A factorization method for products of holomorphic matrix functions

A class of matrix functions defined on a contour which bounds a finitely connected domain in the complex plane is considered. It is assumed that each matrix function in this class can be explicitly represented as a product of two matrix functions holomorphic in the outer and the inner part of the contour, respectively. The problem of factoring matrix functions in the class under consideration is studied. A constructive method reducing the factorization problem to finitely many explicitly written systems of linear algebraic equations is proposed. In particular, explicit formulas for partial indices are obtained.     

Approximation of rank function and its application to the nearest low-rank correlation matrix

The rank function rank(.) is neither continuous nor convex which brings much difficulty to the solution of rank minimization problems. In this paper, we provide a unified framework to construct the approximation functions of rank(.), and study their favorable properties. Particularly, with two families of approximation functions, we propose a convex relaxation method for the rank minimization problems with positive semidefinite cone constraints, and illustrate its application by computing the nearest low-rank correlation matrix. Numerical results indicate that this convex relaxation method is comparable with the sequential semismooth Newton method (Li and Qi in SIAM J Optim 21:1641–1666, 2011) and the majorized penalty approach (Gao and Sun, 2010) in terms of the quality of solutions.     

Basis problems for matrix valued functions

Basis problems for self-adjoint matrix valued functions are studied. We suggest a new and nonstandard method to solve basis problems both in finite and infinite dimensional spaces. Although many results in this paper are given for operator functions in infinite dimensional Hilbert spaces, but to demonstrate practicability of this method and to present a full solution of basis problems, in this paper we often restrict ourselves to matrix valued functions which generate Rayleigh systems on the n-dimensional complex space Cn. The suggested method is an improvement of an approach given recently in our paper [M. Hasanov, A class of nonlinear equations in Hilbert space and its applications to completeness problems, J. Math. Anal. Appl. 328 (2007) 1487-1494], which is based on the extension of the resolvent of a self-adjoint operator function to isolated eigenvalues and the properties of quadratic forms of the extended resolvent. This approach is especially useful for nonanalytic and nonsmooth operator functions when a suitable factorization formula fails to exist.     

Inverse spectral problem of fourth‐order boundary value problems with finite spectrum

In the present paper, we consider a class of inverse spectral problem of fourth‐order boundary value problems. Under the so‐called “Atkinson type” conditions, the problem has finite spectrum and corresponding matrix representations. By using the method of inverse matrix eigenvalue problems of two‐banded matrix, the leading coefficient and potential functions are reconstructed from the given three sets of interlacing real numbers satisfying certain conditions.     

A new class of test functions for global optimization

In this paper we propose a new class of test functions for unconstrained global optimization problems. The class depends on some parameters through which the difficulty of the test problems can be controlled. As a basis for future comparison, we propose a selected set of these functions, with increasing difficulty, and some computational experiments with two simple global optimization algorithms.     

Iterative adaptive RBF methods for detection of edges in two-dimensional functions

Radial basis functions have gained popularity for many applications including numerical solution of partial differential equations, image processing, and machine learning. For these applications it is useful to have an algorithm which detects edges or sharp gradients and is based on the underlying basis functions. In our previous research, we proposed an iterative adaptive multiquadric radial basis function method for the detection of local jump discontinuities in one-dimensional problems. The iterative edge detection method is based on the observation that the absolute values of the expansion coefficients of multiquadric radial basis function approximation grow exponentially in the presence of a local jump discontinuity with fixed shape parameters but grow only linearly with vanishing shape parameters. The different growth rate allows us to accurately detect edges in the radial basis function approximation. In this work, we extend the one-dimensional iterative edge detection method to two-dimensional problems. We consider two approaches: the dimension-by-dimension technique and the global extension approach. In both cases, we use a rescaling method to avoid ill-conditioning of the interpolation matrix. The global extension approach is less efficient than the dimension-by-dimension approach, but is applicable to truly scattered two-dimensional points, whereas the dimension-by-dimension approach requires tensor product grids. Numerical examples using both approaches demonstrate that the two-dimensional iterative adaptive radial basis function method yields accurate results.     

On explicit factorization and applications

This paper is devoted to two topics connected with factorization of triangular 2 by 2 matrix functions. The first application is concerned with explicit factorization of a class of matrices of Daniel-Khrapkov type and the second is related to inversion of finite Toeplitz matrices. In the first section we present the scheme of factorization of triangular 2 by 2 matrix functions.     

New second derivative multistep methods for stiff systems

In this paper we present details of a new class of second derivative multistep methods. Stability analysis is discussed and an improvement in stability region is obtained. With a simple modification we take advantage of calling for the solution of algebraic equations with the same coefficient matrix in each step. Moreover, using IOM to solve the resulting linear systems, the coefficient matrix is not needed explicitly. Some numerical experiments and comparison with several popular codes are given, showing strong superiority of this new class of methods.     

Interpolation problems,extensions of symmetric operators and reproducing kernel spaces II

The aim of part I and this paper is to study interpolation problems for pairs of matrix functions of the extended Nevanlinna class using two different approaches and to make explicit the various links between them. In part I we considered the approach via the Kreîn-Langer theory of extensions of symmetric operators. In this paper we adapt Dym's method to solve interpolation problems by means of the de Branges theory of Hilbert spaces of analytic functions. We also show here how the two solution methods are connected.     

Computing the matrix exponential and other matrix functions

In this note we present an algorithm for the computation of matrix functions, especially the matrix exponential. It is a new application of the so-called ‘elimination method’, which was presented for the scalar case in previous papers (e.g. [5]).     

On a family of singular measures related to Minkowski's?(x) function

In the present paper we are investigating a certain point measure of a distribution function arising in a paper by Grabner et al. [Combinatorica 22 (2002) 245-267]. This distribution function is defined by means of the subtractive Euclidean algorithm and bears a striking resemblance to the singular?(x)-function of H. Minkowski. Beyond it, we will also consider a whole family of distribution functions arising in a natural way from the above ones. Nevertheless we will prove that all of the corresponding measures of the mentioned functions are mutually singular by using dynamical systems and the ergodic theorem.     

Extrema detection of bivariate spline functions

In this paper, we develop a systematic method for detecting the extrema of bivariate spline functions and of their derivatives. It is assumed that the splines are constituted by employing normalized, uniform B-splines as the basis functions, and the detection procedure fully utilizes the spline properties. All the extrema can be found except those with singular Hessian matrix. By numerical examples, we demonstrate the effectiveness of the method.     

求解时谐涡流模型鞍点问题的分块交替分裂隐式迭代算法的改进

分块交替分裂隐式迭代方法是求解具有鞍点结构的复线性代数方程组的一类高效迭代法.本文通过预处理技巧得到原方法的一种加速改进方法,称之为预处理分块交替分裂隐式迭代方法·理论分析给出了新方法的收敛性结果.对于一类时谐涡旋电流模型问题,我们给出了若干满足收敛条件的迭代格式.数值实验验证了新型算法是对原方法的有效改进.     

A systematic method of constructing Boolean functions with optimal algebraic immunity based on the generator matrix of the Reed–Muller code

Because of the recent algebraic attacks, optimal algebraic immunity is now an absolutely necessary (but not sufficient) property for Boolean functions used in stream ciphers. In this paper, we firstly determine the concrete coefficients in the linear expression of the column vectors with respect to a given basis of the generator matrix of Reed–Muller code, which is an important tool for constructing Boolean functions with optimal algebraic immunity. Secondly, as applications of the determined coefficients, we provide simpler and direct proofs for two known constructions. Further, we construct new Boolean functions on odd variables with optimal algebraic immunity based on the generator matrix of Reed–Muller code. Most notably, the new constructed functions possess the highest nonlinearity among all the constructions based on the generator matrix of Reed–Muller code, although which is not as good as the nonlinearity of Carlet–Feng function. Besides, the ability of the new constructed functions to resist fast algebraic attacks is also checked for the variable \(n=11,13\) and 15.     

无约束优化问题的对角稀疏拟牛顿法

对无约束优化问题提出了对角稀疏拟牛顿法,该算法采用了Armijo非精确线性搜索,并在每次迭代中利用对角矩阵近似拟牛顿法中的校正矩阵,使计算搜索方向的存贮量和工作量明显减少,为大型无约束优化问题的求解提供了新的思路．在通常的假设条件下,证明了算法的全局收敛性,线性收敛速度并分析了超线性收敛特征。数值实验表明算法比共轭梯度法有效,适于求解大型无约束优化问题．     

Robust PCA and subspace tracking from incomplete observations using \ell _0-surrogates

Many applications in data analysis rely on the decomposition of a data matrix into a low-rank and a sparse component. Existing methods that tackle this task use the nuclear norm and \(\ell _1\) -cost functions as convex relaxations of the rank constraint and the sparsity measure, respectively, or employ thresholding techniques. We propose a method that allows for reconstructing and tracking a subspace of upper-bounded dimension from incomplete and corrupted observations. It does not require any a priori information about the number of outliers. The core of our algorithm is an intrinsic Conjugate Gradient method on the set of orthogonal projection matrices, the so-called Grassmannian. Non-convex sparsity measures are used for outlier detection, which leads to improved performance in terms of robustly recovering and tracking the low-rank matrix. In particular, our approach can cope with more outliers and with an underlying matrix of higher rank than other state-of-the-art methods.     

P-proper splittings

In this article we introduce the notion of P-proper splitting for square matrices. For an inconsistent linear system of equations \(Ax =b\), we associate an iterative method based on a P-proper splitting of A, which if convergent, converges to the best least squares solution of this system. We extend a result of Stein, using which we prove that if A is positive semidefinite, then the said iterative method converges. Also, we generalize Sylvester’s law of inertia and as an application of this generalization we establish some properties of P-proper splittings. Finally, we prove a comparison theorem for iterative methods associated with P-proper splittings of a positive semidefinite matrix.     

A modified Newton method with cubic convergence: the multivariate case

Recently, a modification of the Newton method for finding a zero of a univariate function with local cubic convergence has been introduced. Here, we extend this modification to the multi-dimensional case, i.e., we introduce a modified Newton method for vector functions that converges locally cubically, without the need to compute higher derivatives. The case of multiple roots is not treated. Per iteration the method requires one evaluation of the function vector and solving two linear systems with the Jacobian as coefficient matrix, where the Jacobian has to be evaluated twice. Since the additional computational effort is nearly that of an additional Newton step, the proposed method is useful especially in difficult cases where the number of iterations can be reduced by a factor of two in comparison to the Newton method. This much better convergence is indeed possible as shown by a numerical example. Also, the modified Newton method can be advantageous in cases where the evaluation of the function is more expensive than solving a linear system with the Jacobian as coefficient matrix. An example for this is given where numerical quadrature is involved. Finally, we discuss shortly possible extensions of the method to make it globally convergent.