A Padé family of iterations for the matrix sign function and related problems

In this paper we consider the Pad'e family of iterations for computing the matrix sign function and the Padé family of iterations for computing the matrix p‐sector function. We prove that all the iterations of the Padé family for the matrix sign function have a common convergence region. It completes a similar result of Kenney and Laub for half of the Padé family. We show that the iterations of the Padé family for the matrix p‐sector function are well defined in an analogous common region, depending on p. For this purpose we proved that the Padé approximants to the function (1?z)^?σ, 0<σ<1, are a quotient of hypergeometric functions whose poles we have localized. Furthermore we proved that the coefficients of the power expansion of a certain analytic function form a positive sequence and in a special case this sequence has the log‐concavity property. Copyright © 2011 John Wiley & Sons, Ltd.     

A new proof of the refined alternating sign matrix theorem

In the early 1980s, Mills, Robbins and Rumsey conjectured, and in 1996 Zeilberger proved a simple product formula for the number of n×n alternating sign matrices with a 1 at the top of the ith column. We give an alternative proof of this formula using our operator formula for the number of monotone triangles with prescribed bottom row. In addition, we provide the enumeration of certain 0-1-(−1) matrices generalizing alternating sign matrices.     

On a new family of high‐order iterative methods for the matrix pth root

The main goal of this paper is to approximate the principal pth root of a matrix by using a family of high‐order iterative methods. We analyse the semi‐local convergence and the speed of convergence of these methods. Concerning stability, it is well known that even the simplified Newton method is unstable. Despite it, we present stable versions of our family of algorithms. We test numerically the methods: we check the numerical robustness and stability by considering matrices that are close to be singular and badly conditioned. We find algorithms of the family with better numerical behavior than the Newton and the Halley methods. These two algorithms are basically the iterative methods proposed in the literature to solve this problem. Copyright © 2015 John Wiley & Sons, Ltd.     

Bounds on the k-th generalized base of a primitive sign pattern matrix

You et al. [L. You, J. Shao, and H. Shan, Bounds on the bases of irreducible generalized sign pattern matrices, Lin. Alg. Appl. 427 (2007), pp. 285–300] extended the concept of the base of a powerful sign pattern matrix to the nonpowerful, irreducible sign pattern matrices. The key to their generalization was to view the relationship A ^l =A ^l?+?p as an equality of generalized sign patterns rather than of sign patterns. You, Shao and Shan showed that for primitive generalized sign patterns, the base is the smallest positive integer k such that all entries of A ^k are ambiguous. In this paper we study the k-th generalized base for nonpowerful primitive sign pattern matrices. For a primitive, nonpowerful sign pattern A, this is the smallest positive integer h such that A^k has h rows consisting entirely of ambiguous entries. Extending the work of You, Shao and Shan, we obtain sharp upper bounds on the k-th generalized base, together with a complete characterization of the equality cases for those bounds. We also show that there exist gaps in the k-th generalized base set of the classes of such matrices.     

A stable test for strict sign regularity

A stable test to check if a given matrix is strictly sign regular is provided. Among other nice properties, we prove that it has an optimal growth factor. The test is compared with other alternative tests appearing in the literature, and its advantages are shown.

     

Stable iterations for the matrix square root

Any matrix with no nonpositive real eigenvalues has a unique square root for which every eigenvalue lies in the open right half-plane. A link between the matrix sign function and this square root is exploited to derive both old and new iterations for the square root from iterations for the sign function. One new iteration is a quadratically convergent Schulz iteration based entirely on matrix multiplication; it converges only locally, but can be used to compute the square root of any nonsingular M-matrix. A new Padé iteration well suited to parallel implementation is also derived and its properties explained. Iterative methods for the matrix square root are notorious for suffering from numerical instability. It is shown that apparently innocuous algorithmic modifications to the Padé iteration can lead to instability, and a perturbation analysis is given to provide some explanation. Numerical experiments are included and advice is offered on the choice of iterative method for computing the matrix square root.     

Correlation functions of the XXZ Heisenberg chain for Δ = 1/2

We present exact results for the correlation functions of the XXZ Heisenberg chain in the thermodynamic limit for the anisotropy parameter Δ = 1/2. We show that the results are given by integers. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 2, pp. 304–310, February, 2007.     

A probing method for computing the diagonal of a matrix inverse

The computation of some entries of a matrix inverse arises in several important applications in practice. This paper presents a probing method for determining the diagonal of the inverse of a sparse matrix in the common situation when its inverse exhibits a decay property, i.e. when many of the entries of the inverse are small. A few simple properties of the inverse suggest a way to determine effective probing vectors based on standard graph theory results. An iterative method is then applied to solve the resulting sequence of linear systems, from which the diagonal of the matrix inverse is extracted. The results of numerical experiments are provided to demonstrate the effectiveness of the probing method. Copyright © 2011 John Wiley & Sons, Ltd.     

Rational Lanczos approximations to the matrix square root and related functions

We consider restricted rational Lanczos approximations to matrix functions representable by some integral forms. A convergence analysis that stresses the effectiveness of the proposed method is developed. Error estimates are derived. Numerical experiments are presented. Copyright © 2008 John Wiley & Sons, Ltd.     

Precise asymptotics for a type of order statistics

This paper obtains some results on the precise asymptotics for the order statistics generated by the random samples of maximum domain of attraction of the Fréchet distribution, which reveal the relations among the boundary function, weight function, convergence rate and limit position in a uniform form.Research supported by National Science Foundation of China (NO. 10271087).     

Newton's method on the complex exponential function

We show that when Newton's method is applied to the product of a polynomial and the exponential function in the complex plane, the basins of attraction of roots have finite area.

     

A computational matrix method for solving systems of high order fractional differential equations

In this paper, we introduced an accurate computational matrix method for solving systems of high order fractional differential equations. The proposed method is based on the derived relation between the Chebyshev coefficient matrix A of the truncated Chebyshev solution u(t)$\mathbf{u}(\mathbf{t})$ and the Chebyshev coefficient matrix A^(ν)${\mathbf{A}}^{(\nu )}$ of the fractional derivative u^(ν)${\mathbf{u}}^{(\nu )}$. The fractional derivatives are presented in terms of Caputo sense. The matrix method for the approximate solution for the systems of high order fractional differential equations (FDEs) in terms of Chebyshev collocation points is presented. The systems of FDEs and their conditions (initial or boundary) are transformed to matrix equations, which corresponds to system of algebraic equations with unknown Chebyshev coefficients. The remaining set of algebraic equations is solved numerically to yield the Chebyshev coefficients. Several numerical examples for real problems are provided to confirm the accuracy and effectiveness of the present method.     

Character expansion method for the first order asymptotics of a matrix integral

The estimation of various matrix integrals as the size of the matrices goes to infinity is motivated by theoretical physics, geometry and free probability questions. On a rigorous ground, only integrals of one matrix or of several matrices with simple quadratic interaction (called AB interaction) could be evaluated so far (see e.g. [19], [17] or [9]). In this article, we follow an idea widely developed in the physics literature, which is based on character expansion, to study more complex interaction. In this context, we derive a large deviation principle for the empirical measure of Young tableaux. We then use it to study a matrix model defined in the spirit of the ‘dually weighted graph model’ introduced in [13], but with a cutoff function such that the matrix integral and its character expansion converge. We prove that the free energy of this model converges as the size of the matrices goes to infinity and study the critical points of the limit.     

A new investigation of the extended Krylov subspace method for matrix function evaluations

For large square matrices A and functions f, the numerical approximation of the action of f(A) to a vector v has received considerable attention in the last two decades. In this paper we investigate theextended Krylov subspace method, a technique that was recently proposed to approximate f(A)v for A symmetric. We provide a new theoretical analysis of the method, which improves the original result for A symmetric, and gives a new estimate for A nonsymmetric. Numerical experiments confirm that the new error estimates correctly capture the linear asymptotic convergence rate of the approximation. By using recent algorithmic improvements, we also show that the method is computationally competitive with respect to other enhancement techniques. Copyright © 2009 John Wiley & Sons, Ltd.     

A new method for calculating general lagrange multiplier in the variational iteration method

In this article, a reliable technique for calculating general Lagrange multiplier operator is suggested. The new algorithm, which is based on the calculus of variations, offers a simple method for calculation of general Lagrange multiplier for all forms. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 996–1001, 2011     

On the choice of the best members of the Kim family and the improvement of its convergence

The best members of the Kim family, in terms of stability, are obtained by using complex dynamics. From this elements, parametric iterative methods with memory are designed. A dynamical analysis of the methods with memory is presented in order to obtain information about the stability of them. Numerical experiments are shown for confirming the theoretical results.     

A family of iterative methods for computing the approximate inverse of a square matrix and inner inverse of a non-square matrix

Based on a quadratical convergence method, a family of iterative methods to compute the approximate inverse of square matrix are presented. The theoretical proofs and numerical experiments show that these iterative methods are very effective. And, more importantly, these methods can be used to compute the inner inverse and their convergence proofs are given by fundamental matrix tools.     

Approximation of inverse operators by a new family of high‐order iterative methods

The main goal of this paper is to approximate inverse operators by high‐order Newton‐type methods with the important feature of not using inverse operators. We analyse the semilocal convergence, the speed of convergence, and the efficiency of these methods. We determine that Chebyshev's method is the most efficient method and test it on two problems: one associated to the heat equation and the other one to a boundary value problem. We consider examples with matrices that are close to be singular and/or are badly conditioned. We check the robustness and the stability of the methods by considering situations with many steps and noised data. Copyright © 2013 John Wiley & Sons, Ltd.     

Successive iterations for unique positive solution of a nonlinear fractional q-integral boundary value problem

In this paper, under certain nonlinear growth conditions, we investigate the existence and successive iterations for the unique positive solution of a nonlinear fractional $q$-integral boundary problem by employing hybrid monotone method, which is a novel approach to nonlinear fractional $q$-difference equation. This paper not only proves the existence of the unique positive solution, but also gives some computable explicit hybrid iterative sequences approximating to the unique positive solution.     

Solutions with constant sign at infinity of a linear functional equation of infinite order

We investigate how the existence and behaviour of solutions φ with a constant sign of the equation