Convergent improvement of SSOR multisplitting method for an H-matrix

Recently, Yun [Jae Heon Yun, Convergence of SSOR multisplitting method for an H$H$-matrix, J. Comput. Appl. Math. 217 (2008) 252–258] studied the convergence of the relaxed multisplitting method associated with SSOR multisplitting for solving a linear system whose coefficient matrix is an H$H$-matrix. In this paper, we improve the main results of Yun’s. Moreover, theoretical analysis and numerical examples clearly show that our new convergent domain is wider.     

Convergence of block iterative methods for linear systems with generalized H-matrices

The paper studies the convergence of some block iterative methods for the solution of linear systems when the coefficient matrices are generalized H$H$-matrices. A truth is found that the class of conjugate generalized H$H$-matrices is a subclass of the class of generalized H$H$-matrices and the convergence results of R. Nabben [R. Nabben, On a class of matrices which arises in the numerical solution of Euler equations, Numer. Math. 63 (1992) 411–431] are then extended to the class of generalized H$H$-matrices. Furthermore, the convergence of the block AOR iterative method for linear systems with generalized H$H$-matrices is established and some properties of special block tridiagonal matrices arising in the numerical solution of Euler equations are discussed. Finally, some examples are given to demonstrate the convergence results obtained in this paper.     

Time discretization and quantization methods for optimal multiple switching problem

In this paper, we study probabilistic numerical methods based on optimal quantization algorithms for computing the solution to optimal multiple switching problems with regime-dependent state process. We first consider a discrete-time approximation of the optimal switching problem, and analyse its rate of convergence. Given a time step h$h$, the error is in general of order (hlog(1/h))^1/2${(hlog(1/h))}^{1/2}$, and of order h^1/2$h^{1/2}$ when the switching costs do not depend on the state process. We next propose quantization numerical schemes for the space discretization of the discrete-time Euler state process. A Markovian quantization approach relying on the optimal quantization of the normal distribution arising in the Euler scheme is analysed. In the particular case of uncontrolled state process, we describe an alternative marginal quantization method, which extends the recursive algorithm for optimal stopping problems as in Bally (2003) [1]. A priori L^p$L^p$-error estimates are stated in terms of quantization errors. Finally, some numerical tests are performed for an optimal switching problem with two regimes.     

Eigenvalue patterned condition numbers: Toeplitz and Hankel cases

We continue the study started in [Noschese and Pasquini, Eigenvalue condition numbers: zero-structured versus traditional. J. Comput. Appl. Math. 185 (2006) 174–189] concerning the sensitivity of simple eigenvalues of a matrix A to perturbations in A   that belong to a chosen subspace of matrices. In [Noschese and Pasquini, Eigenvalue condition numbers: zero-structured versus traditional. J. Comput. Appl. Math. 185 (2006) 174–189] the zero-structured perturbations have been considered. Here we focus on patterned perturbations, and the cases of the Toeplitz and of the Hankel matrices are investigated in detail. Useful expressions of the absolute patterned condition number of the eigenvalue λ$\lambda$ and of the analogue of the matrix yx^H${\mathit{yx}}^{\mathrm{H}}$, which leads to the traditional condition number of λ$\lambda$, are given. MATLAB codes are defined to compare traditional, zero-structured and patterned condition numbers. A report on significant numerical tests is included.     

On topologizable and non-topologizable groups

A group G   is called hereditarily non-topologizable if, for every H?G$H?G$, no quotient of H admits a non-discrete Hausdorff topology. We construct first examples of infinite hereditarily non-topologizable groups. This allows us to prove that c-compactness does not imply compactness for topological groups. We also answer several other open questions about c-compact groups asked by Dikranjan and Uspenskij. On the other hand, we suggest a method of constructing topologizable groups based on generic properties in the space of marked k-generated groups. As an application, we show that there exist non-discrete quasi-cyclic groups of finite exponent; this answers a question of Morris and Obraztsov.     

Analysis and application of the IIPG method to quasilinear nonstationary convection–diffusion problems

We develop a numerical method for the solution of convection–diffusion problems with a nonlinear convection and a quasilinear diffusion. We employ the so-called incomplete interior penalty Galerkin (IIPG) method which is suitable for a discretization of quasilinear diffusive terms. We analyse a use of the IIPG technique for a model scalar time-dependent convection–diffusion equation and derive hp$hp$ a priori error estimates in the L²$L^2$-norm and the H¹$H^1$-seminorm. Moreover, a set of numerical examples verifying the theoretical results is performed. Finally, we present a preliminary application of the IIPG method to the system of the compressible Navier–Stokes equations.     

Quadratically constrained least squares and quadratic problems

Summary We consider the following problem: Compute a vectorx such that Ax–b₂=min, subject to the constraint x₂=. A new approach to this problem based on Gauss quadrature is given. The method is especially well suited when the dimensions ofA are large and the matrix is sparse.It is also possible to extend this technique to a constrained quadratic form: For a symmetric matrixA we consider the minimization ofx ^T A x–2b ^T x subject to the constraint x₂=.Some numerical examples are given.This work was in part supported by the National Science Foundation under Grant DCR-8412314 and by the National Institute of Standards and Technology under Grant 60NANB9D0908.     

Trigonometric Hermite wavelet approximation for the integral equations of second kind with weakly singular kernel

This paper is concerned with a trigonometric Hermite wavelet Galerkin method for the Fredholm integral equations with weakly singular kernel. The kernel function of this integral equation considered here includes two parts, a weakly singular kernel part and a smooth kernel part. The approximation estimates for the weakly singular kernel function and the smooth part based on the trigonometric Hermite wavelet constructed by E. Quak [Trigonometric wavelets for Hermite interpolation, Math. Comp. 65 (1996) 683–722] are developed. The use of trigonometric Hermite interpolant wavelets for the discretization leads to a circulant block diagonal symmetrical system matrix. It is shown that we only need to compute and store O(N)$\mathrm{O}(N)$ entries for the weakly singular kernel representation matrix with dimensions N²$N^2$ which can reduce the whole computational cost and storage expense. The computational schemes of the resulting matrix elements are provided for the weakly singular kernel function. Furthermore, the convergence analysis is developed for the trigonometric wavelet method in this paper.     

A sharp version of Bauer–Fike’s theorem

In this paper, we present a sharp version of Bauer–Fike’s theorem. We replace the matrix norm with its spectral radius or sign-complex spectral radius for diagonalizable matrices; 1-norm and ∞$\infty$-norm for non-diagonalizable matrices. We also give the applications to the pole placement problem and the singular system.     

Solution of augmented linear systems using orthogonal factorizations

We consider solvingx+Ay=b andA ^T x=c for givenb, c andm ×n A of rankn. This is called the augmented system formulation (ASF) of two standard optimization problems, which include as special cases the minimum 2-norm of a linear underdetermined system (b=0) and the linear least squares problem (c=0), as well as more general problems. We examine the numerical stability of methods (for the ASF) based on the QR factorization ofA, whether by Householder transformations, Givens rotations, or the modified Gram-Schmidt (MGS) algorithm, and consider methods which useQ andR, or onlyR. We discuss the meaning of stability of algorithms for the ASF in terms of stability of algorithms for the underlying optimization problems.We prove the backward stability of several methods for the ASF which useQ andR, inclusing a new one based on MGS, and also show under what circumstances they may be regarded as strongly stable. We show why previous methods usingQ from MGS were not backward stable, but illustrate that some of these methods may be acceptable-error stable. We point out that the numerical accuracy of methods that do not useQ does not depend to any significant extent on which of of the above three QR factorizations is used. We then show that the standard methods which do not useQ are not backward stable or even acceptable-error stable for the general ASF problem, and discuss how iterative refinement can be used to counteract these deficiencies.Dedicated to Carl-Eric Fröberg on the occasion of his 75th birthdayThis research was partially supported by NSERC of Canada Grant No. A9236.     

An implicit shift bidiagonalization algorithm for ill-posed systems

Iterative methods based on Lanczos bidiagonalization with full reorthogonalization (LBDR) are considered for solving large-scale discrete ill-posed linear least-squares problems of the form min _x Ax–b₂. Methods for regularization in the Krylov subspaces are discussed which use generalized cross validation (GCV) for determining the regularization parameter. These methods have the advantage that no a priori information about the noise level is required. To improve convergence of the Lanczos process we apply a variant of the implicitly restarted Lanczos algorithm by Sorensen using zero shifts. Although this restarted method simply corresponds to using LBDR with a starting vector (AA ^T) ^p b, it is shown that carrying out the process implicitly is essential for numerical stability. An LBDR algorithm is presented which incorporates implicit restarts to ensure that the global minimum of the CGV curve corresponds to a minimum on the curve for the truncated SVD solution. Numerical results are given comparing the performance of this algorithm with non-restarted LBDR.This work was partially supported by DARPA under grant 60NANB2D1272 and by the National Science Foundation under grant CCR-9209349.     

A structured rank-revealing method for Sylvester matrix

We propose a fast algorithm for computing the numeric ranks of Sylvester matrices. Let S denote the Sylvester matrix and H   denote the Hankel-like-Sylvester matrix. The algorithm is based on a fast Cholesky factorization of S^TS$S^{\mathrm{T}}S$ or H^TH$H^{\mathrm{T}}H$ and relies on a stabilized version of the generalized Schur algorithm for matrices with displacement structure. All computations can be done in O(r(n+m))$\mathrm{O}(r(n+m))$, where n+m$n+m$ and r denote the size and the numerical rank of the Sylvester matrix, respectively.     

The symmetric procrustes problem

The following symmetric Procrustes problem arises in the determination of the strain matrix of an elastic structure: find the symmetric matrixX which minimises the Frobenius (or Euclidean) norm ofAX — B, whereA andB are given rectangular matrices. We use the singular value decomposition to analyse the problem and to derive a stable method for its solution. A perturbation result is derived and used to assess the stability of methods based on solving normal equations. Some comparisons with the standard, unconstrained least squares problem are given.     

Block tridiagonal reduction of perturbed normal and rank structured matrices

It is well known that if a matrix A∈C^n×n$A\in {\mathbb{C}}^{n\times n}$ solves the matrix equation f(A,A^H)=0$f(A,A^H)=0$, where f(x,y)$f(x,y)$ is a linear bivariate polynomial, then A is normal; A   and A^H$A^H$ can be simultaneously reduced in a finite number of operations to tridiagonal form by a unitary congruence and, moreover, the spectrum of A is located on a straight line in the complex plane. In this paper we present some generalizations of these properties for almost normal matrices which satisfy certain quadratic matrix equations arising in the study of structured eigenvalue problems for perturbed Hermitian and unitary matrices.     

Stability of IMEX Runge–Kutta methods for delay differential equations

Stability of IMEX (implicit–explicit) Runge–Kutta methods applied to delay differential equations (DDEs) is studied on the basis of the scalar test equation du/dt=λu(t)+μu(t-τ)$\mathrm{d}u/\mathrm{d}t=\lambda u(t)+\mu u(t-\tau )$, where τ$\tau$ is a constant delay and λ,μ$\lambda ,\mu$ are complex parameters. More specifically, P-stability regions of the methods are defined and analyzed in the same way as in the case of the standard Runge–Kutta methods. A new IMEX method which possesses a superior stability property for DDEs is proposed. Some numerical examples which confirm the results of our analysis are presented.     

A fourth-order Runge–Kutta method based on BDF-type Chebyshev approximations

In this paper we consider a new fourth-order method of BDF-type for solving stiff initial-value problems, based on the interval approximation of the true solution by truncated Chebyshev series. It is shown that the method may be formulated in an equivalent way as a Runge–Kutta method having stage order four. The method thus obtained have good properties relatives to stability including an unbounded stability domain and large α$\alpha$-value concerning A(α)$A(\alpha )$-stability. A strategy for changing the step size, based on a pair of methods in a similar way to the embedding pair in the Runge–Kutta schemes, is presented. The numerical examples reveals that this method is very promising when it is used for solving stiff initial-value problems.     

Asymptotic resurgences for ideals of positive dimensional subschemes of projective space

Recent work of Ein–Lazarsfeld–Smith and Hochster–Huneke raised the containment problem of determining which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bocci–Harbourne defined a quantity called the resurgence to address this problem for homogeneous ideals in polynomial rings, with a focus on zero-dimensional subschemes of projective space. Here we take the first steps toward extending this work to higher dimensional subschemes. We introduce new asymptotic versions of the resurgence and obtain upper and lower bounds on them for ideals I$I$ of smooth subschemes, generalizing what is done in Bocci and Harbourne (2010)  [5]. We apply these bounds to ideals of unions of general lines in P^N${\mathbb{P}}^N$. We also pose a Nagata type conjecture for symbolic powers of ideals of lines in P³${\mathbb{P}}^3$.     

Analysis of a splitting method for incompressible inviscid rotational flow problems

This paper is devoted to analyze a splitting method for solving incompressible inviscid rotational flows. The problem is first recast into the velocity–vorticity–pressure formulation by introducing the additional vorticity variable, and then split into three consecutive subsystems. For each subsystem, the L²$L^2$ least-squares finite element approach is applied to attain accurate numerical solutions. We show that for each time step this splitting least-squares approach exhibits an optimal rate of convergence in the H¹$H^1$ norm for velocity and pressure, and a suboptimal rate in the L²$L^2$ norm for vorticity. A numerical example in two dimensions is presented, which confirms the theoretical error estimates.