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In this paper we propose and study a new Lagrange finite element method for the two-dimensional Maxwell’s equations. Its solution may be singular because of the nonsmooth domain with reentrant corners. The proposed method allows the standard Lagrange elements on barycentric refinements of any order greater than or equal to two. We analyze the proposed method for the eigenvalue problem and the indefinite source problem, obtaining the well-posedness and optimal error estimates. Numerical results are presented for confirming the theoretical results.

A coercive mixed variational formulation on H 0 ( curl ; Ω ) × H ( div ; Ω ) is proposed for the generalized Maxwell problem which typically arises from computational electromagnetism. The mixed variables are the electric field and a pseudo electric displacement field. The well-posedness of the mixed variational problem is proven in the general settings (multiply connected domain of Lipschitz-continuous boundary with a number of connected components, filling with discontinuous, anisotropic and inhomogeneous media); more importantly, the coercivity is established. A conforming finite element discretization is further proposed, where the electric field is approximated by H ( curl ; Ω ) -conforming edge element while the pseudo electric displacement field by H ( div ; Ω ) -conforming flux element. Error estimates are obtained, and in particular, the method produces an L 2 curl-convergent approximation and more importantly, an L 2 div-convergent approximation for the solution.

In this paper, we propose and analyze a mixed $H^1$-conforming finite element method for solving Maxwell's equations in terms of electric field and Lagrange multiplier, where the multiplier is intr...

In this paper, we consider the eigenvalue embedding problem of the undamped vibroacoustic system with no-spillover (EEP-UVA), which is to update the original system to a new undamped vibroacoustic system, such that some eigen-structures are replaced with newly measured ones, while the remaining eigen-structures are kept unchanged. We provide a set of parametric solutions to the EEP-UVA. The freedoms in the parametric matrices can be further exploited to achieve some other desirable properties. The performance of the proposed algorithms are illustrated by numerical examples.

Linear discriminant analysis (LDA) as a well-known supervised dimensionality reduction method has been widely applied in many fields. However, the lack of sparsity in the LDA solution makes interpretation of the results challenging. In this paper, we propose a new model for sparse uncorrelated LDA (ULDA). Our model is based on the characterization of all solutions of the generalized ULDA. We incorporate sparsity into the ULDA transformation by seeking the solution with minimum $\ell _{1}$ -norm from all minimum dimension solutions of the generalized ULDA. The problem is then formulated as an $\ell _{1}$ -minimization problem with orthogonality constraint. To solve this problem, we devise two algorithms: 1) by applying the linearized alternating direction method of multipliers and 2) by applying the accelerated linearized Bregman method. Simulation studies and high-dimensional real data examples demonstrate that our algorithms not only compute extremely sparse solutions but also perform well in classification.

In this paper we propose and study a finite element method for a curlcurl-graddiv eigenvalue interface problem. Its solution may be of piecewise non-$H^1$. We would like to approximate such a solution in an $H^1$-conforming finite element space. With the discretizations of both curl and div operators of the underlying eigenvalue problem in two finite element spaces, the proposed method is essentially a standard $H^1$-conforming element method, up to element bubbles which can be statically eliminated at element levels. We first analyze the proposed method for the related source interface problem by establishing the stability and the error bounds. We then analyze the underlying eigenvalue interface problem, and we obtain the error bounds $\mathcal{O}(h^{2r_0})$ for eigenvalues which correspond to eigenfunctions in $\prod_{j=1}^J (H^r(\Omega_j))^3\hookrightarrow (H^{r_0}(\Omega))^3$ space, where the piecewise regularity $r$ and the global regularity $r_0$ may belong to the most interesting interval $[0,1]$.

Over the past few decades, much attention has been drawn to large-scale incremental data analysis, where researchers are faced with huge amounts of high-dimensional data acquired incrementally. In such a case, conventional algorithms that compute the result from scratch whenever a new sample comes are highly inefficient. To conquer this problem, we propose a new incremental algorithm incremental regularized least squares (IRLS) that incrementally computes the solution to the regularized least squares (RLS) problem with multiple columns on the right-hand side. More specifically, for an RLS problem with $c$ ($c>1$) columns on the right-hand side, we update its unique solution by solving an RLS problem with a single column on the right-hand side whenever a new sample arrives, instead of solving an RLS problem with $c$ columns on the right-hand side from scratch. As a direct application of IRLS, we consider the supervised dimensionality reduction of large-scale data and focus on linear discriminant analysis (...

In this paper we study the existence of analytic eigenvalue functions of an analytic matrix-valued function $L(\lambda,\rho)$. Instead of proposing sufficient conditions for each individual case as in the literature, we propose a systematic scheme to discuss the existence of analytic eigenvalue functions of $L(\lambda,\rho)$ when $\lambda_0$ is a semisimple eigenvalue of $L(\lambda,0)$. We show that $\lambda(\rho)=\lambda_0+\rho\mu(\rho)$ is an eigenvalue of $L(\lambda,\rho)$ if and only if $\mu(\rho)$ is an eigenvalue of another analytic matrix-valued function $P(\mu,\rho)$ which is constructed based on the first order (partial) derivatives of $L(\lambda,\rho)$ at $(\lambda_0,0)$. Based on this result, a systematic scheme is proposed to check whether there exist analytic eigenvalue functions of $L(\lambda,\rho)$. This systematic scheme covers existing sufficient conditions in the literature, and can lead to much more general conditions.

This paper develops an adaptive edge finite element method for a Maxwell interface problem: $$ {\mathbf{curl}}\mu ^{-1} {\mathbf{curl}}u_\delta +\delta \varepsilon u_\delta =f, $$curlμ-1curlu?+??u?=f, where $$\delta >0$$?>0 is allowed to degenerate to zero. A residual-based a posteriori error estimator is analyzed, with $$\delta $$?-uniform lower and (global and local) upper error bounds. $$\delta $$?-uniform convergence and optimality of the adaptive algorithm are also established. Numerical results are also presented.

In this paper, we study the analyticity of the semisimple eigenvalue and the corresponding eigenvector functions of general analytic matrix-valued functions which are very important in many engineering applications such as the optimum design of dynamical structures, model updating, damage detecting, quantum mechanics, diffraction grating theory, medical imaging, and social network theory. We establish some new sufficient conditions on existence of analytic eigenvalue and eigenvector functions corresponding to semisimple eigenvalues of general analytic matrix-valued functions, which relaxes the condition in [P. Lancaster, A. S. Markus, and F. Zhou, SIAM J. Matrix Anal. Appl., 25 (2003), pp. 606--626]. We also present a numerical method to compute the derivatives of the eigenvalue and eigenvector functions. Numerical performance of the method is illustrated by some numerical examples.

In this paper, taking the double obstacle as the control variable, we consider an optimal control problem for a quasilinear elliptic variational inequality with source term. We establish the existence theorem and derive the optimality system for the underlying problem.

In this paper, we propose and analyze a nodal-continuous and H 1 -conforming finite element method for the numerical computation of Maxwell's equations, with singular solution in a fractional order Sobolev space H r ( Ω ) , where r may take any value in the most interesting interval ( 0 , 1 ) . The key feature of the method is that mass-lumping linear finite element L 2 projections act on the curl and divergence partial differential operators so that the singular solution can be sought in a setting of L 2 ( Ω ) space. We shall use the nodal-continuous linear finite elements, enriched with one element bubble in each element, to approximate the singular and non- H 1 solution. Discontinuous and nonhomogeneous media are allowed in the method. Some error estimates are given and a number of numerical experiments for source problems as well as eigenvalue problems are presented to illustrate the superior performance of the proposed method.

In this paper, we propose and analyze a novel stabilized finite element method (FEM) for the system of generalized Stokes equations arising from the time-discretization of transient Stokes problem. The system involves a small viscosity, which is proportional to the inverse of large Reynolds number, and a large reaction coefficient, which is the inverse of small time step. The proposed stabilized FEM employs the C 0 piecewise linear elements for both velocity field and pressure on the same mesh and uses the residuals of the momentum equation and the divergence-free equation to define the stabilization terms. The stabilization parameters are fixed and element-independent, without a comparison of the viscosity, the reaction coefficient and the mesh size. Using the finite element solution of an auxiliary boundary value problem as the interpolating function for velocity and the H 1 -seminorm projection for pressure, instead of the usual nodal interpolants, we derive error estimates for the stabilized finite element approximations to velocity and pressure in the L 2 and H 1 norms and most importantly, we explicitly establish the dependence of error bounds on the viscosity, the reaction coefficient and the mesh size. Our analysis reveals that this stabilized FEM is particularly suitable for the generalized Stokes system with a small viscosity and a large reaction coefficient, which has never been achieved before in the error analysis of other stabilization methods in the literature. We then numerically confirm the effectiveness of the proposed stabilized FEM. Comparisons made with other existing stabilization methods show that the newly proposed method can attain better accuracy and stability.

Some inverse eigenvalue problems for Hermitian and generalized Hamiltonian/skew-Hamiltonian matrices are discussed, including best approximation problems and eigenvalue updating problems. Based on the spectral decomposition theory on Hermitian and Hamiltonian/skew-Hamiltonian matrices, simpler parametric expressions of solutions to the general inverse eigenvalue problems are given, and then the best approximation problems and eigenvalue updating problems are solved by using these parametric expressions.

We consider quadratic eigenvalue problems in which the coefficient matrices, and hence the eigenvalues and eigenvectors, are functions of a real parameter. Our interest is in cases in which these functions remain differentiable when eigenvalues coincide. Many papers have been devoted to numerical methods for computing derivatives of eigenvalues and eigenvectors, but most require the eigenvalues to be well separated. The few that consider close or repeated eigenvalues place severe restrictions on the eigenvalue derivatives. We propose, analyze, and test new algorithms for computing first and higher order derivatives of eigenvalues and eigenvectors that are valid much more generally. Numerical results confirm the effectiveness of our methods for tightly clustered eigenvalues.

In this paper, a theoretical framework with a completely new theory is presented for the general curlcurl-graddiv second-order elliptic eigenproblem when discretized by the recently developed local $L^2$ projected $C^0$ finite element method. The theoretical framework consists of two Fortin-type interpolations and an Inf-Sup inequality associated with a trilinear curl/div form. From this framework, error estimates are readily derived for the source problem as well as the eigenproblem. The new theory is verified for the local $L^2$ projected $C^0$ finite element method with elementwise linear element $L^2$ projections applied to div and curl operators and the $C^0$ linear elements enriched with some face-bubbles and element-bubbles for the singular solution. The challenging question is whether $C^0$ elements can give spectrally correct approximations when eigenfunctions are singular (not in $(H^1(\Omega))^d$ space). It is shown that optimal error bounds can be obtained from this theoretical framework with ...

In this paper, we propose and analyze a new stabilized finite element method using continuous piecewise linear (or bilinear) elements for solving 2D reaction–convection–diffusion equations. The equation under consideration involves a small diffusivity e and a large reaction coefficient σ , leading to high Peclet number and high Damkohler number. In addition to giving error estimates of the approximations in L 2 and H 1 norms, we explicitly establish the dependence of error bounds on the diffusivity, the L ∞ norm of convection field, the reaction coefficient and the mesh size. Our analysis shows that the proposed method is particularly suitable for problems with a small diffusivity and a large reaction coefficient, or more precisely, with a large mesh Peclet number and a large mesh Damkohler number. Several numerical examples exhibiting boundary or interior layers are given to illustrate the high accuracy and stability of the proposed method. The results obtained are also compared with those of existing stabilization methods.

In this paper, we develop a continuous finite element method for the curlcurl-grad div vector second-order elliptic problem in a three-dimensional polyhedral domain occupied by discontinuous nonhomogeneous anisotropic materials. In spite of the fact that the curlcurl-grad div interface problem is closely related to the elliptic interface problem of vector Laplace operator type, the continuous finite element discretization of the standard variational problem of the former generally fails to give a correct solution even in the case of homogeneous media whenever the physical domain has reentrant corners and edges. To discretize the curlcurl-grad div interface problem by the continuous finite element method, we apply an element-local $$L^2$$ projector to the curl operator and a pseudo-local $$L^2$$ projector to the div operator, where the continuous Lagrange linear element enriched by suitable element and face bubbles may be employed. It is shown that the finite element problem retains the same coercivity property as the continuous problem. An error estimate $$\mathcal{O }(h^r)$$ in an energy norm is obtained between the analytical solution and the continuous finite element solution, where the analytical solution is in $$\prod _{l=1}^L (H^r(\Omega _l))^3$$ for some $$r\in (1/2,1]$$ due to the domain boundary reentrant corners and edges (e.g., nonconvex polyhedron) and due to the interfaces between the different material domains in $$\Omega =\bigcup _{l=1}^L \Omega _l$$ .

In this paper we develop the C 0 finite element method for a generalized curlcurl-grad div indefinite Maxwell problem in a Lipschitz domain such as nonconvex polygon for which the solution of the problem may be nonsmooth and only have the H r regularity for some r

To deal with the divergence-free constraint in a double curl problem, ${\rm curl\,} \mu^{-1} {\rm curl\,} u=f$ and ${\rm div\,} \varepsilon u=0$ in $\Omega$, where $\mu$ and $\varepsilon$ represent the physical properties of the materials occupying $\Omega$, we develop a $\delta$-regularization method, ${\rm curl\,} \mu^{-1} {\rm curl\,} u_\delta +\delta \varepsilon u_\delta=f$, to completely ignore the divergence-free constraint ${\rm div\,} \varepsilon u=0$. We show that $u_\delta$ converges to $u$ in $H({\rm curl\,};\Omega)$ norm as $\delta\rightarrow 0$. The edge finite element method is then analyzed for solving $u_\delta$. With the finite element solution $u_{\delta,h}$, a quasioptimal error bound in the $H({\rm curl\,};\Omega)$ norm is obtained between $u$ and $u_{\delta,h}$, including a uniform (with respect to $\delta$) stability of $u_{\delta,h}$ in the $H({\rm curl\,};\Omega)$ norm. All the theoretical analysis is done in a general setting, where $\mu$ and $\varepsilon$ may be discontinuous, an...

This paper is devoted to the state-constrained optimal control problem of evolutionary variational inequality. In this paper, the control domain is not necessarily convex. Moreover, since our state constraint is quite general and, in many cases, it requires pointwise behavior of the state, the framework of the partial differential equation (instead of the abstract framework) is used. Some optimality conditions (in the form of Pontryagin’s principle) for optimal controls are established.

The purpose of this paper is to propose a proof for the Poincare-Friedrichs inequality for piecewise H 1 functions on anisotropic meshes. By verifying suitable assumptions involved in the newly proposed proof, we show that the Poincare-Friedrichs inequality for piecewise H 1 functions holds independently of the aspect ratio which characterizes the shape-regular condition in finite element analysis. In addition, under the maximum angle condition, we establish the Poincare-Friedrichs inequality for the Crouzeix-Raviart non-conforming linear finite element. Counterexamples show that the maximum angle condition is only sufficient.

SUMMARY One of the most successful methods for solving the least-squares problem minxAx−b� 2 with a highly ill-conditioned or rank deficient coefficient matrix A is the method of Tikhonov regularization. In this paper, we derive the normwise, mixed and componentwise condition numbers and componentwise perturbation bounds for the Tikhonov regularization. Our results are sharper than the known results. Some numerical examples are given to illustrate our results. Copyright q 2010 John Wiley & Sons, Ltd.

A new least-squares finite element method is analyzed for the thin plate problem subject to various boundary conditions (clamped, simply supported and free). The unknown variables are deflection, slope, moment and shear force. The coercivity property is established. As a result, all variables can be approximated by any conforming finite elements. In particular, an H^1-ellipticity is proven for the free thin plate. This indicates that optimal error bounds hold for all variables with the use of equal-order continuous elements. Numerical experiments are performed to confirm the theoretical results obtained.

The state feedback decoupling problem with stability for general proper systems described by $(A,B,C,D)$ quadruples has been studied for a long time. But, it is still an open problem in the sense that there is still a lack of numerically verifiable solvability conditions and numerically implementable methods for solving it in the existing literature. In this paper numerically verifiable solvability conditions and a numerical method for solving this open problem are developed. The proposed method is based on orthogonal transformations and hence can be implemented in a numerically reliable manner.

In this paper we study an extended J-spectral factorization problem, i.e., J-spectral factorization problem for descriptor systems which may not have full column normal rank and may exhibit poles and zeros on the extended imaginary axis. We present necessary and sufficient solvability conditions and develop a numerically reliable method for the underlying problem. Our method is implemented by using only orthogonal transformations and has an acceptable computational complexity.

In this paper, algebraic characterizations for the positive realness of descriptor systems are studied. It is shown that the positive realness of descriptor systems can be determined by solving a linear matrix inequality, and hence the celebrated positive real lemma for standard state space systems is extended to descriptor systems. In addition, the lossless positive realness of both standard state space systems and descriptor systems is characterized explicitly based on the controllable staircase forms of standard state space systems and the generalized controllable staircase forms of descriptor systems, respectively.

In this paper, we introduce a new adaptive method for computing the numerical solutions of a class of quenching parabolic equations which exhibit a solution with one singularity. Our method systematically generates an irregular mesh with mesh-dependent temporal increments based on the solution behavior from which an implicit finite difference scheme associated with the irregular mesh is constructed. The convergence and stability of the finite difference scheme is analyzed for the solution before quenching. An equivalent linearized model is used to justify the stability of the method near quenching as well. A numerical example is provided to demonstrate the viability of the proposed method.

A new least-squares finite element method is developed for the curl-div magnetostatic problem in Lipschitz and multiply connected domains filled with anisotropic nonhomogeneous materials. In order to deal with possibly low regularity of the solution, local $L^2$ projectors are introduced to standard least-squares formulation and applied to both curl and div operators. Coercivity is established by adding suitable mesh-dependent bilinear terms. As a result, any continuous finite elements (lower-order elements are enriched with suitable bubbles) can be employed. A desirable error bound is obtained: $||{\bf u}-{\bf u}_h||_0\le C\,||{\bf u}-\tilde{{\bf u}}||_0$, where ${\bf u}_h$ and $\tilde{{\bf u}}$ are the finite element approximation and the finite element interpolant of the exact solution ${\bf u}$, respectively. Numerical tests confirm the theoretical results.

More than a decade ago, Bramble, Pasciak and Xu developed a framework in analyzing the multigrid methods with nonnested spaces or non-inherited quadratic forms. It was subsequently known as the BPX multigrid framework, which was widely used in the analysis of multigrid and domain decomposition methods. However, the framework has an apparent limit in the analysis of nonnested V-cycle methods, and it produces a variable V-cycle, or nonuniform convergence rate V-cycle methods, or other nonoptimal results in analysis thus far. This paper completes a long-time effort in extending the BPX multigrid framework so that it truly covers the nonnested V-cycle. We will apply the extended BPX framework to the analysis of many V-cycle nonnested multigrid methods. Some of them were proven previously only for two-level and W-cycle iterations. Some numerical results are presented to support the theoretical analysis of this paper.

In this paper, the optimality system as well as the existence theorem for an obstacle optimal control problem are established, in which the governing system is a parabolic bilateral variational inequality and the input control is the pair of upper and lower obstacles.

In this paper a nearest doubly stochastic matrix problem is studied. This problem is to find the closest doubly stochastic matrix with the prescribed $(1,1)$ entry to a given matrix. According to the well-established dual theory in optimization, the dual of the underlying problem is an unconstrained differentiable, but not twice differentiable, convex optimization problem. A Newton-type method is used for solving the associated dual problem, and then the desired nearest doubly stochastic matrix is obtained. Under some mild assumptions, the quadratic convergence of the proposed Newton method is proved. The numerical performance of the method is also demonstrated by numerical examples.

Derivatives of eigenvalues and eigenvectors of parameter-dependent matrix eigenproblems play a key role in the optimum design of structures in engineering, and in the solution of inverse problems, such as the problem of model updating, which arises, for example, when information on the normal modes of vibration of a structure is used to detect structural damage. Both these applications often involve quadratic eigenvalue problems. Most existing methods for the computation of derivatives of quadratic eigenvalue problems are based on the assumption that repeated eigenvalues have well separated first order derivatives. In this paper we propose new algorithms for computing derivatives of eigenvalues and eigenvectors for quadratic eigenvalue problems under much more general conditions than existing methods, whose effectiveness for repeated or tightly clustered eigenvalues are confirmed by some numerical examples.

This paper studies the numerical solution of multi-dimensional nonlinear degenerate reaction-diffusion differential equations with a singular force term over a rectangular domain. The equations may generate strong quenching singularities. Our work focuses on a variable temporal step Peaceman-Rachford splitting method with an adaptive moving mesh in space. The temporal and spatial adaptation is implemented based on arc-length type of estimations of the time derivative of the solution since the time derivative of the solution approaches infinity when the quenching occurs. The multi-dimensional problem is split into a few one-dimensional problems and the splitting procedure can also be parallelized so that the computational time is significantly reduced. The physical monotonicity of the solution and stability of this variable step moving mesh scheme are analyzed for the time away from the quenching. As stability analysis may not be valid when it is very close to the quenching, thus an exact linear problem is introduced to justify the stability near the quenching time. Finally we provide some numerical examples to illustrate our results as well as to demonstrate the viability and efficiency of the method for the quenching problem or other problems with point singularities. We will also show the significant reduction in computational time required with parallel implementation of the algorithm on a computer with multi-CPU.

In this paper we develop a numerical method for computing the semistabilizing solution of a generalized algebraic Riccati equation (GARE). The semistabilizing solution of such a GARE has been used to characterize the solvability of the $(J, J^\prime)$-spectral factorization problem for general rational matrices which have poles and zeros on the extended imaginary axis. The main difficulty for solving such a GARE is that its associated skew-Hamiltonian/Hamiltonian pencil has eigenvalues on the extended imaginary axis; consequently, it is not clear which eigenspace of the associated skew-Hamiltonian/Hamiltonian pencil can characterize the desired semistabilizing solution; i.e., it is not clear which eigenvectors and principal vectors corresponding to the eigenvalues on the extended imaginary axis should be contained in the eigenspace that we wish to compute, and hence the well-known generalized eigenvalue approach for the classical algebraic Riccati equations cannot be directly employed for it. Our proposed method consists of computations of the eigendecomposition of the system pencil corresponding to the eigenvalues on the extended imaginary axis and the stable eigenspace of an augmented matrix pencil; hence, it is a generalization of the generalized eigenvalue approach for the classical algebraic Riccati equations.

Two new L2 least-squares (LS) finite element methods are developed for the velocity-pressure-vorticity first-order system of the Stokes problem with Dirichlet velocity boundary condition. A key feature of these new methods is that a local or almost local L2 projector is applied to the residual of the momentum equation. Such L2 projection is always defined onto the linear finite element space, no matter which finite element spaces are used for velocity-pressure-vorticity variables. Consequently, the implementation of this L2-projected LS method is almost as easy as that of the standard L2 LS method. More importantly, the former has optimal error estimates in L2-norm, with respect to both the order of approximation and the required regularity of the exact solution for velocity using equal-order interpolations and for all three variables (velocity, pressure, and vorticity) using unequal-order interpolations. Numerical experiments are given to demonstrate the theoretical results.

This paper is concerned with an optimal control problem for quasi-linear elliptic variational inequality in which the bilateral obstacles are the control. The cost functional of this optimal control problem is of Lagrange type in which the pth power of Laplacian of the control appears. This feature leads to the fact that it is hard to derive the optimality system for the underlying problem. In this paper, the optimality system is established by utilizing the special structure of the approximate optimality system including the monotonicity of the leading differential operator.

This paper studies the numerical solution of two-dimensional nonlinear degenerate reaction-diffusion differential equations with singular forcing terms over rectangular domains. The equations considered may generate strong quenching singularities. This investigation focuses on a variable time step Peaceman--Rachford splitting method for the aforementioned problem. The time adaptation is implemented based on arc-length estimations of the first time derivative of the solution. The two-dimensional problem is split into several one-dimensional problems so that the computational cost is significantly reduced. The monotonicity and localized linear stability of the variable step scheme are investigated. We give some numerical examples to illustrate our results as well as to demonstrate the viability and efficiency of the method over existing methods for the quenching problem. It is also shown that the numerical solution obtained preserves important properties of the physical solution of the given problem.

In this paper, we develop a new numerical method for a special coordinate basis of a linear time invariant system. Such a special coordinate basis is essentially a structural decomposition which explicitly displays the finite and infinite zero structures, as well as the invertibility structures of the given system. The technique is playing important roles in numerous topics in system and control theory, such as robust control, H/sub /spl infin// and H/sub 2/ optimal control almost disturbance decoupling, and zero placement of linear systems, just to name a few. Our method consists of three steps: reduction by orthogonal transformations, reduction by generalized Sylvester equations, and extraction of infinite zero structure. The performance of our method is illustrated by some numerical examples.

This is the first of two papers on the row by row decoupling problem and the triangular decoupling problem. In this paper we study the row by row decoupling problem with stability in control theory. We first prove a nice reduction property for the row by row decoupling problem with stability and then develop a numerically reliable method for solving it. The basis of our main results is some condensed forms under orthogonal transformations, which can be implemented in numerically stable ways. Hence our results lead to numerically reliable methods for solving the studied problem using existing numerical linear algebra software such as MATLAB. In the sequel [SIAM J. Matrix Anal. Appl., 23 (2002), pp. 1171--1182], we will consider a related problem---the triangular decoupling problem---and parameterize all solutions for it.

In general, the solution of a Maxwell interface problem would not belong to H 1 space and the standard C 0 finite element method fails. With the help of local L 2 projections applied to both the curl and div operators, we propose a new C 0 finite element method which can correctly converge to the non H 1 space solution. Stability and error estimates are given.

This paper is concerned with iterative methods for computing partial derivatives of eigenvalues and eigenvectors of matrix-valued functions of several real variables. First, an analysis is given of a previously announced method which computes mixed partial derivatives of simple eigenvalues and the corresponding eigenvectors and also second order mixed partial derivatives of repeated eigenvalues. Next a new method for computing third order partial derivatives of repeated eigenvalues and second order derivatives of the corresponding eigenvectors is presented and its key properties are established. Efficiency and numerical stability are considered as well as theoretical convergence.

This paper presents and analyzes new algorithms for computing the numerical values of derivatives, of arbitrary order, and of eigenvalues and eigenvectors of ${\bf A}(\rho){\bf x}(\rho) = \lambda(\rho){\bf B}(\rho){\bf x}(\rho)$ at a point $\rho=\rho_0$ at which the eigenvalues considered are multiple. Here ${\bf A}(\rho)$ and ${\bf B}(\rho)$ are hermitian matrices which depend analytically on a single real variable $\rho,$ and ${\bf B}(\rho_0)$ is positive definite. The algorithms are valid under more general conditions than previous algorithms. Numerical results support the theoretical analysis and show that the algorithms are also useful when eigenvalues are merely very close rather than coincident.

An iterative method is introduced for computing second-order partial derivatives (sensitivities) of eigenvalues and eigenvectors of matrices which depend on a number of real design parameters. Numerical tests confirm the viability of the method and support our theoretical analysis. Alternative methods are reviewed briefly and compared with the one proposed here.

Let[Figure not available: see fulltext.] be thek-dimensional subspace spanned by the translates ?(·?2?j/k),j=0, 1, ...,k?1, of a continuous, piecewise smooth, complexvalued, 2?-periodic function ?. For a given functionf?L 2(??, ?), its least squares approximantS kf from[Figure not available: see fulltext.] can be expressed in terms of an orthonormal basis. Iff is continuous,S kf can be computed via its discrete analogue by fast Fourier transform. The discrete least squares approximant is used to approximate Fourier coefficients, and this complements the works of Gautschi on attenuation factors. Examples of[Figure not available: see fulltext.] include the space of trigonometric polynomials where ? is the de la Vallee Poussin kernel, algebraic polynomial splines where ? is the periodic B-spline, and trigonometric polynomial splines where ? is the trigonometric B-spline.

A gap is closed in a previously published proof of a result concerning direct computation of derivatives of eigenvalues and eigenvectors of parameter-dependent matrices. The result is strengthened and the problem discussed.

An earlier simultaneous iteration method for computing derivatives of eigenvalues and the corresponding eigenvectors of nonsymmetric matrices depending smoothly on some real parameters is extended to the case of repeated eigenvalues

This paper presents an algorithm for solving a class of large scale nonlinear programming which is originally derived from the multi-stage stochastic convex nonlinear programming. With the Lagrangian-dual method and the Moreau-Yosida regularization, the primal problem is transformed into a smooth convex problem. By introducing a self-concordant barrier function, an approximate generalized Newton method is designed in this paper. The algorithm is shown to be of superlinear convergence. At last, some preliminary numerical results are provided.