一类Nevanlinna-Pick问题的Toeplitz向量方法

首次应用改进的Ｔｏｅｐｌｉｔｚ向量方法刻划Ｃａｒａｔｈｅｏｄｏｒｙ函数类中含多重零插值点的Ｎｅｖａｎ ｌｉｎｎａ Ｐｉｃｋ问题与截断的三角矩量问题之间的内在联系，从而给出这类Ｎｅｖａｎｌｉｎｎａ Ｐｉｃｋ问题的可解性准则和通解的参数化表示．     

On rank variation of block matrices generated by Nevanlinna matrix functions

Within the framework of the multiple Nevanlinna–Pick matrix interpolation and its related matrix moment problem, we study the rank of block moment matrices of various kinds, generalized block Pick matrices and generalized block Loewner matrices, as well as their Potapov modifications, generated by Nevanlinna matrix functions, and derive statements either on rank (or inertia) invariance in different senses or on rank variation of such types of block matrices (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solution of a multiple Nevanlinna–Pick problem for Carathéodory functions via orthogonal rational functions in the matrix case

We consider an interpolation problem of Nevanlinna–Pick type for matrix‐valued Carathéodory functions, where the values of the functions and its derivatives up to certain orders are given at finitely many points of the open unit disk. For the non‐degenerate case, i.e., in the particular situation that a specific block matrix (which is formed by the given data in the problem) is positive Hermitian, the solution set of this problem is described in terms of orthogonal rational matrix‐valued functions. These rational matrix functions play here a similar role as Szegő's orthogonal polynomials on the unit circle in the classical case of the trigonometric moment problem. In particular, we present and use a connection between Szegő and Schur parameters for orthogonal rational matrix‐valued functions which in the primary situation of orthogonal polynomials was found by Geronimus. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The boundary Carathéodory-Fejér interpolation problem

We give a new solvability criterion for the boundary Carathéodory-Fejér problem: given a point x∈R and, a finite set of target values, to construct a function f in the Pick class such that the first few derivatives of f take on the prescribed target values at x. We also derive a linear fractional parametrization of the set of solutions of the interpolation problem with real target values. The proofs are based on a reduction method due to Julia and Nevanlinna.     

E_m函数类中Nevanlinna-Pick插值与广义Stieltjes矩量问题

令E\-m=(-∞,∞)＼∪[DD(]m[]j=1[DD)](α\-j,β\-j).函数类[WTHT]N[WTBX](E\-m)表示在上半复平面解析且虚部非负,在诸(α\-j,β\-j)(j=1,…,m)内解析且为实值的函数全体.该文用Hankel 向量方法建立[WTHT]N[WTBX](E\-m)函数类 中含有限(或无限可数)插值点的Nevanlinna Pick 问题与集合E\-m上 相关的非标准截断(或全)广义Stieltjes 矩量问题解集之间的一一对应.用类似于Riesz的办法建立E\-m上非标准截断广义Stieltjes矩量问题的可解性准则,从而获得了[WTHT]N[WTBX](E\-m)函数类中Nevanlinna Pick问题的可解性准则.     

Tangential Nevanlinna-Pick Interpolation and Its Connection with Hamburger Matrix Moment Problem

This paper is concerned with the solution of a certain tangential Nevanlinna-Pick interpolation for Nevanlinna functions. We use the so-called block Hankel vector method to establish two intrinsic connections between the tangential Nevanlinna-Pick interpolation in the Nevanlinna class and the truncated Hamburger matrix moment problem associated with the block Hankel vector under consideration: one is a congruent relationship between their information matrices, and the other is a divisor-remainder connection between their solutions. These investigations generalize our previous work on the Nevanlinna-Pick interpolation and power matrix moment problem.     

Step-by-step solving of ordered interpolation problem for stieltjes functions

We investigate a step-by-step solving of ordered generalized interpolation problems for Stieltjes matrix functions and obtain a multiplicative representation for the sequence of resolvent matrices. Thematrix factors inmultiplicative representations of the resolventmatrices are expressed through the Schur–Stieltjes parameters, for which we obtain explicit formulas and give an algorithm of step-by-step solving of Stieltjes type interpolation problems. As examples, we consider step-by-step solutions of the Stieltjes matrix moment problem and the problems by Nevanlinna–Pick and Caratheodory.     

Boundary Nevanlinna�CPick interpolation via reduction and augmentation

We give an elementary proof of Sarason??s solvability criterion for the Nevanlinna?CPick problem with boundary interpolation nodes and boundary target values. We also give a concrete parametrization of all solutions of such a problem. The proofs are based on a reduction method due to Julia and Nevanlinna. Reduction of functions corresponds to Schur complementation of the corresponding Pick matrices.     

Matrix valued interpolation and truncated Hamburger moment problems

A solvability condition for matrix valued directional single-node interpolation problems of Loewner type is established, in terms of properties of Pick kernel. As a consequence, a solvability condition for matrix valued directional truncated Hamburger moment problems is obtained.     

On a class of extremal solutions of a moment problem for rational matrix‐valued functions in the nondegenerate case I

The main theme of this paper is the discussion of a parametrized family of solutions of a finite moment problem for rational matrix‐valued functions in the nondegenerate case. We will show that each member of this family is extremal in several directions concerning some point of the open unit disk. These investigations are inspired by the authors' paper [23], where a similar topic is studied in the context of the matricial Carathéodory problem. We will see that larger parts of the results presented there can be extended to the rational case studied here.The main theme of this paper is the discussion of a parametrized family of solutions of a finite moment problem for rational matrix‐valued functions in the nondegenerate case. We will show that each member of this family is extremal in several directions concerning some point of the open unit disk. These investigations are inspired by the authors' paper [23], where a similar topic is studied in the context of the matricial Carathéodory problem. We will see that larger parts of the results presented there can be extended to the rational case studied here (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Inverse generating function approach for the preconditioning of Toeplitz‐block systems

As proposed by R. H. Chan and M. K. Ng (1993), linear systems of the form T [ f ] x = b , where T [ f ] denotes the n×n Toeplitz matrix generated by the function f, can be solved using iterative solvers with as a preconditioner. This article aims at generalizing this approach to the case of Toeplitz‐block matrices and matrix‐valued generating functions F . We prove that if F is Hermitian positive definite, most eigenvalues of the preconditioned matrix T [ F ⁻¹]T[ F ] are clustered around one. Numerical experiments demonstrate the performance of this preconditioner.     

A note on the complex semi‐definite matrix Procrustes problem

This note outlines an algorithm for solving the complex ‘matrix Procrustes problem’. This is a least‐squares approximation over the cone of positive semi‐definite Hermitian matrices, which has a number of applications in the areas of Optimization, Signal Processing and Control. The work generalizes the method of Allwright (SIAM J. Control Optim. 1988; 26 (3):537–556), who obtained a numerical solution to the real‐valued version of the problem. It is shown that, subject to an appropriate rank assumption, the complex problem can be formulated in a real setting using a matrix‐dilation technique, for which the method of Allwright is applicable. However, this transformation results in an over‐parametrization of the problem and, therefore, convergence to the optimal solution is slow. Here, an alternative algorithm is developed for solving the complex problem, which exploits fully the special structure of the dilated matrix. The advantages of the modified algorithm are demonstrated via a numerical example. Copyright © 2007 John Wiley & Sons, Ltd.     

Matrix representations of Sturm‐Liouville problems with coupled eigenparameter‐dependent boundary conditions and transmission conditions

We study matrix representations of Sturm‐Liouville problems with coupled eigenparameter‐dependent boundary conditions and transmission conditions. Meanwhile, given any matrix eigenvalue problem with coupled eigenparameter‐dependent boundary conditions and transmission conditions, we construct a class of Sturm‐Liouville problems with given boundary conditions and transmission conditions such that they have the same eigenvalues.     

A Jacobi–Davidson method for two‐real‐parameter nonlinear eigenvalue problems arising from delay‐differential equations

The critical delays of a delay‐differential equation can be computed by solving a nonlinear two‐parameter eigenvalue problem. The solution of this two‐parameter problem can be translated to solving a quadratic eigenvalue problem of squared dimension. We present a structure preserving QR‐type method for solving such quadratic eigenvalue problem that only computes real‐valued critical delays; that is, complex critical delays, which have no physical meaning, are discarded. For large‐scale problems, we propose new correction equations for a Newton‐type or Jacobi–Davidson style method, which also forces real‐valued critical delays. We present three different equations: one real‐valued equation using a direct linear system solver, one complex valued equation using a direct linear system solver, and one Jacobi–Davidson style correction equation that is suitable for an iterative linear system solver. We show numerical examples for large‐scale problems arising from PDEs. Copyright © 2012 John Wiley & Sons, Ltd.     

Functional calculus under Kreiss type conditions

It is shown that for an algebraic Banach space operator T , the Kreiss condition, ‖(zI – T )^–1‖ ≤ , |z | > 1, implies the following functional calculus estimate $$ \Vert f (T) \Vert \le {16 \over {\pi} }\, C \cdot {\rm deg} (T) \, \Vert f \Vert _{\infty}\, , $$ where deg(T ) is the degree of the minimal polynomial annihilating T . This result extends the known estimates of the powers of T for Kreiss operators on finite dimensional spaces. In the case of a general Kreiss operator, an estimate of the rational calculus is proved: $$ \Vert r(T) \Vert \le {16 \over {\pi} }\, C ( {\rm deg}(r) + 1) \, \Vert r \Vert _{\infty} \, . $$ Similar estimates hold for the polynomial calculus under generalized Kreiss conditions. A link is also established between the sharp constant in the first estimate and the norm of the best solution for a Nevanlinna–Pick type interpolation problem in analytic Besov classes. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mixed antiplane boundary‐value problem for a piecewise‐homogeneous elastic body with a semi‐infinite interfacial crack

Antiplane stress state of a piecewise‐homogeneous elastic body with a semi‐infinite crack along the interface is considered. The longitudinal displacements along one of the crack edges on a finite interval, adjacent to the crack tip, are known. Shear stresses are applied to the body along the crack edges and at infinity. The problem reduces to a Riemann–Hilbert boundary‐value matrix problem with a piecewise‐constant coefficient for a complex potential in the class of symmetric functions. The complex potential is found explicitly using a Gaussian hypergeometric function. The stress state of the body close to the singular points is investigated. The stress intensity factors are determined. Copyright © 2016 John Wiley & Sons, Ltd.     

On a sub‐Stiefel Procrustes problem arising in computer vision

A sub‐Stiefel matrix is a matrix that results from deleting simultaneously the last row and the last column of an orthogonal matrix. In this paper, we consider a Procrustes problem on the set of sub‐Stiefel matrices of order n. For n = 2, this problem has arisen in computer vision to solve the surface unfolding problem considered by R. Fereirra, J. Xavier and J. Costeira. An iterative algorithm for computing the solution of the sub‐Stiefel Procrustes problem for an arbitrary n is proposed, and some numerical experiments are carried out to illustrate its performance. For these purposes, we investigate the properties of sub‐Stiefel matrices. In particular, we derive two necessary and sufficient conditions for a matrix to be sub‐Stiefel. We also relate the sub‐Stiefel Procrustes problem with the Stiefel Procrustes problem and compare it with the orthogonal Procrustes problem. Copyright © 2015 John Wiley & Sons, Ltd.     

An inexact shift‐and‐invert Arnoldi algorithm for Toeplitz matrix exponential

We revisit the shift‐and‐invert Arnoldi method proposed in [S. Lee, H. Pang, and H. Sun. Shift‐invert Arnoldi approximation to the Toeplitz matrix exponential, SIAM J. Sci. Comput., 32: 774–792, 2010] for numerical approximation to the product of Toeplitz matrix exponential with a vector. In this approach, one has to solve two large‐scale Toeplitz linear systems in advance. However, if the desired accuracy is high, the cost will be prohibitive. Therefore, it is interesting to investigate how to solve the Toeplitz systems inexactly in this method. The contribution of this paper is in three regards. First, we give a new stability analysis on the Gohberg–Semencul formula (GSF) and define the GSF condition number of a Toeplitz matrix. It is shown that when the size of the Toeplitz matrix is large, our result is sharper than the one given in [M. Gutknecht and M. Hochbruck. The stability of inversion formulas for Toeplitz matrices, Linear Algebra Appl., 223/224: 307–324, 1995]. Second, we establish a relation between the error of Toeplitz systems and the residual of Toeplitz matrix exponential. We show that if the GSF condition number of the Toeplitz matrix is medium‐sized, then the Toeplitz systems can be solved in a low accuracy. Third, based on this relationship, we present a practical stopping criterion for relaxing the accuracy of the Toeplitz systems and propose an inexact shift‐and‐invert Arnoldi algorithm for the Toeplitz matrix exponential problem. Numerical experiments illustrate the numerical behavior of the new algorithm and show the effectiveness of our theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.