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Corresponding Eigenvectors (corresponding + eigenvector)
Selected AbstractsTwo characterizations of matrices with the Perron,Frobenius propertyNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 11-12 2009Abed Elhashash Abstract Two characterizations of general matrices for which the spectral radius is an eigenvalue and the corresponding eigenvector is either positive or nonnegative are presented. One is a full characterization in terms of the sign of the entries of the spectral projector. In another case, different necessary and sufficient conditions are presented that relate to the classes of the matrix. These characterizations generalize well-known results for nonnegative matrices. Copyright © 2009 John Wiley & Sons, Ltd. [source] A fast algorithm for computing the smallest eigenvalue of a symmetric positive-definite Toeplitz matrixNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 4 2008N. Mastronardi Abstract Recent progress in signal processing and estimation has generated considerable interest in the problem of computing the smallest eigenvalue of a symmetric positive-definite (SPD) Toeplitz matrix. An algorithm for computing upper and lower bounds to the smallest eigenvalue of a SPD Toeplitz matrix has been recently derived (Linear Algebra Appl. 2007; DOI: 10.1016/j.laa.2007.05.008). The algorithm relies on the computation of the R factor of the QR factorization of the Toeplitz matrix and the inverse of R. The simultaneous computation of R and R,1 is efficiently accomplished by the generalized Schur algorithm. In this paper, exploiting the properties of the latter algorithm, a numerical method to compute the smallest eigenvalue and the corresponding eigenvector of SPD Toeplitz matrices in an accurate way is proposed. Copyright © 2008 John Wiley & Sons, Ltd. [source] On singularities in the solution of three-dimensional Stokes flow and incompressible elasticity problems with cornersINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2004A. Dimitrov Abstract In this paper, a numerical procedure is presented for the computation of corner singularities in the solution of three-dimensional Stokes flow and incompressible elasticity problems near corners of various shape. For obtaining the order and mode of singularity, a neighbourhood of the singular point is considered with only local boundary conditions. The weak formulation of this problem is approximated using a mixed u, p Galerkin,Petrov finite element method. Additionally, a separation of variables is used to reduce the dimension of the original problem. As a result, the quadratic eigenvalue problem (P+,Q+,2R)d=0 is obtained, where the saddle-point-type matrices P, Q, R are defined explicitly. For a numerical solution of the algebraic eigenvalue problem an iterative technique based on the Arnoldi method in combination with an Uzawa-like scheme is used. This technique needs only one direct matrix factorization as well as few matrix,vector products for finding all eigenvalues in the interval ,,(,) , (,0.5, 1.0), as well as the corresponding eigenvectors. Some benchmark tests show that this technique is robust and very accurate. Problems from practical importance are also analysed, for instance the surface-breaking crack in an incompressible elastic material and the three-dimensional viscous flow of a Newtonian fluid past a trihedral corner. Copyright © 2004 John Wiley & Sons, Ltd. [source] Efficient computation of order and mode of corner singularities in 3D-elasticityINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2001A. Dimitrov Abstract A general numerical procedure is presented for the efficient computation of corner singularities, which appear in the case of non-smooth domains in three-dimensional linear elasticity. For obtaining the order and mode of singularity, a neighbourhood of the singular point is considered with only local boundary conditions. The weak formulation of the problem is approximated by a Galerkin,Petrov finite element method. A quadratic eigenvalue problem (P+,Q+,2R) u=0 is obtained, with explicitly analytically defined matrices P,Q,R. Moreover, the three matrices are found to have optimal structure, so that P,R are symmetric and Q is skew symmetric, which can serve as an advantage in the following solution process. On this foundation a powerful iterative solution technique based on the Arnoldi method is submitted. For not too large systems this technique needs only one direct factorization of the banded matrix P for finding all eigenvalues in the interval ,e(,),(,0.5,1.0) (no eigenpairs can be ,lost') as well as the corresponding eigenvectors, which is a great improvement in comparison with the normally used determinant method. For large systems a variant of the algorithm with an incomplete factorization of P is implemented to avoid the appearance of too much fill-in. To illustrate the effectiveness of the present method several new numerical results are presented. In general, they show the dependence of the singular exponent on different geometrical parameters and the material properties. Copyright © 2001 John Wiley & Sons, Ltd. [source] The reaction between ethyl and molecular oxygen II: Further analysisINTERNATIONAL JOURNAL OF CHEMICAL KINETICS, Issue 11 2001James A. Miller The present investigation is a rather substantial extension and elaboration of our previous work on the same reaction. In this article we accomplish four primary objectives: 1.We show quantitatively how sensitive the high-temperature rate coefficient k(T) is to E02, the threshold energy of the transition state for direct molecular elimination of HO2 from ethylperoxy radical (C2H5O2), thus deducing a value of E02=,3.0 kcal/mol (measured from reactants). 2.We derive the result that k0(T) , k,,(T) in the high-temperature regime, where k0(T) is the zero-pressure rate coefficient, and k,,(T) is the infinite-pressure rate coefficient for the bimolecular channel. 3.Most importantly, we discuss the three different regimes of the reaction (low-temperature, transition, and high-temperature) in terms of the eigenvectors and eigenvalues of G, the transition matrix of the master equation. The transition regime is shown to be a region of avoided crossing between the two chemically significant eigenvalue curves in which the thermal rate coefficient k (T ,p) jumps from one eigenvalue to the other. This jump is accompanied by a "mixing" of the corresponding eigenvectors, through which both eigenvectors deplete the reactant. The onset of the high-temperature regime is triggered by reaching the "stabilization limit" of the ethylperoxy adduct, a limit that is induced by a shift in equilibrium of the stabilization reaction. Our identification of the prompt and secondary HO2 formed by the reaction with these eigenvalue/eigenvector pairs leads to good agreement between theory and the experiments of Clifford et al. (J Phys Chem A 2000, 104, 11549,11560). 4.Lastly, we describe the master equation results in terms of a set of elementary reactions and phenomenological rate coefficients. © 2001 John Wiley & Sons, Inc. Int J Chem Kinet 33: 732,740, 2001 [source] Density functional and vibrational spectroscopic analysis of ,-caroteneJOURNAL OF RAMAN SPECTROSCOPY, Issue 6 2003S. Schlücker Abstract We report a computational study on the structural, energetic and vibrational spectroscopic characteristics of ,-carotene employing density functional theory (DFT). The optimized geometry and the complete vibrational spectrum calculated at the BPW91/6,31G* level, including infrared (IR) intensities and Raman activities, are presented. The centrosymmetric structure of ,-carotene is verified both theoretically and experimentally, by identifying a stable calculated structure with Ci symmetry and the mutually exclusive occurrence of bands in the experimental Fourier transform IR and Raman spectrum, respectively. The calculated vibrational spectra reflect the major characteristic features observed experimentally. Differences in the calculated IR intensities and Raman activities for a few dominant modes of two ,-carotene configuration isomers, the all- trans and the natural abundant (C6,C7) s- cis form, are explained qualitatively by the corresponding eigenvectors. At the level of theory employed, s- cis -,-carotene was found to be 8.8 kJ mol,1 more stable than the all- trans form. Calculations on ,-carotene model systems were performed to separate electronic from steric contributions. The higher stability of s- cis -,-carotene is explained by an energetically favored ,-ionone ring conformation, compensating for its shorter conjugation length in comparison with the all- trans form. Copyright © 2003 John Wiley & Sons, Ltd. [source] | ||||||||||