Convergence of a generalized Newton and an inexact generalized Newton algorithms for solving nonlinear equations with nondifferentiable terms

In this paper, we consider two versions of the Newton-type method for solving a nonlinear equations with nondifferentiable terms, which uses as iteration matrices, any matrix from B-differential of semismooth terms. Local and global convergence theorems for the generalized Newton and inexact generalized Newton method are proved. Linear convergence of the algorithms is obtained under very mild assumptions. The superlinear convergence holds under some conditions imposed on both terms of equation. Some numerical results indicate that both algorithms works quite well in practice.      

Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation

We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in R ⁿ . In these formulas, p-planes are represented as the column space of n×p matrices. The Newton method on abstract Riemannian manifolds proposed by Smith is made explicit on the Grassmann manifold. Two applications – computing an invariant subspace of a matrix and the mean of subspaces – are worked out.     

Inequalities for linear combinations of monomials in p-Newton sequences

The partial order on monomials that corresponds to domination when evaluated at positive Newton sequences is fully understood. Here we take up the corresponding partial order on linear combinations of monomials. In part using analysis based upon the cone structure of the exponents in p-Newton sequences, an array of conditions is given for this new partial order. It appears that a characterization in general will be difficult. Within the case in which all coefficients are 1, the situation in which, for general sequence length, there are two monomials, each of length two and nonnegative integer exponents, the partial order is fully characterized. The characterization is combinatorial, in terms of indices in the monomials, and, already here there is much more than term-wise domination.     

Specifying attracting cycles for Newton maps of polynomials

We show that for any set of n distinct points in the complex plane, there exists a polynomial p of degree at most n+1 so that the corresponding Newton map, or even the relaxed Newton map, for p has the given points as a super-attracting cycle. This improves the result in Plaza and Romero [6], which shows how to find such a polynomial of degree 2n. Moreover, we show that in general one cannot improve upon degree n+1. Our methods allow us to give a simple, constructive proof of the known result that for each cycle length n ≥ 2 and degree d ≥ 3, there exists a polynomial of degree d whose Newton map has a super-attracting cycle of length n.     

Semilocal and global convergence of the Newton‐HSS method for systems of nonlinear equations

Newton‐HSS methods, which are variants of inexact Newton methods different from the Newton–Krylov methods, have been shown to be competitive methods for solving large sparse systems of nonlinear equations with positive‐definite Jacobian matrices (J. Comp. Math. 2010; 28 :235–260). In that paper, only local convergence was proved. In this paper, we prove a Kantorovich‐type semilocal convergence. Then we introduce Newton‐HSS methods with a backtracking strategy and analyse their global convergence. Finally, these globally convergent Newton‐HSS methods are shown to work well on several typical examples using different forcing terms to stop the inner iterations. Copyright © 2010 John Wiley & Sons, Ltd.     

Local Convergence of the Interior-Point Newton Method for General Nonlinear Programming

In this paper, a formulation for an interior-point Newton method of general nonlinear programming problems is presented. The formulation uses the Coleman-Li scaling matrix. The local convergence and the q-quadratic rate of convergence for the method are established under the standard assumptions of the Newton method for general nonlinear programming.     

Reduced Vertex Set Result for Interval Semidefinite Optimization Problems

In this paper we propose a reduced vertex result for the robust solution of uncertain semidefinite optimization problems subject to interval uncertainty. If the number of decision variables is m and the size of the coefficient matrices in the linear matrix inequality constraints is n×n, a direct vertex approach would require satisfaction of 2 ^{n(m+1)(n+1)/2} vertex constraints: a huge number, even for small values of n and m. The conditions derived here are instead based on the introduction of m slack variables and a subset of vertex coefficient matrices of cardinality 2 ⁿ⁻¹, thus reducing the problem to a practically manageable size, at least for small n. A similar size reduction is also obtained for a class of problems with affinely dependent interval uncertainty. This work is supported by MIUR under the FIRB project “Learning, Randomization and Guaranteed Predictive Inference for Complex Uncertain Systems,” and by CNR RSTL funds.     

Equilibrium problem for the eigenvalues of banded block Toeplitz matrices

We consider banded block Toeplitz matrices T_n with n block rows and columns. We show that under certain technical assumptions, the normalized eigenvalue counting measure of T_n for n → ∞ weakly converges to one component of the unique vector of measures that minimizes a certain energy functional. In this way we generalize a recent result of Duits and Kuijlaars for the scalar case. Along the way we also obtain an equilibrium problem associated to an arbitrary algebraic curve, not necessarily related to a block Toeplitz matrix. For banded block Toeplitz matrices, there are several new phenomena that do not occur in the scalar case: (i) The total masses of the equilibrium measures do not necessarily form a simple arithmetic series but in general are obtained through a combinatorial rule; (ii) The limiting eigenvalue distribution may contain point masses, and there may be attracting point sources in the equilibrium problem; (iii) More seriously, there are examples where the connection between the limiting eigenvalue distribution of T_n and the solution to the equilibrium problem breaks down. We provide sufficient conditions guaranteeing that no such breakdown occurs; in particular we show this if T_n is a Hessenberg matrix.     

Reflecting the pascal matrix about its main antidiagonal

Let B denote either of two varieties of order n Pascal matrix, i.e., one whose entries are the binomial coefficients. Let B_R denote the reflection of B about its main antidiagonal. The matrix B is always invertible modulo n; our main result asserts that B^-1 ≡ B^R mod n if and only if n is prime. In the course of motivating this result we encounter and highlight some of the difficulties with the matrix exponential under modular arithmetic. We then use our main result to extend the "Fibonacci diagonal" property of Pascal matrices.     

Base tensorielle des matrices de Hankel (ou de Toeplitz) Applications

Summary The set of all Hankel (or Toeplitz) matrices of dimensionn, is shown to possess tensorial bases: bases made ofn rank one matrices. Four families of such tensorial bases are possible. From this result, we deduce that the following computations can be performed with a number of multiplications of ordern instead of ordern ²: evaluation of the 2n+1 coefficients of the polynomial product of two polynomials of degreen, evaluation of the inverse of a lower triangular toeplitz matrix, evaluation of the quotient and of the remainder in the division of a polynomial of degree 2n by a polynomial of degreen.     

Factorization of moving-average spectral densities by state-space representations and stacking

To factorize a spectral density matrix of a vector moving average process, we propose a state space representation. Although this state space is not necessarily of minimal dimension, its associated system matrices are simple and most matrix multiplications involved are nothing but index shifting. This greatly reduces the complexity of computation. Moreover, in this article we stack every q consecutive observations of the original process MA(q) and generate a vector MA(1) process. We consider a similar state space representation for the stacked process. Consequently, the solution hinges on a surprisingly compact discrete algebraic Riccati equation (DARE), which involves only one Toeplitz and one Hankel block matrix composed of autocovariance functions. One solution to this equation is given by the so-called iterative projection algorithm. Each iteration of the stacked version is equivalent to q iterations of the unstacked one. We show that the convergence behavior of the iterative projection algorithm is characterized by the decreasing rate of the partial correlation coefficients for the stacked process. In fact, the calculation of the partial correlation coefficients via the Whittle algorithm, which takes a very simple form in this case, offers another solution to the problem. To achieve computational efficiency, we apply the general Newton procedure given by Lancaster and Rodman to the DARE and obtain an algorithm of quadratic convergence rate. One immediate application of the new algorithms is polynomial stabilization. We also discuss various issues such as check of positivity and numerical implementation.     

The difference between doubly nonnegative and completely positive matrices

The convex cone of n×n completely positive (CP) matrices and its dual cone of copositive matrices arise in several areas of applied mathematics, including optimization. Every CP matrix is doubly nonnegative (DNN), i.e., positive semidefinite and component-wise nonnegative, and it is known that, for n4 only, every DNN matrix is CP. In this paper, we investigate the difference between 5×5 DNN and CP matrices. Defining a bad matrix to be one which is DNN but not CP, we: (i) design a finite procedure to decompose any n×n DNN matrix into the sum of a CP matrix and a bad matrix, which itself cannot be further decomposed; (ii) show that every bad 5×5 DNN matrix is the sum of a CP matrix and a single bad extreme matrix; and (iii) demonstrate how to separate bad extreme matrices from the cone of 5×5 CP matrices.     

MULTIVARIATE NEWTON SUMS: IDENTITIES AND GENERATING FUNCTIONS

ABSTRACT

This paper is devoted to present, first, a family of formulas extending to the multivariate case the classical Newton (or Newton–Girard) Identities relating the coefficients of a univariate polynomial equation with its roots through the Newton Sums and, secondly, the Generating Functions associated to the new introduced Newton Sums of the multivariate case. As a by-product the kinds of systems accepting these Newton Identities are also characterized together with those allowing the Newton Sums to be computed in an inductive way directly from the coefficients of the polynomial system under consideration.     

A discrete Newton algorithm for minimizing a function of many variables

A Newton-like method is presented for minimizing a function ofn variables. It uses only function and gradient values and is a variant of the discrete Newton algorithm. This variant requires fewer operations than the standard method whenn > 39, and storage is proportional ton rather thann ².     

On pointwise complete pairs of linear transformations

We prove the pointwise completeness of the order n system with constant coefficients under the assumption that the matrices of the system split into square blocks of the same size so that the collection of all blocks embeds into a finite dimensional associative division algebra; the block rank of the passive matrix is at most 2.     

Products of involutory matrices. I

It is shown that, for every integer ?1 and every field F, each n×n matrix over F of determinant ±1 is the product of four involutory matrices over F. Products of three ×n involutory matrices over F are characterized for the special cases where n?4 or F has prime order ?5. It is also shown for every field F that every matrix over F of determinant ±1 having no more than two nontrivial invariant factors is a product of three involutory matrices over F.     

Algebraic properties of Manin matrices 1

We study a class of matrices with noncommutative entries, which were first considered by Yu.I. Manin in 1988 in relation with quantum group theory. They are defined as “noncommutative endomorphisms” of a polynomial algebra. More explicitly their defining conditions read: (1) elements in the same column commute; (2) commutators of the cross terms are equal: [M_ij,M_kl]=[M_kj,M_il] (e.g. [M₁₁,M₂₂]=[M₂₁,M₁₂]). The basic claim is that despite noncommutativity many theorems of linear algebra hold true for Manin matrices in a form identical to that of the commutative case. Moreover in some examples the converse is also true, that is, Manin matrices are the most general class of matrices such that linear algebra holds true for them. The present paper gives a complete list and detailed proofs of algebraic properties of Manin matrices known up to the moment; many of them are new. In particular we provide complete proofs that an inverse to a Manin matrix is again a Manin matrix and for the Schur formula for the determinant of a block matrix; we generalize the noncommutative Cauchy–Binet formulas discovered recently arXiv:0809.3516, which includes the classical Capelli and related identities. We also discuss many other properties, such as the Cramer formula for the inverse matrix, the Cayley–Hamilton theorem, Newton and MacMahon–Wronski identities, Plücker relations, Sylvester's theorem, the Lagrange–Desnanot–Lewis Carroll formula, the Weinstein–Aronszajn formula, some multiplicativity properties for the determinant, relations with quasideterminants, calculation of the determinant via Gauss decomposition, conjugation to the second normal (Frobenius) form, and so on and so forth. Finally several examples and open question are discussed. We refer to [A. Chervov, G. Falqui, Manin matrices and Talalaev's formula, J. Phys. A 41 (2008) 194006; V. Rubtsov, A. Silantiev, D. Talalaev, Manin matrices, elliptic commuting families and characteristic polynomial of quantum gl_n elliptic Gaudin model, in press] for some applications in the realm of quantum integrable systems.     

A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities

In this paper we take a new look at smoothing Newton methods for solving the nonlinear complementarity problem (NCP) and the box constrained variational inequalities (BVI). Instead of using an infinite sequence of smoothing approximation functions, we use a single smoothing approximation function and Robinson’s normal equation to reformulate NCP and BVI as an equivalent nonsmooth equation H(u,x)=0, where H:ℜ ²ⁿ →ℜ ²ⁿ , u∈ℜ ⁿ is a parameter variable and x∈ℜ ⁿ is the original variable. The central idea of our smoothing Newton methods is that we construct a sequence {z ^k =(u ^k ,x ^k )} such that the mapping H(·) is continuously differentiable at each z ^k and may be non-differentiable at the limiting point of {z ^k }. We prove that three most often used Gabriel-Moré smoothing functions can generate strongly semismooth functions, which play a fundamental role in establishing superlinear and quadratic convergence of our new smoothing Newton methods. We do not require any function value of F or its derivative value outside the feasible region while at each step we only solve a linear system of equations and if we choose a certain smoothing function only a reduced form needs to be solved. Preliminary numerical results show that the proposed methods for particularly chosen smoothing functions are very promising. Received June 23, 1997 / Revised version received July 29, 1999?Published online December 15, 1999     

Tournament matrices and their generalizations,I.

If M is any complex matrix with rank (M + M ^* + I) = 1, we show that any eigenvalue of M that is not geometrically simple has 1/2 for its real part. This generalizes a recent finding of de Caen and Hoffman: the rank of any n × n tournament matrix is at least n ? 1. We extend several spectral properties of tournament matrices to this and related types of matrices. For example, we characterize the singular real matrices M with 0 diagonal for which rank (M + M^T + I) = 1 and we characterize the vectors that can be in the kernels of such matrices. We show that singular, irreducible n × n tournament matrices exist if and only n? {2,3,4,5} and exhibit many infinite families of such matrices. Connections with signed digraphs are explored and several open problems are presented.     

Matrix relation algebras

Square matrices over a relation algebra are relation algebras in a natural way. We show that for fixed n, these algebras can be characterized as reducts of some richer kind of algebra. Hence for fixed n, the class of n × n matrix relation algebras has a first–order characterization. As a consequence, homomorphic images and proper extensions of matrix relation algebras are isomorphic to matrix relation algebras. Received July 18, 2001; accepted in final form April 24, 2002.