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1.
In this paper, by applying the real representations of complex matrices, the particular structure of the real representations and the Moore–Penrose generalized inverse, we obtain the explicit expression of the minimal norm least squares Hermitian solution of the complex matrix equation AXB+CXD=E. And we also derive the minimal norm least squares Hermitian solution of the complex matrix equation AXB=E. Our proposed formulas only involve real matrices, and therefore are more effective and portable than those reported in Yuan and Liao (2014). The corresponding algorithms only perform real arithmetic which also consider the particular structure of the real representations of complex matrices. Two numerical examples are provided to demonstrate the effectiveness of our algorithms.  相似文献   

2.
Suppose that AXB=C is a consistent matrix equation and partition its solution X into a 2×2 block form. In this article we give some formulas for the maximal and minimal ranks of the submatrices in a solution X to AXB=C. From these formulas, we derive necessary and sufficient conditions for the submatrices to be zero and nonsingular, respectively. As applications, we give a group of formulas for the maximal and minimal ranks of submatrices in generalized inverses of matrices and their properties.  相似文献   

3.
In this note, the problem of solution to the matrix equation AX+XTC=B is considered by the Moore-Penrose generalized inverse matrix. A general solution to this equation is obtained. At the same time, some useful conclusions are made, which play important roles in the linear system theories and applications.  相似文献   

4.
The purpose of this paper is deriving the minimal residual (MINIRES) algorithm for finding the symmetric least squares solution on a class of Sylvester matrix equations. We prove that if the system is inconsistent, the symmetric least squares solution can be obtained within finite iterative steps in the absence of round-off errors. Furthermore, we provide a method for choosing the initial matrix to obtain the minimum norm least squares symmetric solution of the problem. Finally, we give some numerical examples to illustrate the performance of MINIRES algorithm.  相似文献   

5.
The generalized diffusion equations, with nonlinear source terms which encompasses the Fisher, Newell-Whitehead, FitzHugh-Nagumo and Allen-Cahn equations as particular forms are solved by the G′/G-expansion method. The exact solutions are in terms of hyperbolic, trigonometric and rational functions with external parameters. This paper concludes with the stationary topological soliton solution of the FitzHugh-Nagumo equation that is obtained by the Ansatz method.  相似文献   

6.
沈光星  黄德超 《科技通报》2006,22(3):283-287
利用多项式矩阵理论,给出了循环线性方程组有解的判定并求出各种情况下的解。若方程组有唯一解,求出其唯一解;若方程组有无穷解,求出其极小范数解;若方程组无解,求出其极小范数最小二乘解。  相似文献   

7.
This paper attempts to show the more suitability of the extended general Struble's technique than the unified Krylov-Bogoliubov-Mitropolskii (KBM) method in solving the problems that occur during the critical conditions. Recently a critically damped condition of an nth, n=2,3, … order weakly nonlinear autonomous ordinary differential equation has been investigated by the unified KBM method, in which the corresponding unperturbed equation has some real (negative) repeated eigenvalues. But there are more important critical conditions, which are still untouched. One of them occurs when a pair of complex eigenvalues is equal to another. It is complicated to formulate as well as to utilize the KBM method to investigate this condition. However, the extended general Struble's technique is applicable to both autonomous and non-autonomous systems. Solutions obtained for different critical conditions as well as for different initial conditions show a good agreement with the numerical solutions. The method is illustrated by an example of a fourth-order nonlinear differential equation whose unperturbed equation has repeated complex eigenvalues. A steady-state solution is determined for the non-autonomous equation. Moreover, a critical condition of a fourth-order nonlinear equation is investigated when two real eigenvalues of the unperturbed equation are non-positive and equal.  相似文献   

8.
A modified and compact form of Krylov-Bogoliubov-Mitropolskii (KBM) unified method is extended to obtain approximate solution of an nth order, n=2,3,…, ordinary differential equation with small nonlinearities when unperturbed equation has some repeated real eigenvalues. The existing unified method is used when the eigenvalues are distinct whether they are purely imaginary or complex or real. The new form is presented generalizing all the previous formulae derived individually for second-, third- and fourth-order equations to obtain undamped, damped, over-damped and critically damped solutions. Therefore, all types of oscillatory and non-oscillatory solutions are determined by suitable substitution of the eigenvalues in a general result. The formulation of the method is very simple and the determination of the solution is easy. The method is illustrated by an example of a fourth-order equation when unperturbed equation has two real and equal eigenvalues. The solution agrees with a numerical solution nicely. Moreover, this solution is useful when the differences between conjugate eigenvalues (real or complex) are small. Thus the method is a complement of the existing modified and compact form of KBM method.  相似文献   

9.
By means of the real linear operator, we establish an iterative algorithm for solving a class of complex generalized coupled Sylvester matrix equations. The finite termination of the proposed algorithm is proved. By representing a complex matrix as a larger real matrix, we present a new method to prove that the minimum-norm solution or minimum-norm least squares solution of the complex generalized coupled Sylvester matrix equations can be obtained by an appropriate selection for the initial matrices, which has not been found in the existing work. Numerical experiments on some randomly generated data and practical image restoration problem show that the proposed algorithm is feasible and effective.  相似文献   

10.
Severely ill-conditioned Hermitian matrices are commonly seen in digital signal processing as well as in other applications where the Galerkin method and the least squares method are employed to seek optimal solutions of linear or nonlinear models. A new treatment for such matrices is proposed in this paper, which is fundamentally different from the well-known conjugate gradient preconditioners, SVD-based methods, and other popular algorithms for solving ill-conditioned systems in the literature. We propose an exact, alternative formula for the inverse of a Hermitian matrix via parametric diagonal perturbation. It reduces the condition number of the original ill-conditioned matrix exponentially. The performance of the proposed algorithm is established via condition analysis and demonstrated over severely ill-conditioned matrix systems from digital filter designs.  相似文献   

11.
A method of using orthogonal shifted Legendre polynomials for identifying the parameters of a process whose behaviour can be modelled by a linear differential equation with time-varying coefficients in the form of finite-order polynomials is presented. It is based on the repeated integration of the differential equation and the representations of ∫0ts(τ) dτ = Ps(t) and ts(t) = Rs(t), where P and R are constant matrices and s(t) is a shifted Legendre vector whose elements are shifted Legendre polynomials. The differential input-output equation is converted into a set of overdetermined linear algebraic equations for a least squares solution. The results of simulation studies are included to illustrate the applicability of the method.  相似文献   

12.
This paper presents a simple and efficient design method for cosine-modulated filter banks with prescribed stopband attenuation, passband ripple, and channel overlap. The method casts the design problem as a linear minimization of filter coefficients such that their value at ω=π/2M is 0.707, which results in a simpler, more direct design procedure. The weighted constrained least squares technique is exploited for designing the prototype filter for cosine modulation (CM) filter banks. Several design examples are included to show the increased efficiency and flexibility of the proposed method over the exiting methods. An application of the proposed method is considered in the area of sub-band coding of the ECG and speech signals.  相似文献   

13.
In this paper, the problem of robust H∞ filtering for uncertain systems with time-varying distributed delays is considered. The uncertainties under discussion are time varying but norm bounded. Based on the Lyapunov stability theory, sufficient condition for the existence of full order H∞ filters is proposed by linear matrix inequality (LMI) approach such that the filtering error system is asymptotically sable and satisfies a prescribed attenuation level of noise. A numerical example is given to demonstrate the availability of the proposed method.  相似文献   

14.
A unified theory is presented for obtaining the transient response of nth order nonlinear systems with small nonlinearities by Krylov-Bogoliubov-Mitropolskii method. The method is a generalization of Bogoliubov's asymptotic method and covers all three cases when the roots of the corresponding linear equation are real, complex conjugate, or purely imaginary. It is shown that by suitable substitution for the roots in the general result that the solution corresponding to each of the three cases can be obtained. The method is illustrated by examples.  相似文献   

15.
A class of matrices with 1, -1 and 0 entries, called the H-matrices, is introduced and properties of such matrices are investigated. It is shown that for an H-matrix K the matrix G = KDKT, where D is a diagonal matrix with nonnegative diagonal entries, has certain properties applicable to topological synthesis of networks. A technique for decomposing a particular symmetrical matrix G, called the Q-matrix, into KDKT, where K is an H-matrix and D is a diagonal matrix with positive diagonal entries, is also developed.  相似文献   

16.
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17.
Confirmatory composite analysis (CCA) is a structural equation modeling (SEM) technique that specifies and assesses composite models. In a composite model, the construct emerges as a linear combination of observed variables. CCA was invented by Jörg Henseler and Theo K. Dijkstra in 2014, was subsequently fully elaborated by Schuberth et al. (2018), and was then introduced into business research by Henseler and Schuberth (2020b). Inspired by Hair et al. (2020), a recent article in the International Journal of Information Management (Motamarri et al., 2020) used the same term ‘confirmatory composite analysis’ as a technique for confirming measurement quality in partial least squares structural equation modeling (PLS-SEM) specifically. However, the original CCA (Henseler et al., 2014; Schuberth et al., 2018) and the Hair et al. (2020) technique are very different methods, used for entirely different purposes and objectives. So as to not confuse researchers, we advocate that the later-published Hair et al. (2020) method of confirming measurement quality in PLS-SEM be termed ‘method of confirming measurement quality’ (MCMQ) or ‘partial least squares confirmatory composite analysis’ (PLS-CCA). We write this research note to clarify the differences between CCA and PLS-CCA.  相似文献   

18.
This paper investigates the Riemannian trust-region method for H2 model reduction of bilinear systems. The H2 error norm is treated as a cost function on the Stiefel manifold such that the orthogonality constraint for the projection matrix is plainly satisfied. The property related to the Euclidean gradient is studied. Then, the inner product associated with the Riemannian Hessian is derived, which can simplify the expression of the trust-region subproblem. The trust-region method for H2 model reduction is accordingly established and the convergence is further discussed. Finally, two numerical examples are employed to demonstrate the performance of the proposed method.  相似文献   

19.
In order to quantize Dirac's classical point electron1 we supplement Einstein's classical equation (E/c)2 ? p2 = b2 with a reciprocal classical equation (CΔt)2 ? (Δr)2 = a2 where b = mc and a is Dirac's signal radius. Δt is the time saved by a light signal in various states of motion of the electron, and a/c is the rest time saved. Our former efforts2 of obtaining an integral equation for the probability amplitude have been rectified by Born.3 There is no solution of the integral equation, however, unless advanced and retarded phases are introduced simultaneously, along with Dirac's advanced and retarded potentials. We have obtained a transcendental equation for the eigen-value μ = αγ where α is the Sommerfeld fine-structure constant, and γ is the numerical factor in Dirac's signal radius a = γe2/mc2. The smallest eigen-value is μ = 0.0299.That is, ab = hγ = h/210.  相似文献   

20.
The identification of linear, discrete time, scalar output systems which are driven exclusively by white, zero mean, inaccessible noise sequences is discussed. Two principal results are presented. First, two methods (least squares and an autocorrelation technique) for identifying the system characteristic equation coefficients are compared. The least squares approach is shown to be biased except for special cases. In general, the bias cannot be removed. If the state transition matrix is of the phase variable form, bias removal requires a knowledge of the measurement noise variance and all but one of the state driving noise variances. The autocorrelation technique is not biased asymptotically and does not require a knowledge of the noise variances.Secondly, it is shown that the m2 elements of the state transition matrix cannot be identified uniquely from the scalar output sequence autocorrelation coefficients if the system order is higher than one. The implication of this uncertainty in the state transition matrix on optimal filtering of the output sequence is briefly discussed.  相似文献   

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