQR proposed in Paige and Saunders [LSQR: An algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Softw. 8(1) (1982), pp. 43–71], we provide two matrix iterative algorithms for finding solution with the least norm to the QLS problem by making use of structure of real representation matrices. Numerical experiments are presented to illustrate the efficiency of our algorithms. [ABSTRACT FROM PUBLISHER]

Algorithms for calculating sequences of upper and lower bounds for an estimate of the optimal backward error for linear least squares problems are developed, based on the Gauss quadrature theory. Numerical results show that the bounds converge quickly and are therefore useful in practice. [ABSTRACT FROM AUTHOR]

In this article, there is offered a parametric class of iterative methods for computing the polar decomposition of a matrix. Each iteration of this class needs only one scalar-by-matrix and three matrix-by-matrix multiplications. It is no use computing inversion, so no numerical problems can be created because of ill-conditioning. Some available methods can be included in this class by choosing a suitable value for the parameter. There are obtained conditions under which this class is always quadratically convergent. The numerical comparison performed among six quadratically convergent methods for computing polar decomposition, and a special method of this class, chosen based on a specific value for the parameter, shows that the number of iterations of the special method is considerably near that of a cubically convergent Halley's method. Ten n×n matrices with n=5, 10, 20, 50, 100 were chosen to make this comparison. [ABSTRACT FROM AUTHOR]

We discuss a parallel library of efficient algorithms for the solution of linear-quadratic optimal control problems involving large-scale systems with state-space dimension up to O(104). We survey the numerical algorithms underlying the implementation of the chosen optimal control methods. The approaches considered here are based on invariant and deflating subspace techniques, and avoid the explicit solution of the associated algebraic Riccati equations in case of possible ill-conditioning. Still, our algorithms can also optionally compute the Riccati solution. The major computational task of finding spectral projectors onto the required invariant or deflating subspaces is implemented using iterative schemes for the sign and disk functions. Experimental results report the numerical accuracy and the parallel performance of our approach on a cluster of Intel Itanium-2 processors. [ABSTRACT FROM AUTHOR]