Mixed,componentwise condition numbers and small sample statistical condition estimation of Sylvester equations

We present a componentwise perturbation analysis for the continuous‐time Sylvester equations. Componentwise, mixed condition numbers and new perturbation bounds are derived for the matrix equations. The small sample statistical method can also be applied for the condition estimation. These condition numbers and perturbation bounds are tested on numerical examples and compared with the normwise condition number. The numerical examples illustrate that the mixed condition number gives sharper bounds than the normwise one. Copyright © 2011 John Wiley & Sons, Ltd.     

Structured condition numbers and statistical condition estimation for the LDU factorization

In this article, we consider the structured condition numbers for LDU, factorization by using the modified matrix-vector approach and the differential calculus, which can be represented by sets of parameters. By setting the specific norms and weight parameters, we present the expressions of the structured normwise, mixed, componentwise condition numbers and the corresponding results for unstructured ones. In addition, we investigate the statistical estimation of condition numbers of LDU factorization using the probabilistic spectral norm estimator and the small-sample statistical condition estimation method, and devise three algorithms. Finally, we compare the structured condition numbers with the corresponding unstructured ones in numerical experiments.     

Small sample statistical condition estimation for the total least squares problem

In this paper, under the genericity condition, we study the condition estimation of the total least squares (TLS) problem based on small sample condition estimation (SCE), which can be incorporated into the direct solver for the TLS problem via the singular value decomposition (SVD) of the augmented matrix [A, b]. Our proposed condition estimation algorithms are efficient for the small and medium size TLS problem because they utilize the computed SVD of [A, b] during the numerical solution to the TLS problem. Numerical examples illustrate the reliability of the algorithms. Both normwise and componentwise perturbations are considered. Moreover, structured condition estimations are investigated for the structured TLS problem.     

Asymptotic estimation for the condition numbers in BEM

Summary. We apply the boundary element methods (BEM) to the interior Dirichlet problem of the two dimensional Laplace equation, and its discretization is carried out with the collocation method using piecewise linear elements. In this paper, some precise asymptotic estimations for the discretization matrix (where denotes the division number) are investigated. We show that the maximum norm of and the condition number of have the forms: and , respectively, as , where the constants and are explicitly given in the proof. Although these estimates indicate illconditionedness of this numerical computation, the -convergence of this scheme with maximum norm is proved as an application of the main results. Received January 25, 1993 / Revised version received March 13, 1995     

A new version of the Smith method for solving Sylvester equation and discrete-time Sylvester equation

Recently, Xue etc. \cite{28} discussed the Smith method for solving Sylvester equation $AX+XB=C$, where one of the matrices $A$ and $B$ is at least a nonsingular $M$-matrix and the other is an (singular or nonsingular) $M$-matrix. Furthermore, in order to find the minimal non-negative solution of a certain class of non-symmetric algebraic Riccati equations, Gao and Bai \cite{gao-2010} considered a doubling iteration scheme to inexactly solve the Sylvester equations. This paper discusses the iterative error of the standard Smith method used in \cite{gao-2010} and presents the prior estimations of the accurate solution $X$ for the Sylvester equation. Furthermore, we give a new version of the Smith method for solving discrete-time Sylvester equation or Stein equation $AXB+X=C$, while the new version of the Smith method can also be used to solve Sylvester equation $AX+XB=C$, where both $A$ and $B$ are positive definite. % matrices. We also study the convergence rate of the new Smith method. At last, numerical examples are given to illustrate the effectiveness of our methods     

Well-posedness and small data scattering for the generalized Ostrovsky equation

We consider the generalized Ostrovsky equation utx=u+(up)_xx. We show that the equation is locally well posed in Hs, s>3/2 for all integer values of p?2. For p?4, we show that the equation is globally well posed for small data in H⁵∩W3,1 and moreover, it scatters small data. The latter results are corroborated by numerical computations which confirm the heuristically expected decay of ‖uLr_‖∼t−(r−2)/(2r).     

Initial condition solutions of the generalized feller equation

Summary Solutions for given initial conditions are established for the generalized (autonomous parabolic) Feller equation in one positive space variable and one positive time variable. The coefficients of this equation are power functions of the space variable and depend on four parameters. In general, the equation is singular at the origin and at infinity. It contains as special cases the special Feller equation, the Kepinski equation, and the heat equation. Areas of application include biology, superradiant emission processes, heat propagation in solids (with special applications in the area of heat shield and ablation material design), and certain chemical reaction-diffusion processes. It is noteworthy that, for particular values of the parameters, the equation allows an evolution theoretic derivation of the fundamental distribution laws of Wien, Maxwell, Poisson, and Gauss. The general initial condition solution will be derived from a fundamental solution and will be given in terms of an integral transform for locally summable functions (singular integral). It is also shown that, for admissible parameter values, there always exist nontrivial solutions which approach zero as the time variable goes to zero and that, for particular parameter ranges, there exist singular solutions, conservative solutions, and delta function initial condition solutions.

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Zusammenfassung Es werden Lösungen für gegebene Anfangsbedingungen für die verallgemeinerte (autonome, parabolische) Fellersche Gleichung in einer positiven Raumvariablen und einer positiven Zeitvariablen aufgestellt. Die Koeffizienten dieser Gleichung sind Potenzen der Raumvariablen und hängen von vier Parametern ab. Die Gleichung ist im allgemeinen singulär am Ursprung und im Unendlichen. Sie umfasst als Spezialfälle die spezielle Fellersche Gleichung, die Kepinskische Gleichung und die Wärmegleichung. Anwendung findet sie in der Biologie, in superstrahlenden Emissionsprozessen, in der Theorie der Wärmeausbreitung in Festkörpern (besonders beim Entwurf von Hitzeschilden und Abschmelzmaterialien) und im Gebiet gewisser chemischer Reaktions-Diffusionsprozesse. Est ist bemerkenswert, dass die Gleichung für besondere Parameterwerte eine evolutionstheoretische Herleitung der grundlegenden Verteilungsgesetze von Wien, Maxwell, Poisson und Gauss ermöglicht. Die allgemeine Lösung für gegebene Anfangsbedingung wird aus einer Grundlösung entwickelt und in der Form einer Integraltransformation gegeben für lokal summierbare Funktionen (singuläres Integral). Es wird weiterhin gezeigt, dass für zulässige Parameterwerte stets nichtriviale Lösungen existeren, die nach Null streben, wenn die Zeitvariable nach Null geht, und dass es für besondere Parameterbereiche singuläre und konservative Lösungen gibt und solche, die einer Deltafunktion als Anfangsbedingung entsprechen.

     

Applications of the generalized Sylvester matrix

Recently a Sylvester matrix for several polynomials has been defined, establishing the relative primeness and the greatest common divisor of polynomials. Using this matrix, we perform qualitative analysis of several polynomials regarding the inners, the bigradients, Trudi's theorem, and the connection of inners and the Schur complement. Also it is shown how the regular greatest common divisor of m+1 (m>1) polynomial matrices can be determined.     

A semilinear equation with generalized Wentzell boundary condition

In this paper we consider a semilinear equation with a generalized Wentzell boundary condition. We prove the local well-posedness of the problem and derive the conditions of the global existence of the solution and the conditions for finite time blow-up. We also derive an estimate for the blow-up time.     

Global existence of small amplitude solution for the generalized IMBq equation

In this paper, we reconsider the problem discussed in [G.W. Chen, S.B. Wang, Small amplitude solutions of the generalized IMBq equation, J. Math. Anal. Appl. 274 (2002) 846-866]. The proof of global existence presented in [G.W. Chen, S.B. Wang, Small amplitude solutions of the generalized IMBq equation, J. Math. Anal. Appl. 274 (2002) 846-866] is very simple in form, but it is a pity that the authors overlooked the bad behavior of low frequency part of B(t)ψ which causes trouble in L^∞ and Hs estimates. In this paper, we will give out a new proof of the global existence under an additional condition on the initial data.     

Convex constrained optimization for large-scale generalized Sylvester equations

We propose and study the use of convex constrained optimization techniques for solving large-scale Generalized Sylvester Equations (GSE). For that, we adapt recently developed globalized variants of the projected gradient method to a convex constrained least-squares approach for solving GSE. We demonstrate the effectiveness of our approach on two different applications. First, we apply it to solve the GSE that appears after applying left and right preconditioning schemes to the linear problems associated with the discretization of some partial differential equations. Second, we apply the new approach, combined with a Tikhonov regularization term, to restore some blurred and highly noisy images.     

Finite iterative algorithm for the complex generalized Sylvester tensor equations

A finite iterative algorithm is proposed to solve a class of complex generalized Sylvester tensor equations. The properties of this proposed algorithm are discussed based on a real inner product of two complex tensors and the finite convergence of this algorithm is obtained. Two numerical examples are offered to illustrate the effectiveness of the proposed algorithm.     

A condition guaranteeing the absence of the blow-up phenomenon for the generalized Burgers equation

The initial boundary value problem for the generalized Burgers equation with nonlinear sources is considered. We formulate a condition guaranteeing the absence of the blow-up of a solution and discuss the optimality of this condition.