A class of marked invariant subspaces with an application to algebraic Riccati equations

Invariant subspaces of a matrix A are considered which are obtained by truncation of a Jordan basis of a generalized eigenspace of A. We characterize those subspaces which are independent of the choice of the Jordan basis. An application to Hamilton matrices and algebraic Riccati equations is given.     

On optimality of approximate low rank solutions of large-scale matrix equations

In this paper, we discuss some optimality results for the approximation of large-scale matrix equations. In particular, this includes the special case of Lyapunov and Sylvester equations, respectively. We show a relation between the iterative rational Krylov algorithm and a Riemannian optimization method which recently has been shown to locally minimize a certain energy norm of the underlying Lyapunov operator. Moreover, we extend the results for a more general setting leading to a slight modification of IRKA. By means of some numerical test examples, we show the efficiency of the proposed methods.     

Controllability of matrix eigenvalue algorithms: the inverse power method

In this paper we initiate a program to study the controllability properties of matrix eigenvalue algorithms arising in numerical linear algebra. Our focus is on a well-known eigenvalue method, the inverse power iteration defined on projective space. A complete characterization of the reachable sets and their closures is given via cyclic invariant subspaces. Moreover, a necessary and sufficient condition for almost controllability of the inverse power method is derived.     

A projection method for the numerical solution of linear systems in separable stiff differential equations

This paper deals with the efficient implementation of implicit methods for solving stiff ODEs, in the case of Jacobians with separable sets of eigenvalues. For solving the linear systems required we propose a method which is particularly suitable when the large eigenvalues of the Jacobian matrix are few and well separated from the small ones. It is based on a combination of an initial iterative procedure, which reduces the components of the vector error along to the nondominant directions of J and a projection Krylov method which reduces the components of the vector error along to the directions corresponding to the large eigenvalues. The technique solves accurately and cheaply the linear systems in the Newton's method, and computes the number of stiff eigenvalues of J when this information is not explicitly available. Numerical results are given as well as comparisons with the LSODE code.     

A method for calculating successive approximate solutions for a class of block banded M/G/1 type Markovian models

This paper presents an efficient equilibrium solution algorithm for a class of infinite block banded M/G/1 type Markov chains. By re-blocking the states, these are a class of the so-called quasi-birth-and-death (QBD) type chains. The proposed algorithm is not based on an iterative approach, so that the exact solution can be computed in a known finite number of steps. The key point on which the algorithm is based is the identification of a linear dependence among variables. This dependence is expressed in terms of a companion matrix. The equilibrium solution of the Markov chain is obtained operating on this matrix.

An attractive feature of the algorithm is that it allows the computation of a succession of approximate solutions with growing accuracy, until the exact solution is obtained in a finite number of steps. The class of block-banded M/G/1 type Markov chains we consider requires that the lower diagonal block is invertible and that the chain is ergodic. However, many models arising from telecommunication systems satisfy this restriction. Results for a case study show that the proposed algorithm is efficient and quite accurate, even when providing approximate solutions.     

Guaranteed roll-off in a class of high-gain feedback design problems

A number of synthesis problems associated with (almost) disturbance decoupling by state or measurement feedback is considered. Starting from a mathematical definition of the notion of high-frequency roll-off, known results on the solvability of these problems are generalized to the situation in which we require their solvability together with a certain guaranteed roll-off between disturbance and control. The conditions are formulated in terms of the solvability and approximate solvability of certain matrix equations in rational functions.     

Intervals of solutions of the discrete-time algebraic Riccati equation

If two solutions YZ of the DARE are given then the set of solutions X with YXZ can be parametrized by invariant subspaces of the closed loop matrix corresponding to Y. The paper extends the geometric theory of Willems from the continuous-time to the discrete-time ARE making the weakest possible assumptions.     

Identification of certain noisy MA models: new results

In this paper, we address the identification problem of p-inputs q-outputs MA models, corrupted by a white noise with unknown covariance matrix, in the case where p<q. Under certain additional conditions, we show that the generating function of the MA model is identifiable (up to a p×p constant orthogonal matrix) from the autocovariance function of the observation. Our results extend those already obtained in Desbouvries et al. [5] and Desbouvries and Loubaton [6].     

On controllability,near-controllability,controllable subspaces,and nearly-controllable subspaces of a class of discrete-time multi-input bilinear systems

This paper considers a class of discrete-time multi-input inhomogeneous bilinear systems. The structure of such systems is most close to linear time-invariant systems’ but they own a strong property. That is, if the systems are uncontrollable, they can still be nearly controllable. Necessary and sufficient conditions for controllability and near-controllability of the systems are established by using a classical decomposition. Furthermore, a geometric characterization is given for the systems such that controllable subspaces and nearly-controllable subspaces are derived and characterized. Similar results on controllability are also obtained for the continuous-time counterparts of the systems. Finally, examples are provided to demonstrate the conceptions and results of this paper.     

A complement about almost controllability subspaces

We show that, for any almost controllability subspace, there exist a maximal input chain and a family of state feedbacks which lead to the well known ‘feedback-chain’ expression for this subspace.     

Parallel eigenvalue reordering in real Schur forms

A parallel algorithm for reordering the eigenvalues in the real Schur form of a matrix is presented and discussed. Our novel approach adopts computational windows and delays multiple outside‐window updates until each window has been completely reordered locally. By using multiple concurrent windows the parallel algorithm has a high level of concurrency, and most work is level 3 BLAS operations. The presented algorithm is also extended to the generalized real Schur form. Experimental results for ScaLAPACK‐style Fortran 77 implementations on a Linux cluster confirm the efficiency and scalability of our algorithms in terms of more than 16 times of parallel speedup using 64 processors for large‐scale problems. Even on a single processor our implementation is demonstrated to perform significantly better compared with the state‐of‐the‐art serial implementation. Copyright © 2009 John Wiley & Sons, Ltd.     

Parallel Partial Stabilizing Algorithms for Large Linear Control Systems

In this paper we present parallel algorithms for stabilizing large linear control systems on multicomputers. Our algorithms first separate the stable part of the linear control system and then compute a stabilizing feedback for the unstable part. Both stages are solved by means of the matrix sign function which presents a high degree of parallelism and scalability.The experimental results on an IBM SP2 platform show the performance of our approach.     

On almost invariant subspaces for implicit linear discrete-time systems

The concept of almost invariant subspace for an implicit linear discrete-time system is introduced and studied in detail. It is shown also that for regular homogeneous implicit systems the so-called deflating subspaces can be identified with almost invariant subspaces.     

Necessary and sufficient conditions for existence of J-spectral factorization for para-Hermitian rational matrix functions

For a para-Hermitian rational matrix function G(λ)=J+C(λI_p−A)⁻¹B, where J=J∗ is invertible, and which has no poles and zeros on the imaginary line, we give necessary and sufficient conditions in terms of A,B,C and J for the existence of a J-spectral factorization, as well as an algorithm to obtain the J-spectral factor in case it exists.     

Further decomposition for singular systems and its properties on geometric subspace

This paper focuses on the relationship between the geometric subspaces and the structural decomposition of continuous-time singular systems. The original structural decomposition is not capable of revealing explicitly the invariant geometric subspaces for singular systems. As such, a further decomposition is necessary and is thus investigated in this paper. Under a new decomposition proposed, the supremal output-nulling (A,E, ImB)-invariant subspace of singular systems can be clearly expressed in an explicit form, and some of its applications are also addressed.     

Geotensity: Combining Motion and Lighting for 3D Surface Reconstruction

This paper is about automatically reconstructing the full 3D surface of an object observed in motion by a single static camera. Based on the two paradigms, structure from motion and linear intensity subspaces, we introduce the geotensity constraint that governs the relationship between four or more images of a moving object. We show that it is possible in theory to solve for 3D Lambertian surface structure for the case of a single point light source and propose that a solution exists for an arbitrary number point light sources. The surface may or may not be textured. We then give an example of automatic surface reconstruction of a face under a point light source using arbitrary unknown object motion and a single fixed camera.     

基于证据推理的多特征融合人脸识别算法

提出了一种基于证据推理的多特征融合人脸识别算法(DSPSA).该算法利用证据推理理论在处理不确定和冲突信息方面的优越性,融合多个面部特征的信息,有效地处理了人脸图像由于光照、旋转、表情等因素造成不确定信息,从而达到改善识别结果以及增强识别系统对训练样本库以外类别的识别能力.算法中提出了新的基本置信指派构造公式.     

Motion-based segmentation of objects using overlapping temporal windows

Motion segmentation refers to the problem of separating the objects in a video sequence according to their motion. It is a fundamental problem of computer vision, since various systems focusing on the analysis of dynamic scenes include motion segmentation algorithms. In this paper we present a novel approach, where a video shot is temporally divided in successive and overlapping windows and motion segmentation is performed on each window respectively. This attribute renders the algorithm suitable even for long video sequences. In the last stage of the algorithm the segmentation results for every window are aggregated into a final segmentation. The presented algorithm can handle effectively asynchronous trajectories on each window even when they have no temporal intersection. The evaluation of the proposed algorithm on the Berkeley motion segmentation benchmark demonstrates its scalability and accuracy compared to the state of the art.     

一种基于加权TOPS的宽带DOA估计新方法

针对投影子空间正交性测试（Test of orthogonality of projected subspaces,TOPS）对宽带信号波达方向估计（Direction?of?arrival, DOA）存在角度分辨率较低,且易出现伪峰的问题,提出了一种加权TOPS的宽带DOA估计新方法。该方法通过最大化各频率点信号子空间与噪声子空间特征值区分度选择参考频点,同时利用信号子空间投影代替其零空间投影;然后利用正交频率子空间测试法（Test of orthogonality of frequency subspaces, TOFS）对平方TOPS法的判定矩阵进行加权修正;最后对判定矩阵求迹实现宽带DOA估计,避免了奇异值分解。与现有的TOPS法、平方TOPS以及TOFS相比,该方法提高DOA估计精度,能够有效剔除伪峰,降低了算法复杂度,且对间隔相近信源DOA估计分辨率更高。仿真实验结果验证了该方法的有效性。