Showing papers on "Matrix (mathematics) published in 1995"

MatInd and MatInspector: new fast and versatile tools for detection of consensus matches in nucleotide sequence data.

Abstract: The identification of potential regulatory motifs in new sequence data is increasingly important for experimental design. Those motifs are commonly located by matches to IUPAC strings derived from consensus sequences. Although this method is simple and widely used, a major drawback of IUPAC strings is that they necessarily remove much of the information originally present in the set of sequences. Nucleotide distribution matrices retain most of the information and are thus better suited to evaluate new potential sites. However, sufficiently large libraries of pre-compiled matrices are a prerequisite for practical application of any matrix-based approach and are just beginning to emerge. Here we present a set of tools for molecular biologists that allows generation of new matrices and detection of potential sequence matches by automatic searches with a library of pre-compiled matrices. We also supply a large library (> 200) of transcription factor binding site matrices that has been compiled on the basis of published matrices as well as entries from the TRANSFAC database, with emphasis on sequences with experimentally verified binding capacity. Our search method includes position weighting of the matrices based on the information content of individual positions and calculates a relative matrix similarity. We show several examples suggesting that this matrix similarity is useful in estimating the functional potential of matrix matches and thus provides a valuable basis for designing appropriate experiments.

Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach

Abstract: An enhanced, numerically stable transmittance matrix approach is developed and is applied to the implementation of the rigorous coupled-wave analysis for surface-relief and multilevel gratings. The enhanced approach is shown to produce numerically stable results for excessively deep multilevel surface-relief dielectric gratings. The nature of the numerical instability for the classic transmission matrix approach in the presence of evanescent fields is determined. The finite precision of the numerical representation on digital computers results in insufficient accuracy in numerically representing the elements produced by inverting an ill-conditioned transmission matrix. These inaccuracies will result in numerical instability in the calculations for successive field matching between the layers. The new technique that we present anticipates and preempts these potential numerical problems. In addition to the full-solution approach whereby all the reflected and the transmitted amplitudes are calculated, a simpler, more efficient formulation is proposed for cases in which only the reflected amplitudes (or the transmitted amplitudes) are required. Incorporating this enhanced approach into the implementation of the rigorous coupled-wave analysis, we obtain numerically stable and convergent results for excessively deep (50 wavelengths), 16-level, asymmetric binary gratings. Calculated results are presented for both TE and TM polarization and for conical diffraction.

Algebraic Riccati equations

Abstract: 1. Preliminaries from the theory of matrices 2. Indefinite scalar products 3. Skew-symmetric scalar products 4. Matrix theory and control 5. Linear matrix equations 6. Rational matrix functions 7. Geometric theory: the complex case 8. Geometric theory: the real case 9. Constructive existence and comparison theorems 10. Hermitian solutions and factorizations of rational matrix functions 11. Perturbation theory 12. Geometric theory for the discrete algebraic Riccati equation 13. Constructive existence and comparison theorems 14. Perturbation theory for discrete algebraic Riccati equations 15. Discrete algebraic Riccati equations and matrix pencils 16. Linear-quadratic regulator problems 17. The discrete Kalman filter 18. The total least squares technique 19. Canonical factorization 20. Hoo control problems 21. Contractive rational matrix functions 22. The matrix sign function 23. Structured stability radius Bibliography List of notations Index

Combined differential and common-mode scattering parameters: theory and simulation

Abstract: A theory for combined differential and common-mode normalized power waves is developed in terms of even and odd mode impedances and propagation constants for a microwave coupled line system. These are related to even and odd-mode terminal currents and voltages. Generalized s-parameters of a two-port are developed for waves propagating in several coupled modes. The two-port s-parameters form a 4-by-4 matrix containing differential-mode, common-mode, and cross-mode s-parameters. A special case of the theory allows the use of uncoupled transmission lines to measure the coupled-mode waves. Simulations verify the concept of these mixed-mode s-parameters, and demonstrate conversion from mode to mode for asymmetric microwave structures. >

Electromagnetic scattering by an aggregate of spheres

Abstract: We present a comprehensive solution to the classical problem of electromagnetic scattering by aggregates of an arbitrary number of arbitrarily configured spheres that are isotropic and homogeneous but may be of different size and composition. The profile of incident electromagnetic waves is arbitrary. The analysis is based on the framework of the Mie theory for a single sphere and the existing addition theorems for spherical vector wave functions. The classic Mie theory is generalized. Applying the extended Mie theory to all the spherical constituents in an aggregate simultaneously leads to a set of coupled linear equations in the unknown interactive coefficients. We propose an asymptotic iteration technique to solve for these coefficients. The total scattered field of the entire ensemble is constructed with the interactive scattering coefficients by the use of the translational addition theorem a second time. Rigorous analytical expressions are derived for the cross sections in a general case and for all the elements of the amplitude-scattering matrix in a special case of a plane-incident wave propagating along the z axis. As an illustration, we present some of our preliminary numerical results and compare them with previously published laboratory scattering measurements.

On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback

Abstract: In this paper, it is shown that the problem of checking the solvability of a bilinear matrix inequality (BMI), is NP-hard. A matrix valued function, F(X,Y), is called bilinear if it is linear with respect to each of its arguments, and an inequality of the form, F(X,Y)>0 is called a bilinear matrix inequality. Recently, it was shown that, the static output feedback problem, fixed order controller problem, reduced order H/sup /spl infin// controller design problem, and several other control problems can be formulated as BMIs. The main result of this paper shows that the problem of checking the solvability of BMIs is NP-hard, and hence it is rather unlikely to find a polynomial time algorithm for solving general BMI problems. As an independent result, it is also shown that simultaneous stabilization with static output feedback is an NP-hard problem, namely for given n plants, the problem of checking the existence of a static gain matrix, which stabilizes all of the n plants, is NP-hard.

Precise solution of few-body problems with the stochastic variational method on a correlated Gaussian basis

Abstract: Precise variational solutions are given for problems involving diverse fermionic and bosonic (N=2--7)-body systems. The trial wave functions are chosen to be combinations of correlated Gaussians, which are constructed from products of the single-particle Gaussian wave packets through an integral transformation, thereby facilitating fully analytical calculations of the matrix elements. The nonlinear parameters of the trial function are chosen by a stochastic technique. The method has proved very efficient, virtually exact, and it seems feasible for any few-body bound-state problems emerging in nuclear or atomic physics.

Heuristic algorithms for the bilevel origin-destination matrix estimation problem

Abstract: Recently, a bilevel programming approach has been used for estimation of origin-destination (O-D) matrix in congested networks This approach integrates the conventional generalized least squares estimation model and the standard network equilibrium model into one process We extend this approach and develop a more general model and efficient heuristic algorithms to handle more realistic situation where link flow interaction cannot be ignored The extended model is formulated in the form of a bilevel programming problem with variational inequality constraints The upper-level problem seeks to minimize the sum of error measurements in traffic counts and O-D matrices, while the lower-level problem represents a network equilibrium problem formulated as variational inequalities, which guarantees that the estimated O-D matrix and corresponding link flows satisfy the network equilibrium conditions Two computational techniques are presented for solving the bilevel O-D matrix estimation model One is a heuristic iterative algorithm between traffic assignment and O-D matrix estimation and the other one is a sensitivity analysis based heuristic algorithm Properties of the two algorithms are analyzed theoretically and compared numerically with small network examples It is concluded that both algorithms can be used as efficient approaches for the bilevel O-D matrix estimation problems

On the use of a working correlation matrix in using generalised linear models for repeated measures

Abstract: SUMMARY In a seminal paper Liang & Zeger (1986) extended the use of generalised linear models to repeated measures data. They based the analysis on specifications for the means and variances of the observations, as usual for generalised linear models, but showed how specifications for the correlations between measurements made on the same unit could be avoided by using a 'working' correlation matrix. In some cases the parameters involved in this matrix are subject to an uncertainty of definition which can lead to a breakdown of the asymptotic properties of the estimators.

NP-hardness of some linear control design problems

Abstract: We show that some basic linear control design problems are NP-hard, implying that, unless P=NP, they cannot be solved by polynomial time algorithms. The problems that we consider include simultaneous stabilization by output feedback, stabilization by state or output feedback in the presence of bounds on the elements of the gain matrix, and decentralized control. These results are obtained by first showing that checking the existence of a stable matrix in an interval family of matrices is an NP-hard problem.

Reduced-Order Modeling of Large Linear Subcircuits via a Block Lanczos Algorithm

Abstract: A method for the efficient computation of accurate reduced-order models of large linear circuits is described. The method, called MPVL, employs a novel block Lanczos algorithm to compute matrix Pade approximations of matrix-valued network transfer functions. The reduced-order models, computed to the required level of accuracy, are used to speed up the analysis of circuits containing large linear blocks. The linear blocks are replaced by their reduced-order models, and the resulting smaller circuit can be analyzed with general-purpose simulators, with significant savings in simulation time and, practically, no loss of accuracy.

On the probability that a random ±1-matrix is singular

Abstract: We report some progress on the old problem of estimating the probability, Pn, that a random n× n ± 1 matrix is singular: Theorem. There is a positive constant ε for which Pn < (1− ε)n. This is a considerable improvement on the best previous bound, Pn = O(1/ √ n), given by Komlós in 1977, but still falls short of the often-conjectured asymptotical formula Pn = (1 + o(1))n 221−n. The proof combines ideas from combinatorial number theory, Fourier analysis and combinatorics, and some probabilistic constructions. A key ingredient, based on a Fourier-analytic idea of Halász, is an inequality (Theorem 2) relating the probability that a ∈ Rn is orthogonal to a random ε ∈ {±1}n to the corresponding probability when ε is random from {−1, 0, 1}n with Pr(εi = −1) = Pr(εi = 1) = p and εi’s chosen independently.

On solving first-kind integral equations using wavelets on a bounded interval

Abstract: The conventional method of moments (MoM), when applied directly to integral equations, leads to a dense matrix which often becomes computationally intractable. To overcome the difficulties, wavelet-bases have been used previously which lead to a sparse matrix. The authors refer to "MoM with wavelet bases" as "wavelet MoM". There have been three different ways of applying the wavelet techniques to boundary integral equations: 1) wavelets on the entire real line which requires the boundary conditions to be enforced explicitly, 2) wavelet bases for the bounded interval obtained by periodizing the wavelets on the real line, and 3) "wavelet-like" basis functions. Furthermore, only orthonormal (ON) bases have been considered. The present authors propose the use of compactly supported semi-orthogonal (SO) spline wavelets specially constructed for the bounded interval in solving first-kind integral equations. They apply this technique to analyze a problem involving 2D EM scattering from metallic cylinders. It is shown that the number of unknowns in the case of wavelet MoM increases by m-1 as compared to conventional MoM, where m is the order of the spline function. Results for linear (m=2) and cubic (m=4) splines are presented along with their comparisons to conventional MoM results. It is observed that the use of cubic spline wavelets almost "diagonalizes" the matrix while maintaining less than 1.5% of relative normed error. The authors also present the explicit closed-form polynomial representation of the scaling functions and wavelets. >

Polynomial roots from companion matrix eigenvalues

Abstract: In classical linear algebra, the eigenvalues of a matrix are sometimes defined as the roots of the characteristic polynomial. An algorithm to compute the roots of a polynomial by computing the eigenvalues of the corresponding companion matrix turns the tables on the usual definition. We derive a first- order error analysis of this algorithm that sheds light on both the underlying geometry of the problem as well as the numerical error of the algorithm. Our error analysis expands on work by Van Dooren and Dewilde in that it states that the algorithm is backward normwise stable in a sense that must be defined carefully. Regarding the stronger concept of a small componentwise backward error, our analysis predicts a small such error on a test suite of eight random polynomials suggested by Toh and Trefethen. However, we construct examples for which a small componentwise relative backward error is neither predicted nor obtained in practice. We extend our results to polynomial matrices, where the result is essentially the same, but the geometry becomes more complicated.

Precise solution of few-body problems with the stochastic variational method

Abstract: A precise variational solution to $N$=2--6-body problems is reported The trial wave functions are chosen to be combinations of correlated Gaussians, which facilitate a fully analytical calculation of the matrix elements The nonlinear parameters of the trial function are selected by a stochastic method Fermionic and bosonic few-body systems are investigated for interactions of different type A comparison of the results with those available in the literature shows that the method is both accurate and efficient

Short wavelets and matrix dilation equations

Abstract: Scaling functions and orthogonal wavelets are created from the coefficients of a lowpass and highpass filter (in a two-band orthogonal filter bank). For "multifilters" those coefficients are matrices. This gives a new block structure for the filter bank, and leads to multiple scaling functions and wavelets. Geronimo, Hardin, and Massopust (see J. Approx. Theory, vol.78, p.373-401, 1994) constructed two scaling functions that have extra properties not previously achieved. The functions /spl Phi//sub 1/ and /spl Phi//sub 2/ are symmetric (linear phase) and they have short support (two intervals or less), while their translates form an orthogonal family. For any single function /spl Phi/, apart from Haar's piecewise constants, those extra properties are known to be impossible. The novelty is to introduce 2/spl times/2 matrix coefficients while retaining orthogonality of the multiwavelets. This note derives the properties of /spl Phi//sub 1/ and /spl Phi//sub 2/ from the matrix dilation equation that they satisfy. Then our main step is to construct associated wavelets: two wavelets for two scaling functions. The properties were derived by Geronimo, Hardin, and Massopust from the iterated interpolation that led to /spl Phi/1 and /spl Phi//sub 2/. One pair of wavelets was found earlier by direct solution of the orthogonality conditions (using Mathematica). Our construction is in parallel with recent progress by Hardin and Geronimo, to develop the underlying algebra from the matrix coefficients in the dilation equation-in another language, to build the 4/spl times/4 paraunitary polyphase matrix in the filter bank. The short support opens new possibilities for applications of filters and wavelets near boundaries. >

Synthesis of coherent two-port lattice-form optical delay-line circuit

Abstract: A method is presented for synthesizing a coherent two-port lattice-form optical delay-line circuit that is composed of optical delay lines, directional couplers, and phase shifters. The two bases of the method are the use of a unimodulus para-unitary matrix as a transfer matrix and the division of the transfer matrix into basic component transfer matrices. We succeeded in obtaining a set of recurrent equations with which to calculate circuit parameters to use for designing an optical delay-line circuit with a desired cross-port (through-port) transfer function. In the developed method, it is shown that two-port optical delay-line circuits can have the same transmission characteristics as finite impulse response digital filters with complex expansion coefficients. Three synthesis examples for optical frequency filters are described: a linear-phase Chebyshev filter, a multi-channel selector, and a group-delay dispersion equalizer. It is confirmed that transmission characteristics with a maximum transmittance of 100% can be always synthesized. The allowable parameter error for the synthesized linear-phase Chebyshev filter is also discussed. >

Geometric Approach to Factor Analysis for the Estimation of Orthogonality and Practical Peak Capacity in Comprehensive Two-Dimensional Separations

Abstract: Procedures were developed for the estimation of orthogonality in two-dimensional (2D) separations. The parameters evaluated include peak spreading angle, retention correlation, and practical peak capacity. Solute retention parameters, such as retention times and capacity factors on both dimensions, were used to establish a correlation matrix, from which a peak spreading angle matrix was calculated using a geometric approach to factor analysis. The orthogonality is defined by the correlation matrix with correlation coefficients that vary from 0 (orthogonal) to 1 (perfect correlation). Equations were derived for the calculation of practical peak capacity in 2D separations. The calculations are based on the peak capacities obtained on each dimension and the peak spreading angle in an orthogonal, 2D retention space. The equations and the procedures can be used to evaluate the performance of a comprehensive 2D separation. Using experimental data from a 2D GC separation, it is demonstrated that the equations are very useful for the comparison, evaluation, and optimization of 2D separations.

Efficient implementation of rigorous coupled-wave analysis for surface-relief gratings

Abstract: A computationally efficient implementation of rigorous coupled-wave analysis is presented The eigenvalue problem for a one-dimensional grating in a conical mounting is reduced to two eigenvalue problems in the corresponding nonconical mounting This reduction yields two n × n matrices to solve for eigenvalues and eigenvectors, where n is the number of orders retained in the computation For a two-dimensional grating, the size of the matrix in the eigenvalue problem is reduced to 2n × 2n These simplifications reduce the computation time for the eigenvalue problem by 8–32 times compared with the original computation time In addition, we show that with rigorous coupled-wave analysis one analytically satisfies reciprocity by retaining the appropriate choice of spatial harmonics in the analysis Numerical examples are given for metallic lamellar gratings, pulse-width-modulated gratings, deep continuous surface-relief gratings, and two-dimensional gratings

Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials

Abstract: A new technique is proposed for the recovery of optical phase from intensity information. The method is based on the decomposition of the transport-of-intensity equation into a series of Zernike polynomials. An explicit matrix formula is derived, expressing the Zernike coefficients of the phase as functions of the Zernike coefficients of the wave-front curvature inside the aperture and the Fourier coefficients of the wave-front boundary slopes. Analytical expressions are given, as well as a numerical example of the corresponding phase retrieval matrix. This work lays the basis for an effective algorithm for fast and accurate phase retrieval.

Mueller matrix imaging polarimetry

Abstract: The design and operation of a Mueller matrix imaging polarimeter is presented. The instrument is configurable to make a wide variety of polarimetric measurements of optical systems and samples. In one configuration, it measures the polarization properties of a set of ray paths through a sample. The sample may comprise a single element, such as a lens, polarizer, retarder, spatial light modulator, or beamsplitter, or an entire optical system containing many elements. In a second configuration, it measures an optical system's point spread matrix, a Mueller matrix relating the polarization state of a point object to the distribution of intensity and polarization across the image. The instrument is described and a number of example measurements are provided that demonstrate the Mueller matrix imaging polarimeter's unique measurement capability.

Self-consistent first-principles technique with linear scaling.

Abstract: An algorithm for first-principles electronic structure calculations having a computational cost which scales linearly with the system size is presented. Our method exploits the real-space localization of the density matrix, and in this respect it is related to the technique of Li, Nunes and Vanderbilt. The density matrix is expressed in terms of localized support functions, and a matrix of variational parameters, L, having a finite spatial range. The total energy is minimized with respect to both the support functions and the elements of the L matrix. The method is variational, and becomes exact as the ranges of the support functions and the L matrix are increased. We have tested the method on crystalline silicon systems containing up to 216 atoms, and we discuss some of these results.