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This note outlines an algorithm for solving the complex ‘matrix Procrustes problem’. This is a least‐squares approximation over the cone of positive semi‐definite Hermitian matrices, which has a number of applications in the areas of Optimization, Signal Processing and Control. The work generalizes the method of Allwright (SIAM J. Control Optim. 1988; 26 (3):537–556), who obtained a numerical solution to the real‐valued version of the problem. It is shown that, subject to an appropriate rank assumption, the complex problem can be formulated in a real setting using a matrix‐dilation technique, for which the method of Allwright is applicable. However, this transformation results in an over‐parametrization of the problem and, therefore, convergence to the optimal solution is slow. Here, an alternative algorithm is developed for solving the complex problem, which exploits fully the special structure of the dilated matrix. The advantages of the modified algorithm are demonstrated via a numerical example. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
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The Structured Total Least Squares (STLS) problem is a natural extension of the Total Least Squares (TLS) approach when structured matrices are involved and a similarly structured rank deficient approximation of that matrix is desired. In many of those cases the STLS approach yields a Maximum Likelihood (ML) estimate as opposed to, e.g., TLS.In this paper we analyze the STLS problem for Hankel matrices (the theory can be extended in a straightforward way to Toeplitz matrices, block Hankel and block Toeplitz matrices). Using a particular parametrisation of rank-deficient Hankel matrices, we show that this STLS problem suffers from multiple local minima, the properties of which depend on the parameters of the new parametrisation. The latter observation makes initial estimates an important issue in STLS problems and a new initialization method is proposed. The new initialization method is applied to a speech compression example and the results confirm the improved performance compared to other previously proposed initialization methods. 相似文献
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基于交替投影算法求解单变量线性约束矩阵方程问题 总被引:2,自引:1,他引:1
研究如下线性约束矩阵方程求解问题:给定A∈R~(m×n),B∈R~(n×p)和C∈R~(m×p),求矩阵X∈R(?)R~(n×n)\"使得A×B=C以及相应的最佳逼近问题,其中集合R为如对称阵,Toeplitz阵等构成的线性子空间,或者对称半(ε)正定阵,(对称)非负阵等构成的闭凸集.给出了在相容条件下求解该问题的交替投影算法及算法收敛性分析.通过大量数值算例说明该算法的可行性和高效性,以及该算法较传统的矩阵形式的Krylov子空间方法(可行前提下)在迭代效率上的明显优势,本文也通过寻求加速技巧进一步提高算法的收敛速度. 相似文献
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In this paper, we discuss basic properties, a least‐squares problem for row extended matrices and the associated approximation problem. First, we obtain their basic properties by applying their particular structure. Then we derive a general representation of the solutions to the least‐squares problem, and we obtain an expression for the solution to the associated approximation problem. Finally, we provide a perturbation analysis and a perturbation bound for the best approximate solution. The results are illustrated by numerical examples. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
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本文利用矩阵的奇异值分解(SVD),给出了在一流形上矩阵方程B^TXB=D的加权最小二乘对称解的通解表达式,并解决了加权最小二乘对称解的最佳逼近问题。 相似文献
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In this paper, we discuss the matrix equation AXB = C. We obtain a terse expression of weighted least-squares solution by applying the canonical correlation decomposition(CCD), we discuss the sufficient and necessary conditions for existence of Hermitian solutions, and derive a general form of the Hermite solutions.We also derive the expression of the optimal approximation solution in the set of Hermite soluti6ns to given matrix. 相似文献
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应用共轭梯度方法和线性投影算子,给出迭代算法求解了线性矩阵方程AX=B在任意线性子空间上的最小二乘解问题.在不考虑舍入误差的情况下,可以证明,所给迭代算法经过有限步迭代可得到矩阵方程AX=B的最小二乘解、极小范数最小二乘解及其最佳逼近.文中的数值例子证实了该算法的有效性. 相似文献
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讨论了矩阵方程组A_1XB_1=D_1,A_2XB_2=D_2反对称最小二乘解的递推算法,该算法不仅能够用于计算反对称最小二乘解,而且在选取特殊的初始矩阵时,算法能够求出矩阵方程组的极小范数反对称最小二乘解,以及对给定的矩阵进行最佳逼近的反对称解. 相似文献
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The Toeplitz pencil conjecture stated in [W. Schmale, P.K. Sharma, Problem 30-3: singularity of a toeplitz matrix, IMAGE 30 (2003); W. Schmale, P.K. Sharma, Cyclizable matrix pairs over and a conjecture on toeplitz pencils, Linear Algebra Appl. 389 (2004) 33-42] is equivalent to a conjecture for n×n Hankel pencils of the form Hn(x)=(ci+j-n+1), where c0=x is an indeterminate, cl=0 for l<0, and for l1. In this paper it is shown to be implied by another conjecture, which we call the root conjecture. The root conjecture asserts a strong relationship between the roots of certain submaximal minors of Hn(x) specialized to have c1=c2=1. We give explicit formulae in the ci for these minors and prove the root conjecture for minors mnn,mn-1,n of degree 6. This implies the Hankel Pencil conjecture for matrices up to size 8×8. The main tools involved are a partial parametrization of the set of solutions of systems of polynomial equations that are both homogeneous and index sum homogeneous, and use of the Sylvester identity for matrices. 相似文献
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利用矩阵的奇异值分解及广义逆,给出了矩阵约束下矩阵反问题AX=B有实对称解的充分必要条件及其通解的表达式.此外,给出了在矩阵方程的解集合中与给定矩阵的最佳逼近解的表达式. 相似文献
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By means of complex representation of a quaternion matrix, we study the relationship between the solutions of the quaternion equality constrained least squares problem and that of complex equality constrained least squares problem, and obtain a new technique of finding a solution of the quaternion equality constrained least squares problem. 相似文献
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The notion of Hankel operators associated with analytic crossed products were introduced and researched in [2]. In this paper,
we study the adjoint of Hankel operators and give necessary and sufficient condition that the adjoint of a Hankel operator
is again a Hankel operator.
This work was supported in part by a Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science. 相似文献
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设H是复数域C上的可分Hilbert空间,B(■)是■上有界线性算子的全体.本文证明B(■)可以由三个投影生成,改进了C.Davis[2]的方法,使得第三个投影E3的取法更为灵活,证明更为简洁,获得了这样{E1,E2,E3}更多的族,并且给出当■是有限维情形时的证明。 相似文献
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结构矩阵低秩逼近在图像压缩、计算机代数和语音编码中有广泛应用.首先给出了几类结构矩阵的投影公式,再利用交替投影方法计算结构矩阵低秩逼近问题.数值试验表明新方法是可行的. 相似文献
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We extend the algorithm of [4], based on Newton's iteration and on the concept of -displacement rank, to the computation of the generalized inverse A
+ of an m×n Toeplitz matrix A. We introduce new strategies for the dynamical control of the truncation level at each step of the iteration. Numerical experiments and an application to a problem of image restoration are shown. An object-oriented implementation in C++ is described. 相似文献
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Zhongzhi Zhang Changrong Liu 《高等学校计算数学学报(英文版)》2006,15(1):60-66
Upon using the denotative theorem of anti-Hermitian generalized Hamiltonian matrices,we solve effectively the least-squares problem min‖AX-B‖over anti-Hermitian generalized Hamiltonian matrices.We derive some necessary and sufficient conditions for solvability of the problem and an expression for general solution of the matrix equation AX=B.In addition,we also obtain the expression for the solution of a relevant optimal approximate problem. 相似文献
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Georg Schneider 《Mathematische Nachrichten》2005,278(3):312-317
We consider the canonical solution operator to restricted to (0, 1)‐forms with coefficients in the generalized Fock‐spaces (1) We will show that the canonical solution operator restricted to (0, 1)‐forms with ‐coefficients can be interpreted as a Hankel‐operator. Furthermore we will show that the canonical solution operator is not compact for m ≥ 2. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
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Thomas Führer Michael Karkulik 《Numerical Methods for Partial Differential Equations》2019,35(5):1777-1800
We provide new insights into the a priori theory for a time‐stepping scheme based on least‐squares finite element methods for parabolic first‐order systems. The elliptic part of the problem is of general reaction‐convection‐diffusion type. The new ingredient in the analysis is an elliptic projection operator defined via a nonsymmetric bilinear form, although the main bilinear form corresponding to the least‐squares functional is symmetric. This new operator allows to prove optimal error estimates in the natural norm associated to the problem and, under additional regularity assumptions, in the L2 norm. Numerical experiments are presented which confirm our theoretical findings. 相似文献