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Several iterative methods for maximal correlation problems (MCPs) have been proposed in the literature. This paper deals with the convergence of these iterations and contains three contributions. Firstly, a unified and concise proof of the monotone convergence of these iterative methods is presented. Secondly, a starting point strategy is analysed. Thirdly, some error estimates are presented to test the quality of a computed solution. Both theoretical results and numerical tests suggest that combining with this starting point strategy these methods converge rapidly and are more likely converging to a global maximizer of MCP. [ABSTRACT FROM AUTHOR]

SUMMARY We study the linearization method to estimate the backward error of approximate solutions to several least squares problems, including the scaled total least squares (STLS) problem, the equality constrained least squares (LSE) problem, and the least squares problem over a sphere (LSS). For the STLS problem, we present several new results about the linearization estimate derived by Chang and Titley-Peloquin. For the LSE and the LSS problems, we derive linearization estimates of the backward errors and compare them with existing backward error bounds by numerical tests. Our experiments show that the linearization estimates are good enough approximations of the backward errors as the approximate solution approaches the exact one. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: Recently, Psarrakos obtained a lower bound, which depends on a real parameter , for the distance between the pseudospectrum of a matrix polynomial and a given point that lies out of it. In this note, we study the best choice of the parameter. As a consequence, the best lower bound among those given by Psarrakos is obtained. [Copyright &y& Elsevier]

The trajectory calculations of He+H2+, He+D2+ and He+T2+ reactions on the Palmieri potential energy surface (Palmieri et al., Mol Phys, 2000, 98, 1835) have been carried out to study the isotope effect on stereodynamics at the collision energy of 50 kcal/mol. The distributions of dihedral angle P (φr) and the distributions of angle between and ′, P (θr) are discussed. The product rotational alignment parameters 〈P2(′&cirf;)> for the title reactions are compared with the mass factors. Furthermore, the angular distributions of the product rotational vectors in the form of polar plot in θr and φr are calculated. The calculations indicate that the stereodynamics properties of the title reactions are sensitive to the mass factor and the long range interaction of ion-molecule reaction. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010 [ABSTRACT FROM AUTHOR]

Abstract: Block numerical ranges of matrix polynomials, especially the quadratic numerical range, are considered. The main results concern spectral inclusion, boundedness of the block numerical range, an estimate of the resolvent in terms of the quadratic numerical range, geometrical properties of the quadratic numerical range, and inclusion between block numerical ranges of the matrix polynomials for refined block decompositions. As an application, we connect the quadratic numerical range with the localization of the spectrum of matrix polynomials. [Copyright &y& Elsevier]

Abstract: The vector correlations between products and reagents for the chemical reactions have been calculated by means of the quasi-classical trajectory (QCT) method at a collision energy of 1.0eV on the AQUILANTI surface [P. Palmieri, C. Puzzarini, V. Aquilanti, G. Capecchi, S. Cavalli, D. De Fazio, A. Aguilar, X. Gimenez, J.M. Lucas, Mol. Phys. 98 (2000) 1835]. The isotope effect on dynamical stereochemistry of the title reactions is discussed in detail. The pronounced isotopic effect may be ascribed to the difference of the mass factor in the two reactions. [Copyright &y& Elsevier]

Abstract: There are known results about the continuity of the important pseudoinverses. We extend some well known results to the {1}-inverse. As for the Drazin inverse, which is not a {1}-inverse, we present new perturbation bounds. [Copyright &y& Elsevier]

Abstract: We consider the two-sided Rayleigh quotient introduced by Ostrowski as convergence acceleration tool for eigenvalue solvers. An interesting property of the two-sided Rayleigh quotient is established: its approximation to eigenvalues depends on the eigenvalue condition number. This fact is valid not only for matrix but also for matrix polynomial, multiparameter system, and periodic matrix pairs. [Copyright &y& Elsevier]

Let m, n, p be positive integers such that m ≥⃒ n+p. Suppose (A, B) ∈ Cm×n × Cm×p, and let The total least square problem concerns the determination of the existence of (E, F ) in having the smallest Frobenius norm. In this article, we characterize elements of the set and derive a formula forfor any unitarily invariant norm |·| on Cm×(n+p), where [E|F] denotes the m× (n+p) matrix formed by the columns of E and F. Furthermore, we give a necessary and sufficient condition on (A, B) and the unitarily invariant norm |·| so that there exists attaining ρ (A,B). The results cover those on the total least square problem, and those of Huang and Yan on the existence of so that [E|F] has the smallest spectral norm. [ABSTRACT FROM AUTHOR]

Abstract: We discuss the backward error analysis of the Hermite eigenproblems. Three kinds of problems are included: generalized eigenproblem, matrix polynomial eigenproblem, and multiparameter eigenproblem. In the case of the approximate eigenvalue being real, these problems have already been solved by Higham and Higham [SIAM J. Matrix Anal. 20 (1998) 493], Tisseur [Lin. Alg. Appl. 309 (2002) 339], and Hachstenbach and Plestenjak [Lin. Alg. Appl. 375 (2003) 63]. This note as a complement of their results considers the complex case. [Copyright &y& Elsevier]

In this paper, we consider a class of alternating coefficient matrices quadratic eigenvalue problem (AQEP) appearing in a wide range of applications. At first, the AQEP is reformulated as a generalized eigenvalue problem (GEP) by applying Tisseur''s linearization technique. Then we point out a property about skew-symmetric tridiagonal matrix eigenvalue problem. Consequently, we give that the corresponding GEP can be solved by applying the Cholesky-QL algorithm or Lanczos algorithm without complex arithmetic. The structured backward error for QEP would be discussed because of the special structure of the polynomial eigenvalue problems. We develop structured condition number, the upper bound and the lower bound are given. Some computational experiments and some concluding remarks are provided. [Copyright &y& Elsevier]

The two matrix equations Xs+ATX−tA=In and Xs−ATX−tA=In are studied. Based on the fixed-point theory, the existence of the symmetric positive definite solutions are proved. Sensitivity analysis of the maximal solution is presented. Some elegant estimates of the positive definite solutions are obtained. Three iterative methods for computing the positive solutions are proposed. [Copyright &y& Elsevier]

The product rotational polarizations of reaction Li+HF→LiF+H at different collision energies, as well as at the different vibrational states and rotational states, are calculated by using the quasi-classical trajectory method based on a new potential energy surface constructed by Aguado et al. [J. Chem. Phys. 119 (2003) 10088]. We investigate the alignment and the orientation of the product molecule by calculating the P(θr, ϕr) distributions describing polar angle distribution, the P(θr) distributions describing the k–j′ correlation and the P(ϕr) distributions describing the k–k′–j′ correlation. We also explore the dependence of reaction probabilities and cross sections on the rotational and vibrational quantum number of the title reaction. It is concluded that the vibrational state has more important impact on the angular distribution, reaction probability and cross section. [ABSTRACT FROM AUTHOR]