Showing papers on "Matrix (mathematics) published in 1992"

The rapid generation of mutation data matrices from protein sequences

Abstract: An efficient means for generating mutation data matrices from large numbers of protein sequences is presented here. By means of an approximate peptide-based sequence comparison algorithm, the set sequences are clustered at the 85% identity level. The closest relating pairs of sequences are aligned, and observed amino acid exchanges tallied in a matrix. The raw mutation frequency matrix is processed in a similar way to that described by Dayhoff et al. (1978), and so the resulting matrices may be easily used in current sequence analysis applications, in place of the standard mutation data matrices, which have not been updated for 13 years. The method is fast enough to process the entire SWISS-PROT databank in 20 h on a Sun SPARCstation 1, and is fast enough to generate a matrix from a specific family or class of proteins in minutes. Differences observed between our 250 PAM mutation data matrix and the matrix calculated by Dayhoff et al. are briefly discussed.

Second-order perturbation theory with a complete active space self-consistent field reference function.

Abstract: The recently implemented second‐order perturbation theory based on a complete active space self‐consistent field reference function has been extended by allowing the Fock‐type one‐electron operator, which defines the zeroth‐order Hamiltonian to have nonzero elements also in nondiagonal matrix blocks. The computer implementation is now less straightforward and more computer time will be needed in obtaining the second‐order energy. The method is illustrated in a series of calculations on N2, NO, O2, CH3, CH2, and F−.

Shape and motion from image streams under orthography: a factorization method

Abstract: Inferring scene geometry and camera motion from a stream of images is possible in principle, but is an ill-conditioned problem when the objects are distant with respect to their size. We have developed a factorization method that can overcome this difficulty by recovering shape and motion under orthography without computing depth as an intermediate step. An image stream can be represented by the 2FxP measurement matrix of the image coordinates of P points tracked through F frames. We show that under orthographic projection this matrix is of rank 3. Based on this observation, the factorization method uses the singular-value decomposition technique to factor the measurement matrix into two matrices which represent object shape and camera rotation respectively. Two of the three translation components are computed in a preprocessing stage. The method can also handle and obtain a full solution from a partially filled-in measurement matrix that may result from occlusions or tracking failures. The method gives accurate results, and does not introduce smoothing in either shape or motion. We demonstrate this with a series of experiments on laboratory and outdoor image streams, with and without occlusions.

Intersection theory on the moduli space of curves and the matrix Airy function

Abstract: We show that two natural approaches to quantum gravity coincide. This identity is nontrivial and relies on the equivalence of each approach to KdV equations. We also investigate related mathematical problems.

A novel discrete variable representation for quantum mechanical reactive scattering via the S-matrix Kohn method

Abstract: A novel discrete variable representation (DVR) is introduced for use as the L2 basis of the S‐matrix version of the Kohn variational method [Zhang, Chu, and Miller, J. Chem. Phys. 88, 6233 (1988)] for quantum reactive scattering. (It can also be readily used for quantum eigenvalue problems.) The primary novel feature is that this DVR gives an extremely simple kinetic energy matrix (the potential energy matrix is diagonal, as in all DVRs) which is in a sense ‘‘universal,’’ i.e., independent of any explicit reference to an underlying set of basis functions; it can, in fact, be derived as an infinite limit using different basis functions. An energy truncation procedure allows the DVR grid points to be adapted naturally to the shape of any given potential energy surface. Application to the benchmark collinear H+H2→H2+H reaction shows that convergence in the reaction probabilities is achieved with only about 15% more DVR grid points than the number of conventional basis functions used in previous S‐matrix Kohn...

Sparse matrices in matlab: design and implementation

Abstract: The matrix computation language and environment MATLAB is extended to include sparse matrix storage and operations. The only change to the outward appearance of the MATLAB language is a pair of commands to create full or sparse matrices. Nearly all the operations of MATLAB now apply equally to full or sparse matrices, without any explicit action by the user. The sparse data structure represents a matrix in space proportional to the number of nonzero entries, and most of the operations compute sparse results in time proportional to the number of arithmetic operations on nonzeros.

Estimating two-dimensional frequencies by matrix enhancement and matrix pencil

Abstract: A new method, called the matrix enhancement and matrix pencil (MEMP) method, is presented for estimating two-dimensional (2-D) frequencies. In the MEMP method, an enhanced matrix is constructed from the data samples, and then the matrix pencil approach is used to extract out the 2-D sinusoids from the principal eigenvectors of the enhanced matrix. The MEMP method yields the estimates of the 2-D frequencies efficiently, without solving the roots of a 2-D polynomial or searching in a 2-D space. It is shown that the MEMP method can be faster than a 2-D FFT method if the number of the 2-D sinusoids is much smaller than the data set. Simulation results are provided to show that the accuracy of the MEMP method can be very close to the Cramer-Rao lower bound. >

The multifrontal method for sparse matrix solution: theory and practice

Abstract: This paper presents an overview of the multifrontal method for the solution of large sparse symmetric positive definite linear systems. The method is formulated in terms of frontal matrices, update matrices, and an assembly tree. Formal definitions of these notions are given based on the sparse matrix structure. Various advances to the basic method are surveyed. They include the role of matrix reorderings, the use of supernodes, and other implementatjon techniques. The use of the method in different computational environments is also described.

Approximation with Kronecker Products

Abstract: Let A be an m-by-n matrix with m=m1m2 and n=n1n2. We consider the problem of finding (mathematical formula omitted) so that (mathematical formula omitted) is minimized. This problem can be solved by computing the largest singular value and associated singular vectors of a permuted version of A. If A is symmetric, definite, non-negative, or banded, then the minimizing B and C are similarly structured. The idea of using Kronecker product preconditioners is briefly discussed.

Large-Scale Sparse Singular Value Computations

Abstract: We present four numerical methods for computing the singular value decomposition SVD of large sparse matrices on a multiprocessor architecture. We emphasize Lanczos and subspace iteration-based methods for determining several of the largest singular triplets singular values and corresponding left- and right-singular vectors for sparse matrices arising from two practical applications: information retrieval and seismic reflection tomography. The target architectures for our implementations are the CRAY-2S/4-128 and Alliant FX/80. The sparse SVD problem is well motivated by recent information-retrieval techniques in which dominant singular values and their corresponding singular vectors of large sparse term-document matrices are desired, and by nonlinear inverse problems from seismic tomography applications which require approximate pseudo-inverses of large sparse Jacobian matrices. This research may help advance the development of future out-of-core sparse SVD methods, which can be used, for example, to handle extremely large sparse matrices 0 ? 106 rows or columns associated with extremely large databases in query-based information-retrieval applications.

An updating algorithm for subspace tracking

Abstract: In certain signal processing applications it is required to compute the null space of a matrix whose rows are samples of a signal with p components. The usual tool for doing this is the singular value decomposition. However, the singular value decomposition has the drawback that it requires O(p/sup 3/) operations to recompute when a new sample arrives. It is shown that a different decomposition, called the URV decomposition, is equally effective in exhibiting the null space and can be updated in O(p/sup 2/) time. The updating technique can be run on a linear array of p processors in O(p) time. >

How fast are nonsymmetric matrix iterations

Abstract: Three leading iterative methods for the solution of nonsymmetric systems of linear equations are CGN (the conjugate gradient iteration applied to the normal equations), GMRES (residual minimization in a Krylov space), and CGS (a biorthogonalization algorithm adapted from the biconjugate gradient iteration). Do these methods differ fundamentally in capabilities? If so, which is best under which circumstances? The existing literature, in relying mainly on empirical studies, has failed to confront these questions systematically. In this paper it is shown that the convergence of CGN is governed by singular values and that of GMRES and CGS by eigenvalues or pseudo-eigenvalues. The three methods are found to be fundamentally different, and to substantiate this conclusion, examples of matrices are presented for which each iteration outperforms the others by a factor of size $O(\sqrt N )$ or $O(N)$ where N is the matrix dimension. Finally, it is shown that the performance of iterative methods for a particular mat...

Quantitative determination of molecular structure in multilayered thin films of biaxial and lower symmetry from photon spectroscopies. I. Reflection infrared vibrational spectroscopy

Abstract: A semitheoretical formalism based on classical electromagnetic wave theory has been developed for application to the quantitative treatment of reflection spectra from multilayered anisotropic films on both metallic and nonmetallic substrates. Both internal and external reflection experiments as well as transmission can be handled. The theory is valid for all wavelengths and is appropriate, therefore, for such experiments as x‐ray reflectivity, uv–visible spectroscopic ellipsometry, and infrared reflection spectroscopy. Further, the theory is applicable to multilayered film structures of variable number of layers, each with any degree of anisotropy up to and including full biaxial symmetry. The reflectivities (and transmissivities) are obtained at each frequency by solving the wave propagation equations using a rigorous 4×4 transfer matrix method developed by Yeh in which the optical functions of each medium are described in the form of second rank (3×3) tensors. In order to obtain optical tensors for materials not readily available in single crystal form, a method has been developed to evaluate tensor elements from the complex scalar optical functions (n) obtained from the isotropic material with the limitations that the molecular excitations are well characterized and obey photon–dipole selection rules.This method is intended primarily for infrared vibrational spectroscopy and involves quantitative decomposition of the isotropic imaginary optical function (k) spectrum into a sum of contributions from fundamental modes, the assignment of a direction in molecular coordinates to the transition dipole matrix elements for each mode, the appropriate scaling of each k vector component in surface coordinates according to a selected surface orientation of the molecule to give a diagonal im(n) tensor, and the calculation of the real(n) spectrum tensor elements by the Kramers–Kronig transformation. Tensors for other surface orientations are generated by an appropriate rotation matrix operation. To test the viability of this approach, three sets of experimentally derived infrared spectra of oriented monolayer assemblies on quite distinctively different substrates were chosen for simulation: (1) n‐alkanethiols self‐ assembled onto gold, (2) n‐alkanoic acid salt Langmuir–Blodgett (LB) monolayers on carbon, and (3) n‐alkanoic acid salt LB monolayers on silica glass. The formalism developed was used to simulate the spectral response and to derive structural features of the monolayers. Good agreement was found where comparisons with independent studies could be made and, in general, the method appears quite useful for structural studies of highly organized thin films.

Observability analysis of piece-wise constant systems. I. Theory

Abstract: For pt.II see ibid., vol.28, no.4, p.1068-75, Oct. 1992. A method for analyzing the observability of time-varying linear systems which can be modeled as piece-wise constant systems (PWCS) is presented. An observability matrix for such systems is developed for continuous and discrete time representations. A stripped observability matrix (SOM) is introduced which simplifies the analysis in cases where the use of this matrix is legitimate. The observability analysis is presented as a step-by-step procedure. >

Planar near-field to far-field transformation using an equivalent magnetic current approach

Abstract: An alternative method is presented for computing far-field antenna patterns from near-field measurements. The method utilizes the near-field data to determine equivalent magnetic current sources over a fictitious planar surface that encompasses the antenna, and these currents are used to ascertain the far fields. Under certain approximations, the currents should produce the correct far fields in all regions in front of the antenna regardless of the geometry over which the near-field measurements are made. An electric field integral equation (EFIE) is developed to relate the near fields to the equivalent magnetic currents. The method of moments is used to transform the integral equation into a matrix one. The matrix equation is solved with the conjugate gradient method, and in the case of a rectangular matrix, a least-squares solution for the currents is found without explicitly computing the normal form of the equation. Near-field to far-field transformation for planar scanning may be efficiently performed under certain conditions. Numerical results are presented for several antenna configurations. >

The periodic Toda chain and a matrix generalization of the Bessel function recursion relations

Abstract: The authors obtain the quantization conditions of the periodic Toda lattice in the Baxter form: Lambda (u)Q(u)=iNQ(u+i)+i-NQ(u-i) Lambda is the 'transfer matrix' containing the information about the spectrum and Q is an integral operator commuting with Lambda . The logarithms of the matrix elements of Q are the generating functions of the canonical Backlund transformation. The requirement that Q is analytic and vanishes when u goes to infinity completely determines the spectrum of Lambda .

An inverse method for determining elastic material properties and a material interface

Abstract: The problem consists of determining the location and size of a circular inclusion in a finite matrix and the elastic material properties of the inclusion and the matrix.

Wavelet methods for fast resolution of elliptic problems

Abstract: This paper shows that the use of wavelets to discretize an elliptic problem with Dirichlet or Neumann boundary conditions has two advantages: an explicit diagonal preconditioning makes the condition number of the corresponding matrix become bounded by a constant and the order of approximation is locally of spectral type (in contrast with classical methods); using a conjugate gradient method, one thus obtains fast numerical algorithms of resolution. A comparison is also drawn between wavelet and classical methods.

Matrix animation and polar decomposition

Abstract: General 3×3 linear or 4×4 homogenous matrices can be formed by composing primitive matrices for translation, rotation, scale, shear, and perspective. Current 3-D computer graphics systems manipulate and interpolate parametric forms of these primitives to generate scenes and motion. For this and other reasons, decomposing a composite matrix in a meaningful way has been a longstanding challenge. This paper presents a theory and method for doing so, proposing that the central issue is rotation extraction, and that the best way to do that is Polar Decomposition. This method also is useful for renormalizing a rotation matrix containing excessive error.

Sampling and efficiency of metric matrix distance geometry: a novel partial metrization algorithm.

Abstract: In this paper, we present a reassessment of the sampling properties of the metric matrix distance geometry algorithm, which is in wide-spread use in the determination of three-dimensional structures from nuclear magnetic resonance (NMR) data. To this end, we compare the conformational space sampled by structures generated with a variety of metric matrix distance geometry protocols. As test systems we use an unconstrained polypeptide, and a small protein (rabbit neutrophil defensin peptide 5) for which only few tertiary distances had been derived from the NMR data, allowing several possible folds of the polypeptide chain. A process called ‘metrization’ in the preparation of a trial distance matrix has a very large effect on the sampling properties of the algorithm. It is shown that, depending on the metrization protocol used, metric matrix distance geometry can have very good sampling properties'indeed, both for the unconstrained model system and the NMR-structure case. We show that the sampling properties are to a great degree determined by the way in which the first few distances are chosen within their bounds. Further, we present a new protocol (‘partial metrization’) that is computationally more efficient but has the same excellent sampling properties. This novel protocol has been implemented in an expanded new release of the program X-PLOR with distance geometry capabilities.

Damage and Failure in Unidirectional Ceramic–Matrix Composites

Abstract: A study of the mechanical characteristics of a unidirectional fiber–reinforced calcium aluminosilicate matrix composite has been conducted. The properties have been related to the individual properties of the matrix, the fibers, and the interfaces, as well as the residual stress, using available models of matrix cracking and fiber fracture. Comparisons have been made with lithium aluminosilicate matrix composites. Predictions of initial matrix cracking and of ultimate strength using the models are found to correlate well with the measured values. However, deficiencies have been noted in the ability of the models to predict the evolution of matrix cracks, plus associated changes in the modulus.

Ab initio calculation of force constants and full phonon dispersions.

Abstract: We present a method to calculate the full phonon spectrum using the local-density approximation and Hellmann-Feynman forces. By a limited number of supercell calculations of the planar force constants, the interatomic force constant matrices are determined. One can then construct the dynamical matrix for any arbitrary wave vector in the Brillouin zone. We describe in detail the procedure for elements in the diamond structure and derive the phonon dispersion curves for Si. The anharmonic effects can also be studied by the present method.

Stability of the method of lines

Abstract: It is well known that a necessary condition for the Lax-stability of the method of lines is that the eigenvalues of the spatial discretization operator, scaled by the time stepk, lie within a distanceO(k) of the stability region of the time integration formula ask?0. In this paper we show that a necessary and sufficient condition for stability, except for an algebraic factor, is that the ?-pseudo-eigenvalues of the same operator lie within a distanceO(?)+O(k) of the stability region ask, ??0. Our results generalize those of an earlier paper by considering: (a) Runge-Kutta and other one-step formulas, (b) implicit as well as explicit linear multistep formulas, (c) weighted norms, (d) algebraic stability, (e) finite and infinite time intervals, and (f) stability regions with cusps. In summary, the theory presented in this paper amounts to a transplantation of the Kreiss matrix theorem from the unit disk (for simple power iterations) to an arbitrary stability region (for method of lines calculations).

BLAF ‐ a robust program for tracking out admittable Bravais lattice(s) from the experimental unit‐cell data

Abstract: BLAF represents an original computer program to devise the Bravais lattice symmetry or possible pseudo-symmetries (with allowance for large axial and angular distortions) of an experimental unit cell The matrix approach to symmetry formulated by Himes & Mighell [Acta Cryst (1987) A43, 375–384] is further developed and employed to analyse admittable mappings of a lattice onto itself Solutions of the matrix equations G = MtGM, where G is the metric tensor of the Buerger reduced lattice, are integral matrices M with det(M) = + 1 and −1 < tr(M) ≤ 3, composing the seven axial hemihedral point groups 432, 622, 422, 32, 222, 2, 1 For non-triclinic symmetries these matrices carry information about important symmetry directions in the lattice, subsequently used in building up an overall transformation matrix to find a conventional (symmetry-conditioned) unit cell The average of the generated G tensors in accordance with the particular point-group rules is a tensor Gav bearing information about the symmetry-constrained lattice parameters Gruber's [Acta Cryst (1989), A45, 123–131] algorithms have been used to evaluate both Buerger cells and the Niggli cell of a triclinic lattice BLAF is realised as a separate module suitable for incorporation in the commonly used crystallographic program packages and in the form of two subroutines: enBLAF – to tackle the lattice symmetry problem by automated single-crystal diffractometers; and rBLAF – to be used for lattice symmetry analysis in, for example, programs for autoindexing of powder data Applications of the three modules are demonstrated in several test examples

Using Results from Replicated Studies to Estimate Linear Models.

Abstract: This article outlines analyses for the results of a series of studies examining intercorrelations among a set of p + 1 variables. A test of whether a common population correlation matrix underlies the set of empirical results is given. Methods are presented for estimating either a pooled or average correlation matrix, depending on whether the studies appear to arise from a single population. A random effects model provides the basis for estimation and testing when the series of correlation matrices may not share a common population matrix. Finally, I show how a pooled correlation matrix (or average matrix) can be used to estimate the standardized coefficients of a regression model for variables measured in the series of studies. Data from a synthesis of relationships among mathematical, verbal, and spatial ability measures illustrate the procedures.