逆H矩阵的新性质

本文在文[4]给出的逆H矩阵定义的基础上,给出了逆H矩阵的新性质．     

A novel method for computation of higher order singular points in nonlinear problems with single parameter

This paper deals with unusual numerical techniques for computation of higher order singular points in nonlinear problems with single parameter. Based on the uniformly extended system, a unified algorithm combining the homotopy and the pseudo-arclength continuation method is given. Properties of the uniformly augmented system are listed and proved. The effectiveness of the proposed algorithm is testified by three numerical examples.     

Generating equations approach for quadratic matrix equations

We show how Van Loan's method for annulling the (2,1) block of skew‐Hamiltonian matrices by symplectic‐orthogonal similarity transformation generalizes to general matrices and provides a numerical algorithm for solving the general quadratic matrix equation: For skew‐Hamiltonian matrices we find their canonical form under a similarity transformation and find the class of all symplectic‐orthogonal similarity transformations for annulling the (2,1) block and simultaneously bringing the (1,1) block to Hessenberg form. We present a structure‐preserving algorithm for the solution of continuous‐time algebraic Riccati equation. Unlike other methods in the literature, the final transformed Hamiltonian matrix is not in Hamiltonian–Schur form. Three applications are presented: (a) for a special system of partial differential equations of second order for a single unknown function, we obtain the matrix of partial derivatives of second order of the unknown function by only algebraic operations and differentiation of functions; (b) for a similar transformation of a complex matrix into a symmetric (and three‐diagonal) one by applying only finite algebraic transformations; and (c) for finite‐step reduction of the eigenvalues–eigenvectors problem of a Hermitian matrix to the eigenvalues– eigenvectors problem of a real symmetric matrix of the same dimension. Copyright © 1999 John Wiley & Sons, Ltd.     

On the relative extrema of a linear combination of elementary symmetric functions

We determine where a linear combination of elementary symmetric functions attains as maximum and minimum over a certain convex set in Rⁿ . We also show that an inequality for elementary symmetric functions proposed by S. Pierce is true.     

The Probability that a Random Real Gaussian Matrix haskReal Eigenvalues, Related Distributions, and the Circular Law

LetAbe annbynmatrix whose elements are independent random variables with standard normal distributions. Girko's (more general) circular law states that the distribution of appropriately normalized eigenvalues is asymptotically uniform in the unit disk in the complex plane. We derive the exact expected empirical spectral distribution of the complex eigenvalues for finiten, from which convergence in the expected distribution to the circular law for normally distributed matrices may be derived. Similar methodology allows us to derive a joint distribution formula for the real Schur decomposition ofA. Integration of this distribution yields the probability thatAhas exactlykreal eigenvalues. For example, we show that the probability thatAhas all real eigenvalues is exactly 2^{−n(n−1)/4}.     

A DIRECT METHOD FOR THE UNIT CIRCLE PROBLEM

The unit circle problem is the problem of finding the number of eigenvalues of a non-Hermitian matrix inside and outside the unit circle . To reduce the cost of computing eigenvalues for the problem, a direct method, which is analogous to that given in [5], is proposed in this paper.     

Simple geometry facilitates iterative solution of a nonlinear equation via a special transformation to accelerate convergence to third order

Direct substitution _xk+1=g(_xk)$x_{k+1}=g(x_k)$ generally represents iterative techniques for locating a root z   of a nonlinear equation f(x)$f(x)$. At the solution, f(z)=0$f(z)=0$ and g(z)=z$g(z)=z$. Efforts continue worldwide both to improve old iterators and create new ones. This is a study of convergence acceleration by generating secondary solvers through the transformation _gm(x)=(g(x)-m(x)x)/(1-m(x))$g_m(x)=(g(x)-m(x)x)/(1-m(x))$ or, equivalently, through partial substitution _gmps(x)=x+G(x)(g-x)$g_{m\mathrm{ps}}(x)=x+G(x)(g-x)$, G(x)=1/(1-m(x))$G(x)=1/(1-m(x))$. As a matter of fact, _gm(x)≡_gmps(x)$g_m(x)\equiv g_{m\mathrm{ps}}(x)$ is the point of intersection of a linearised g   with the g=x$g=x$ line. Aitken's and Wegstein's accelerators are special cases of _gm$g_m$. Simple geometry suggests that m(x)=(g^′(x)+g^′(z))/2$m(x)=(g^{\prime }(x)+g^{\prime }(z))/2$ is a good approximation for the ideal slope of the linearised g  . Indeed, this renders a third-order _gm$g_m$. The pertinent asymptotic error constant has been determined. The theoretical background covers a critical review of several partial substitution variants of the well-known Newton's method, including third-order Halley's and Chebyshev's solvers. The new technique is illustrated using first-, second-, and third-order primaries. A flexible algorithm is added to facilitate applications to any solver. The transformed Newton's method is identical to Halley's. The use of m(x)=(g^′(x)+g^′(z))/2$m(x)=(g^{\prime }(x)+g^{\prime }(z))/2$ thus obviates the requirement for the second derivative of f(x)$f(x)$. Comparison and combination with Halley's and Chebyshev's solvers are provided. Numerical results are from the square root and cube root examples.     

On interpolation of Banach algebras and factorization of weakly compact homomorphisms

We show a necessary and sufficient condition on the lattice Γ for the general real method to preserve the Banach-algebra structure. As an application we derive factorization of weakly compact homomorphisms through interpolation properties of weakly compact operators.     

Sabitov polynomials for volumes of polyhedra in four dimensions

In 1996 I.Kh. Sabitov proved that the volume of a simplicial polyhedron in a 3-dimensional Euclidean space is a root of certain monic polynomial with coefficients depending on the combinatorial type and on edge lengths of the polyhedron only. Moreover, the coefficients of this polynomial are polynomials in edge lengths of the polyhedron. This result implies that the volume of a simplicial polyhedron with fixed combinatorial type and edge lengths can take only finitely many values. In particular, this yields that the volume of a flexible polyhedron in a 3-dimensional Euclidean space is constant. Until now it has been unknown whether these results can be obtained in dimensions greater than 3. In this paper we prove that all these results hold for polyhedra in a 4-dimensional Euclidean space.     

On the length of longest chordless cycles

Upper bounds for the length of a longest (circuit) cycle without chords in a (directed) graph are given in terms of the rank of the adjacency matrix and in terms of its eigenvalues.Received: October 2003, Revised: January 2005, AMS classification: 05C