Schur Algorithm Research Articles

Why Do Field-Based Methods Fail to Model Plasmonics?

The paper studies plasmonics modeling issues and examines the reasons behind the failure of the field-based methods relying on Pade approximations widely used in the analysis of photonic devices based on dielectric materials. Through a study of evanescent, radiation, guided, and surface modes of a plasmonic structure where the failure appears clearly, we demonstrate the physical explanation of this failure and suggest some remedies. We developed a Bidirectional Beam Propagation Method (BiBPM) by adopting a Blocked Schur (BS) algorithm to introduce an unconditionally stable method for plasmonic structures with strong discontinuities. Central to BiBPMs is the accurate calculation of the square root operators that is very widely performed using Pade approximations. However, recent reports demonstrate convergence of Pade that is too slow to lend itself a stable solver in plasmonics [1] . Moreover, Pade approximations completely fail in handling such a strong discontinuity between dielectric and plasmonic waveguides, where a very-wide spectrum of modes could be excited. Alternatively, we propose calculating these operators by the twice faster BS algorithm. Beyond the computational speed, our suggested approach overbears the Pade-based BiBPMs instability and accuracy problems, thanks to the proper physical treatment of surface and evanescent waves: the notorious sources of instability. Through the plasmonic discontinuity problems, the superiority of BS approach has been determined numerically and explained physically.

On Szegő's theorem for a nonclassical case

In this paper we prove Szegő's Theorem for the case when a finite number of Verblunsky coefficients lie outside the closed unit disk. Although a form of this result was already proved by A.L. Sakhnovich, we use a very different method, which shows that the OPUC machinery can still be applied to deal with such nonclassical cases. The basic tool we use is Khrushchev's formula that in the classical case relates the absolutely continuous part of the measure and the N-th iterate of the Schur algorithm. It is noteworthy that Khrushchev's formula makes the proof short and extremely transparent. Also, we discuss Verblunsky's theorem for the case in question.

Performance of a Newton–Krylov–Schur Algorithm for Solving Steady Turbulent Flows

A methodology is presented for characterizing flow solver performance. The methodology can be applied to assess the efficiency of a given approach, where efficiency is defined in terms of accuracy per unit cost measured in central processing unit time. The procedure is presented by demonstrating its application to the parallel Newton–Krylov–Schur finite difference flow solver known as Diablo. The benchmark cases to which the procedure is applied are two-dimensional turbulent flows modeled using the Reynolds-averaged Navier–Stokes equations on three families of NACA 0012 grids with three sets of operating conditions. Performance statistics are presented in a variety of ways that show the relationships between central processing unit time, grid spacing, and accuracy in ways that are informative for both flow solver users and developers.

Efficient analysis of electron waveguides with multiple discontinuities

Blocked Schur finite-element bidirectional beam propagation method (BS-FE-BiBPM) is introduced for the solution of electron waveguides with multiple discontinuities. Scattering properties could be accurately calculated using BiBPM based on time-independent Schrodinger equation while Blocked Schur algorithm is used for accurate computation of the characteristic matrix square root. The suggested approach substantially reduces the computational time while preserving very high efficiency.

A new framework for implicit restarting of the Krylov–Schur algorithm

SummaryThis paper introduces a new framework for implicit restarting of the Krylov–Schur algorithm. It is shown that restarting with arbitrary polynomial filter is possible by reassigning some of the eigenvalues of the Rayleigh quotient through a rank‐one correction, implemented using only the elementary transformations (translation and similarity) of the Krylov decomposition. This framework includes the implicitly restarted Arnoldi (IRA) algorithm and the Krylov–Schur algorithm with implicit harmonic restart as special cases. Further, it reveals that the IRA algorithm can be turned into an eigenvalue assignment method. Copyright © 2014 John Wiley & Sons, Ltd.

Automatic speech signal segmentation based on the innovation adaptive filter

Abstract Speech segmentation is an essential stage in designing automatic speech recognition systems and one can find several algorithms proposed in the literature. It is a difficult problem, as speech is immensely variable. The aim of the authors’ studies was to design an algorithm that could be employed at the stage of automatic speech recognition. This would make it possible to avoid some problems related to speech signal parametrization. Posing the problem in such a way requires the algorithm to be capable of working in real time. The only such algorithm was proposed by Tyagi et al., (2006), and it is a modified version of Brandt’s algorithm. The article presents a new algorithm for unsupervised automatic speech signal segmentation. It performs segmentation without access to information about the phonetic content of the utterances, relying exclusively on second-order statistics of a speech signal. The starting point for the proposed method is time-varying Schur coefficients of an innovation adaptive filter. The Schur algorithm is known to be fast, precise, stable and capable of rapidly tracking changes in second order signal statistics. A transfer from one phoneme to another in the speech signal always indicates a change in signal statistics caused by vocal track changes. In order to allow for the properties of human hearing, detection of inter-phoneme boundaries is performed based on statistics defined on the mel spectrum determined from the reflection coefficients. The paper presents the structure of the algorithm, defines its properties, lists parameter values, describes detection efficiency results, and compares them with those for another algorithm. The obtained segmentation results, are satisfactory.

Comparative Study on Zero Forcing Block Linear Equalization for TD-SCDMA

3G standards such as TD-SCDMA use multi-user detection (MUD) at the base station to combat multi address interference (MAI) and inter-symbol interference (ISI). We review Cholesky factorization and Schur algorithm for Zero Forcing Block Linear Equalization. We compare the solutions of different algorithm for ZF-BLE by computation, and analyze the complexity of those methods.

파라메트릭 배열을 이용한 해저지층 탐사 알고리즘

In this paper, we propose an threshold-based Schur algorithm for estimating the media characteristics of sub-bottom multi-layers by using the signal generated by a parametric array transducer. We use the KZK model to generate a parametric array signal, and use the proposed threshold-based Schur algorithm for estimating the reflection coefficients of multiple sea bottom layers. Using computer simulation, we verify that the difference frequency component generated by the KZK model prevails over the signals of primary frequencies at long range. For the simulation, we use the transmit signal generated by the KZK and the reflected signal obtained from a lattice filter model for the seawater and sub-bottom of multi-level non-homogeneous layers. Through the simulation, we verify that the proposed threshold-based Schur algorithm can give much more accurate and efficient estimates of the reflection coefficients than methods using received signal, matched filter output signal, and normal Schur algorithm output.

해저지층 탐사를 위한 Schur 알고리즘

본 논문은 송신신호와 다층의 해수층 및 해저지층으로 구성된 매질의 퇴적물의 배치 및 형태에 의해 반사된 수신 신호를 이용하여 매질 각층의 특성을 계산 할 수 있는 알고리즘을 제안한다. 제안된 알고리즘은 이종의 다층 매질로 구성되어 있는 해수층 및 해저지층의 구조 특성을 추정하는 것으로, 일반적으로 Schur 알고리즘이라 부른다. 제안된 알고리즘은 주어진 역산란(inverse scattering) 문제를 matrix factorization을 수행하여 효율적으로 매질의 반사계수(reflection coefficient)를 추정한다. 본 논문에서는 제안된 알고리즘의 성능을 분석하기 위하여 해수층 및 해저지층의 이종 다층 매질 구조를 고려한 lattice 필터 모델을 구성하였고, 이 필터 모델로부터 얻어진 송신 및 반사 신호를 이용하여 다양한 컴퓨터 모의실험을 수행한 결과, 제안된 알고리즘이 효율적으로 매질을 추정함을 확인하였다.

Parallel Newton–Krylov–Schur Flow Solver for the Navier–Stokes Equations

The objective of the present paper is to demonstrate the effectiveness of a spatial discretization based on summation-by-parts operators with simultaneous approximation terms in combination with a parallel Newton–Krylov–Schur algorithm for solving the three-dimensional Reynolds-averaged Navier–Stokes equations coupled with the Spalart–Allmaras one-equation turbulence model. The algorithm employs second-order summation-by-parts operators on multiblock structured grids with simultaneous approximation terms to enforce block interface coupling and boundary conditions. The discrete equations are solved iteratively with an inexact-Newton method, while the linear system at each Newton iteration is solved using a flexible Krylov subspace iterative method with an approximate-Schur parallel preconditioner. The algorithm is verified and validated through the solution of two-dimensional model problems, highlighting the correspondence of the current algorithm with several established flow solvers. A transonic solution over the ONERA M6 wing on a mesh with 15.1 million nodes shows good agreement with experiment. Using 128 processors, the residual is reduced by 12 orders of magnitude in 86 min. The solution of transonic flow over the common research model wing–body geometry exhibits the expected grid convergence behavior. The algorithm performs well in solving flows around nonplanar geometries and flows with explicitly specified laminar-to-turbulent transition locations. Parallel scaling studies highlight the excellent scaling characteristics of the algorithm on cases with up to 6656 processors and grids with over 150 million nodes.

The Schur Algorithm Applied to the One-Dimensional Continuous Inverse Scattering Problem

The one-dimensional continuous inverse scattering problem can be solved by the Schur algorithm in the discrete-time domain using sampled scattering data. The sampling rate of the scattering data should be increased to reduce the discretization error, but the complexity of the Schur algorithm is proportional to the square of the sampling rate. To improve this tradeoff between the complexity and the accuracy, we propose a Schur algorithm with the Richardson extrapolation (SARE). The asymptotic expansion of the Schur algorithm, necessary for the Richardson extrapolation, is derived in powers of the discretization step, which shows that the accuracy order (with respect to the discretization step) of the Schur algorithm is 1. The accuracy order of the SARE with the N-step Richardson extrapolation is increased to N+1 with comparable complexity to the Schur algorithm. Therefore, the discretization error of the Schur algorithm can be decreased in a computationally efficient manner by the SARE.

Attitude controller for reentry vehicles using state-dependent Riccati equation method

To get better tracking performance of attitude command over the reentry phase of vehicles, the use of state-dependent Riccati equation (SDRE) method for attitude controller design of reentry vehicles was investigated. Guidance commands are generated based on optimal guidance law. SDRE control method employs factorization of the nonlinear dynamics into a state vector and state dependent matrix valued function. State-dependent coefficients are derived based on reentry motion equations in pitch and yaw channels. Unlike constant weighting matrix Q, elements of Q are set as the functions of state error so as to get satisfactory feedback and eliminate state error rapidly, then formulation of SDRE is realized. Riccati equation is solved real-timely with Schur algorithm. State feedback control law u(x) is derived with linear quadratic regulator (LQR) method. Simulation results show that SDRE controller steadily tracks attitude command, and impact point error of reentry vehicle is acceptable. Compared with PID controller, tracking performance of attitude command using SDRE controller is better with smaller control surface deflection. The attitude tracking error with SDRE controller is within 5°, and the control deflection is within 30°.

New techniques of local damage detection in machinery based on stochastic modelling using adaptive Schur filter

Vibration signal analysis is one of the most effective techniques of monitoring machinery and detecting local damage in their parts, e.g. bearings and gearboxes. However, such detection is sometimes difficult, especially in heavy industrial machines, because of a small proportion of damage-induced components in relation to the remaining components of registered signals. Therefore, more effective signal processing algorithms are being looked for. Moreover, local damage (cracking, pitting, spalling, breakage, etc.) in bearings and gearboxes generates broad-spectrum impulse signals, while the other type can be effectively modelled as a sum of narrowband signals. In this article, techniques based on Schur adaptive filter are proposed for local damage detection. In such an approach, the analysed signal is modelled by means of autoregressive process and the filter is described by so-called reflection coefficients. Schur algorithm is an effective algorithm with very good numerical properties and it is capable of tracking rapid changes in second order statistics of the analysed signal. Thus, the method is well-suited to analysing non-stationary signals and it is potentially interesting for use in bearing and gearbox monitoring.Reflection coefficients describing the signal model, defined with the use of Schur algorithm, may be applied in a variety of ways, giving a chance of employing different solutions in different conditions. In the first proposed solution, detection is based on the weighted sum of derivatives of reflection coefficients, while in the other one – on new signal obtained as power in frequency bands calculated from a parametric spectrogram, whose starting point are reflection coefficients. All these operations are aimed at enhancing changes that occur in the signal at the moments when damage-induced impulses appear. The article also presents guidelines for methods of determining parameter values in the employed analyses. The proposed solutions have been applied for analysing signals coming from a two-stage gearbox of a large machine driving a mining belt conveyor and the obtained results were analysed. They prove the effectiveness of the proposed techniques. It is worth emphasizing that these techniques can be easily adapted for monitoring machinery in varying operating conditions.

Continuous-time lossless systems, boundary interpolation and pivot structures

Balanced realizations of lossless systems can be generated from the tangential Schur algorithm using linear fractional transformations. In discrete-time, canonical forms with pivot structures are obtained by interpolation at infinity. In continuous-time case interpolation at infinity is on the stability boundary, leading to an angular derivative interpolation problem. In the scalar case, the balanced canonical form of Ober can be recovered in this way. Here we generalize to the multivariable case. It is shown that boundary interpolation can be regarded as a limit of classical interpolation with interpolation points tending to the imaginary axis. Some pivot structures can be generated, but no complete atlas is obtained. However, for input normal pairs an atlas of admissible pivot structures can be generated in a closely related way.

Schur Algorithm in The Class $${\mathcal{SI}}$$ of J-contractive Functions Intertwining Solutions of Linear Differential Equations

In the PhD thesis of the second author under the supervision of the third author was defined the class SI of J-contractive functions, depending on a parameter and arising as transfer functions of overdetermined conservative 2D systems invariant in one direction. In this paper we extend and solve in the class SI, a number of problems originally set for the class SC of functions contractive in the open right-half plane, and unitary on the imaginary line with respect to some preassigned signature matrix J. The problems we consider include the Schur algorithm, the partial realization problem and the Nevanlinna-Pick interpolation problem. The arguments rely on a correspondence between elements in a given subclass of SI and elements in SC. Another important tool in the arguments is a new result pertaining to the classical tangential Schur algorithm.

A Divide and Conquer Algorithm for the Superfast Solution of Toeplitz-like Systems

In this paper a new $O(N \,\log^3N)$ solver for $N\times N$ Toeplitz-like systems, based on a divide and conquer technique, is presented. Similarly to the superfast algorithm MBA for the inversion of a Toeplitz-like matrix [R. R. Bitmead and B. D. O. Anderson, Linear Algebra Appl., 34 (1980), pp. 103--116; M. Morf, Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, 1980, pp. 954--959], it exploits the displacement properties. In order to avoid the well-known numerical instability of the explicit inversion, the new algorithm relies on the triangular factorization and back-substitution formula for the system seen as a $2\times 2$ block system with blocks of half size. This idea is the one used in [M. Stewart, SIAM J. Matrix Anal. Appl., 25 (2003), pp. 669--693] to improve the numerical stability of superfast methods based on the generalized Schur algorithm for positive definite Toeplitz matrices, but the algorithm we propose can be applied also to nonsymmetric Toeplitz-like systems. The stability of the algorithm is examined through numerical experiments.

Multipoint Schur algorithm and orthogonal rational functions, I: Convergence properties

Classical Schur analysis is intimately related to the theory of orthogonal polynomials on the circle. We investigate the connection between multipoint Schur analysis and orthogonal rational functions. Specifically, we study the convergence of the Wall rational functions via the development of a rational analogue to the Szegő theory in the case that interpolation points may accumulate on the unit circle. This leads to a generalization of results in [22, 10], and new types of asymptotics.

ON HYPONORMALITY OF TOEPLITZ OPERATORS WITH POLYNOMIAL AND SYMMETRIC TYPE SYMBOLS

In [6], it was shown that hyponormality for Toeplitz operators with polynomial symbols can be reduced to classical Schur's algorithm in function theory. In [6], Zhu has also given the explicit values of the Schur's functions <TEX>${\Phi}_0$</TEX>, <TEX>${\Phi}_1$</TEX> and <TEX>${\Phi}_2$</TEX>. Here we explicitly evaluate the Schur's function <TEX>${\Phi}_3$</TEX>. Using this value we find necessary and sufficient conditions under which the Toeplitz operator <TEX>$T_{\varphi}$</TEX> is hyponormal, where <TEX>${\varphi}$</TEX> is a trigonometric polynomial given by <TEX>${\varphi}(z)$</TEX> = <TEX>${\sum}^N_{n=-N}a_nz_n(N{\geq}4)$</TEX> and satisfies the condition <TEX>$\bar{a}_N\(\array{a_{-1}\\a_{-2}\\a_{-4}\\{\vdots}\\a_{-N}}\)=a_{-N}\;\(\array{\bar{a}_1\\\bar{a}_2\\\bar{a}_4\\{\vdots}\\\bar{a}_N}\)$</TEX>. Finally we illustrate the easy applicability of the derived results with a few examples.

A fast solver for linear systems with displacement structure

We describe a fast solver for linear systems with reconstructable Cauchy-like structure, which requires O(rn^2) floating point operations and O(rn) memory locations, where n is the size of the matrix and r its displacement rank. The solver is based on the application of the generalized Schur algorithm to a suitable augmented matrix, under some assumptions on the knots of the Cauchy-like matrix. It includes various pivoting strategies, already discussed in the literature, and a new algorithm, which only requires reconstructability. We have developed a software package, written in Matlab and C-MEX, which provides a robust implementation of the above method. Our package also includes solvers for Toeplitz(+Hankel)-like and Vandermonde-like linear systems, as these structures can be reduced to Cauchy-like by fast and stable transforms. Numerical experiments demonstrate the effectiveness of the software.

A binary powering Schur algorithm for computing primary matrix roots

An algorithm for computing primary roots of a nonsingular matrix A is presented. In particular, it computes the principal root of a real matrix having no nonpositive real eigenvalues, using real arithmetic. The algorithm is based on the Schur decomposition of A and has an order of complexity lower than the customary Schur based algorithm, namely the Smith algorithm.