Triangularizing matrix polynomials

For an algebraically closed field F$\mathbb{F}$, we show that any matrix polynomial P(λ)∈F[λ]^n×m$P(\lambda )\in \mathbb{F}{[\lambda ]}^{n\times m}$, n?m$n?m$, can be reduced to triangular form, preserving the degree and the finite and infinite elementary divisors. We also characterize the real matrix polynomials that are triangularizable over the real numbers and show that those that are not triangularizable are quasi-triangularizable with diagonal blocks of sizes 1×1$1\times 1$ and 2×2$2\times 2$. The proofs we present solve the structured inverse problem of building up triangular matrix polynomials starting from lists of elementary divisors.     

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.     

Newton’s method and high-order algorithms for the nth root computation

Two modifications of Newton’s method to accelerate the convergence of the n$n$th root computation of a strictly positive real number are revisited. Both modifications lead to methods with prefixed order of convergence p∈N,p≥2$p\in \mathbb{N},p\ge 2$. We consider affine combinations of the two modified p$p$th-order methods which lead to a family of methods of order p$p$ with arbitrarily small asymptotic constants. Moreover the methods are of order p+1$p+1$ for some specific values of a parameter. Then we consider affine combinations of the three methods of order p+1$p+1$ to get methods of order p+1$p+1$ again with arbitrarily small asymptotic constants. The methods can be of order p+2$p+2$ with arbitrarily small asymptotic constants, and also of order p+3$p+3$ for some specific values of the parameters of the affine combination. It is shown that infinitely many p$p$th-order methods exist for the n$n$th root computation of a strictly positive real number for any p≥3$p\ge 3$.     

Improved Newton’s method with exact line searches to solve quadratic matrix equation

In this paper, we study the matrix equation AX²+BX+C=0$A{X}^2+BX+C=0$, where A,B$A,B$ and C$C$ are square matrices. We give two improved algorithms which are better than Newton’s method with exact line searches to calculate the solution. Some numerical examples are reported to illustrate our algorithms.     

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.     

Approximate martingale estimating functions for stochastic differential equations with small noises

An approximate martingale estimating function with an eigenfunction is proposed for an estimation problem about an unknown drift parameter for a one-dimensional diffusion process with small perturbed parameter ε$\epsilon$ from discrete time observations at n$n$ regularly spaced time points k/n$k/n$, k=0,1,…,n$k=0,1,\dots ,n$. We show asymptotic efficiency of an M$M$-estimator derived from the approximate martingale estimating function as ε→0$\epsilon \to 0$ and n→∞$n\to \infty$ simultaneously.     

On minors of the compound matrix of a Laplacian

Let L   be an n×n$n\times n$ matrix with zero row and column sums, n?3$n?3$. We obtain a formula for any minor of the (n−2)$(n-2)$-th compound of L. An application to counting spanning trees extending a given forest is given.     

On matrices over an arbitrary semiring and their generalized inverses

In this paper, we consider matrices with entries from a semiring S. We first discuss some generalized inverses of rectangular and square matrices. We establish necessary and sufficient conditions for the existence of the Moore–Penrose inverse of a regular matrix. For an m×n$m\times n$ matrix A  , an n×m$n\times m$ matrix P and a square matrix Q of order m, we present necessary and sufficient conditions for the existence of the group inverse of QAP   with the additional property that P(QAP)^#Q$P{(QAP)}^{\mathrm{\#}}Q$ is a {1,2}$\{1,2\}$ inverse of A  . The matrix product used here is the usual matrix multiplication. The result provides a method for generating elements in the set of {1,2}$\{1,2\}$ inverses of an m×n$m\times n$ matrix A starting from an initial {1} inverse of A  . We also establish a criterion for the existence of the group inverse of a regular square matrix. We then consider a semiring structure (_Mm×n(S),+,°)$({\mathbf{M}}_{m\times n}(\mathbf{S}),+,°)$ made up of m×n$m\times n$ matrices with the addition defined entry-wise and the multiplication defined as in the case of the Hadamard product of complex matrices. In the semiring (_Mm×n(S),+,°)$({\mathbf{M}}_{m\times n}(\mathbf{S}),+,°)$, we present criteria for the existence of the Drazin inverse and the Moore–Penrose inverse of an m×n$m\times n$ matrix. When S is commutative, we show that the Hadamard product preserves the Hermitian property, and provide a Schur-type product theorem for the product A°(CC^?)$A°(C{C}^{?})$ of a positive semidefinite n×n$n\times n$ matrix A   and an n×n$n\times n$ matrix C.     

A p-adic algorithm for computing the inverse of integer matrices

A method for computing the inverse of an (n×n)$(n\times n)$ integer matrix A$A$ using p$p$-adic approximation is given. The method is similar to Dixon’s algorithm, but ours has a quadratic convergence rate. The complexity of this algorithm (without using FFT or fast matrix multiplication) is O(n⁴(logn)²)$O\left(n^4{(logn)}^2\right)$, the same as that of Dixon’s algorithm. However, experiments show that our method is faster. This is because our methods decrease the number of matrix multiplications but increase the digits of the components of the matrix, which suits modern CPUs with fast integer multiplication instructions.     

On the length of an external branch in the Beta-coalescent

In this paper, we consider Beta(2−α,α)$(2-\alpha ,\alpha )$ (with 1<α<2$1<\alpha <2$) and related Λ$\Lambda$-coalescents. If T⁽ⁿ⁾$T^{\left(n\right)}$ denotes the length of a randomly chosen external branch of the n$n$-coalescent, we prove the convergence of n^α−1T⁽ⁿ⁾$n^{\alpha -1}{T}^{\left(n\right)}$ when n$n$ tends to ∞$\infty$, and give the limit. To this aim, we give asymptotics for the number σ⁽ⁿ⁾${\sigma }^{\left(n\right)}$ of collisions which occur in the n$n$-coalescent until the end of the chosen external branch, and for the block counting process associated with the n$n$-coalescent.     

Estimation for stochastic differential equations with a small diffusion coefficient

We consider a multidimensional diffusion X$X$ with drift coefficient b(_Xt,α)$b(X_t,\alpha )$ and diffusion coefficient εa(_Xt,β)$\epsilon a(X_t,\beta )$ where α$\alpha$ and β$\beta$ are two unknown parameters, while ε$\epsilon$ is known. For a high frequency sample of observations of the diffusion at the time points k/n$k/n$, k=1,…,n$k=1,\dots ,n$, we propose a class of contrast functions and thus obtain estimators of (α,β)$(\alpha ,\beta )$. The estimators are shown to be consistent and asymptotically normal when n→∞$n\to \infty$ and ε→0$\epsilon \to 0$ in such a way that ε⁻¹n^−ρ${\epsilon }^{-1}{n}^{-\rho }$ remains bounded for some ρ>0$\rho >0$. The main focus is on the construction of explicit contrast functions, but it is noted that the theory covers quadratic martingale estimating functions as a special case. In a simulation study we consider the finite sample behaviour and the applicability to a financial model of an estimator obtained from a simple explicit contrast function.     

Generalized frieze pattern determinants and higher angulations of polygons

Frieze patterns (in the sense of Conway and Coxeter) are in close connection to triangulations of polygons. Broline, Crowe and Isaacs have assigned a symmetric matrix to each polygon triangulation and computed the determinant. In this paper we consider d-angulations of polygons and generalize the combinatorial algorithm for computing the entries in the associated symmetric matrices; we compute their determinants and the Smith normal forms. It turns out that both are independent of the particular d  -angulation, the determinant is a power of d−1$d-1$, and the elementary divisors only take values d−1$d-1$ and 1. We also show that in the generalized frieze patterns obtained in our setting every adjacent 2×2$2\times 2$-determinant is 0 or 1, and we give a combinatorial criterion for when they are 1, which in the case d=3$d=3$ gives back the Conway–Coxeter condition on frieze patterns.     

A new method for computing Moore–Penrose inverse matrices

The Moore–Penrose inverse of an arbitrary matrix (including singular and rectangular) has many applications in statistics, prediction theory, control system analysis, curve fitting and numerical analysis. In this paper, an algorithm based on the conjugate Gram–Schmidt process and the Moore–Penrose inverse of partitioned matrices is proposed for computing the pseudoinverse of an m×n$m\times n$ real matrix A$A$ with m≥n$m\ge n$ and rank r≤n$r\le n$. Numerical experiments show that the resulting pseudoinverse matrix is reasonably accurate and its computation time is significantly less than that of pseudoinverses obtained by the other methods for large sparse matrices.     

Representation of solutions of n-order Riccati equation via generalized trigonometric functions

In this work we suggest a systematic method of construction of solutions of the n-order Riccati equation with constant coefficients in a field from the set of generalized trigonometric functions. The generalized trigonometric functions satisfy the system of evolution equations generated by the companion matrix of n  -order polynomial. The set of trigonometric functions depend of (n−1)$(n-1)$ variables formally expressed by series of exponential functions. In a particular case, the second order Riccati equation with constant coefficients is isomorphic to the evolution equation generated by the companion matrix of the associated quadratic polynomial. It is shown that the n>2$n>2$ order Riccati equation with coefficients in a field is derived from a linear system of evolution equations generated by companion matrix of the associated n  -order polynomial under (n−2)$(n-2)$ constraints.     

Convergence of the modified Mann’s iteration method for asymptotically strict pseudo-contractions

Let C$C$ be a closed convex subset of a real Hilbert space H$H$ and assume that T$T$ is an asymptotically κ$\kappa$-strict pseudo-contraction on C$C$ with a fixed point, for some 0≤κ<1$0\le \kappa <1$. Given an initial guess _x0∈C$x_0\in C$ and given also a real sequence {_αn}$\left\{{\alpha }_n\right\}$ in (0, 1), the modified Mann’s algorithm generates a sequence {_xn}$\left\{x_n\right\}$ via the formula: _xn+1=_αnxn+(1−_αn)Tⁿ_xn$x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)T^n{x}_n$, n≥0$n\ge 0$. It is proved that if the control sequence {_αn}$\left\{{\alpha }_n\right\}$ is chosen so that κ+δ<_αn<1−δ$\kappa +\delta <{\alpha }_n<1-\delta$ for some δ∈(0,1)$\delta \in (0,1)$, then {_xn}$\left\{x_n\right\}$ converges weakly to a fixed point of T$T$. We also modify this iteration method by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strongly convergent sequence.     

Eigenvalue estimates for saddle point matrices of Hermitian and indefinite leading blocks

We study the eigenvalue bounds for the nonsingular saddle point matrices of Hermitian and indefinite (1,1)$(1,1)$ and (2,2)$(2,2)$ blocks without imposing the restrictions that the (1,1)$(1,1)$ blocks are positive definite on the kernels of the (2,1)$(2,1)$ blocks.     

Asymptotic distribution of the CLSE in a critical process with immigration

It is known that in the critical case the conditional least squares estimator (CLSE) of the offspring mean of a discrete time branching process with immigration is not asymptotically normal. If the offspring variance tends to zero, it is normal with normalization factor n^2/3$n^{2/3}$. We study a situation of its asymptotic normality in the case of non-degenerate offspring distribution for the process with time-dependent immigration, whose mean and variance vary regularly with non-negative exponents α$\alpha$ and β$\beta$, respectively. We prove that if β<1+2α$\beta <1+2\alpha$, the CLSE is asymptotically normal with two different normalization factors and if β>1+2α$\beta >1+2\alpha$, its limit distribution is not normal but can be expressed in terms of the distribution of certain functionals of the time-changed Wiener process. When β=1+2α$\beta =1+2\alpha$ the limit distribution depends on the behavior of the slowly varying parts of the mean and variance.