Mutual Orthogonality Research Articles

The Lanczos algorithm with partial reorthogonalization

The Lanczos algorithm is becoming accepted as a powerful tool for finding the eigenvalues and for solving linear systems of equations. Any practical implementation of the algorithm suffers however from roundoff errors, which usually cause the Lanczos vectors to lose their mutual orthogonality. In order to maintain some level of orthogonality, full reorthogonalization (FRO) and selective orthogonalization (SO) have been used in the past as a remedy. Here partial reorthogonalization (PRO) is proposed as a new method for maintaining semiorthogonality among the Lanczos vectors. PRO is based on a simple recurrence, which allows us to monitor the loss of orthogonality among the Lanczos vectors directly without computing the inner products. Based on the information from the recurrence, reorthogonalizations occur only when necessary. Thus substantial savings are made as compared to FRO. In some numerical examples we apply the Lanczos algorithm with PRO to the solution of large symmetric systems of linear equations and show that it is a robust and efficient algorithm for maintaining semiorthogonality among the Lanczos vectors. The results obtained compare favorably with the conjugate gradient method.

The Lanczos Algorithm With Partial Reorthogonalization

The Lanczos algorithm is becoming accepted as a powerful tool for finding the eigenvalues and for solving linear systems of equations. Any practical implementation of the algorithm suffers however from roundoff errors, which usually cause the Lanczos vectors to lose their mutual orthogonality. In order to maintain some level of orthogonality, full reorthogonalization (FRO) and selective orthogonalization (SO) have been used in the past as a remedy. Here partial reorthogonalization (PRO) is proposed as a new method for maintaining semiorthogonality among the Lanczos vectors. PRO is based on a simple recurrence, which allows us to monitor the loss of orthogonality among the Lanczos vectors directly without computing the inner products. Based on the information from the recurrence, reorthogonalizations occur only when necessary. Thus substantial savings are made as compared to FRO. In some numerical examples we apply the Lanczos algorithm with PRO to the solution of large symmetric systems of linear equations and show that it is a robust and efficient algorithm for maintaining semiorthogonality among the Lanczos vectors. The results obtained compare favorably with the conjugate gradient method.

部分モード法による建築構造物の固有値解析に関する研究 : 各層を部分構造とした取り扱いと層モードの連成効果について

In this paper, the eigenvalue problem of multi-story structures is considered by means of the Component Mode Method. In the method, a multi-story structre is divided at each story into single-story substructures. The resuing component mode is the mode shape of each story (story mode). From the consideration in terms of the story mode, the following results are obtained. (1) Saving of Computation Time This method needs a certain time in the process of dividing a structure into substructures, but the total computation time including eigenvalue analyses is much shorter, as the size of the eigenvalue analyses is sufficiently small in comparison with that of the orignal eigenvalue problem. Since, in the eigenvalue analysis of each story, basic modes such as translational and torsional ones are easily obtained in a few lowest modes, and since the modes of the whole structure can be constructed by the combination of such basic modes, the computation time will be sufficiently saved. (2) Extration of Necessary Modes The mode shapes of a structure depend on its story modes. It means that the mode shapes of the whole strure can be changed by selecting its story modes. For example, if only a translational mode is adopted at each story, only translational modes appear in the synthesized modes. Thus, the necessary modes which are important for the evaluation of structural responses can easily be exracted by the appropriate selection of story modes. Accordingly, for this Component Mode Method, one can expect a kind of filtering effects in order to extract the necessary modes. (3) Evaluation of Inter-story Relations in terms of Story Modes In the process of the Component Mode Method, inter-story relations are stored only in a matrix M, which shows the mutual orthogonality of story modes. Therefore, the coupling relations among stories are grasped as the coupling relations among story modes. The evaluation of such coupling effects is possible if and only if the Component Mode Method with substructures of each story is introduced

Jahn-Teller resonance states in theVbmetal hydrides: A model calculation

A simple tight-binding model representing hydrogen in the Vb metals and including a special electron-phonon interaction is shown to give a molecular Jahn-Teller resonance state for physically reasonable values of the various energy parameters. This unusual resonance state is pinned to the shell of metal atoms adjacent to hydrogen by an effective Coulombic attraction that exists because of the nearly equal but opposite electron displacements on that shell, caused by the direct interaction of hydrogen on the one hand and by the Jahn-Teller interaction on the other. The mutual orthogonality of these two interactions implies that the Jahn-Teller state is an intrinsic feature of the host metal.

The Tennessee Self-Concept Scale and the normal personality sphere (16PF).

An examination of the internal structure of the Tennessee Self-Concept Scale (TSCS) and the interrelationship among the TSCS scales and the secondary dimensions of the Sixteen Personality Factor Questionnaire (16PF) supported three conclusions: (a) The primary dimension underlying the TSCS is positive self-evaluation, freedom from neurotic symptoms, or the absence of anxiety, (b) this central dimension of the TSCS aligns with the 16PF secondary Anxiety vs. Adjustment, and is virtually independent of the other dimensions of the normal personality sphere, and (c) the mutual orthogonality of extraversion, anxiety or neuroticism, and an empirically derived psychoticism scale provided some support for Eysenck's PEN theory of personality organization.

A Simple Method for Evaluating the System Observability in Satellite Orbit Determination

A method is described for evaluating the system observability in satellite orbit determination. The method is simple and rapid in computation time, because it uses the calculation of the mutual orthogonality between the observation partial vectors rather than the calculation of eigenvalues of the Grammian matrix.

Pressure shifts of F-center hyperfine interaction parameters in alkali chlorides

We report here calculated values of the pressure shift in the shell I hyperfine interaction constants for F-centers in KCl, NaCl and LiCl. The calculation is based on a modified version of the pseudopotential approach of Bartram et al. with electronic polarization included by means of an r-dependent polarization potential. The results agree reasonably well with the available experimental data, and the strong dependence of the pressure shift on ion size ratio is shown to be mainly due to changes in the indirect-overlap contributions to the spin density. These terms result from the requirement of mutual orthogonality of core states on neighbouring ions, and their dependence on the lattice spacing is stronger than that of other contributions to the spin density. Our calculations further indicate that although the predicted value of the pressure shift is sensitive to errors in the pseudopotential and polarization parameters, the relative importance of the indirect overlap contributions is evident in spite of such errors.

Magnetic separation of the second kind: Magnetogravimetric, magnetohydrostatic, and magnetohydrodynamic separations

Magnetogravimetric, magnetohydrostatic, and magnetohydrodynamic separation techniques can be classified as magnetic separations of the second kind. Magnetic separation of the first kind (ordinary magnetic separation) relies on the inherent magnetic susceptibility of the material to be separated. When the medium of separation rather than the separated particles is made magnetizable, a new system of gravity separations can result (magnetic separation of the second kind). In magnetogravimetry, a colloidal solution of a ferro- or ferrimagnetic substance (magnetic fluid) acts as the separation medium. Magnetohydrostatic separations are conducted in an aqueous solution (or melt) of a strongly paramagnetic salt. Magnetohydrodynamics applies the Faraday effect (mutual orthogonality of the force thrust, electric, and magnetic fields) on suspended conducting minerals in an electrolytic solution placed in crossed electric and magnetic fields. The first technique was pioneered mainly in the United States, while the last two techniques were pioneered by Bunin and Andres in the Soviet Union and introduced to the West by Andres. The principles underlying the three separation techniques will be discussed.

Compact-cluster expansion II: Finite nuclei

The compact-cluster formalism, described previously for an infinite system (nuclear matter of liquid 3He), is now extended to the case of closed-shell nuclei. The main concern here is with the new features arising from the fact that the one-body operators of the theory (self-energy insertions and propagator weighting factors) are no longer diagonal in the one-body indices. Two different aspects re explored: (i) the orthogonality of the intermediate-state orbitals to the occupied-state orbitals, and (ii) the mutual orthogonality among the occupied-state orbitals. The concept of self-consistency is also carefully examined. Formally exact treatments of both aspects are presented, as well as some approximations suitable for practical calcultions. For (i), it is demonstrated by explicit calculation that an “orthogonalized plane-wave” approximation should be quite satisfactory for most purposes. This provides a detailed justification for the “ Qe R −1 Q” prescription for the treatment of the exclusion principle, as first proposed by C. W. Wong. Regarding (ii), it is shown that consistent “renormalized” formulation are possible with either a nonorthogonal set of orbitals, or with a strictly orthogonal set. The nonorthogonal formulation is conceptually simpler, and is the one best suited for Brueckner-Hartree-Fock calculations carried out in r space. The orthogonal formulation is more advantageous for calculations in which the occupied orbitals are expanded in terms of a harmonic-oscillator basis. The latter formulation is quite similar to a previous result of Balian and DeDominicis. A formal error in our previous discussion, pointed out by the latter authors, is also examined. It is shown that the set of all skeleton diagrams must be supplemented by certain “compensation skeletons” in order to ensure that the final energy expression includes every Goldstone diagram once and only once. It is argued that this complication does not invalidate any of the important and useful features of the compact-cluster approach.

On the Theory of Layer Structures

RECENTLY, Shinada and Sugano1 and also Ralph2 have published theoretical calculations on the optical absorption in layer structures. These calculations are based on a two-dimensional effective mass approximation and the implications are that conclusions drawn from such an approach apply to the actual physical situation met with in layer structures. In our opinion, this view is at variance with the basic principles of quantum mechanics. Indeed, any two-dimensional representation fails to satisfy the requirements of completeness necessary to describe the states of a physical system and, therefore, is liable to miss physically significant facts. Although the electron wave-functions in layer structures may often be represented with sufficient accuracy as products: of two separate components3,4, this does not imply that the two-dimensional component χ(x,y) alone covers the whole situation. The proper representation of the system involves both components and they must appear explicitly in the definition of any physical parameter. In particular, we note that the optical properties of layer structures are subject to selection rules which depend on the mutual orthogonality properties of the two components, and which cannot, a priori, be ignored. This example illustrates that it is necessary to make a careful assessment of the physical significance of two-dimensional approximations as applied to layer structures.