special h matrices and their schur and diagonal schur complements it is well known see d carlson t markham schur complements of diagonally dominant matrices czech math j 29 104 1979 246 251 2 j liu j li z huang x kong some properties of schur complements and diagonal schur complements of diagonally dominant matrices linear alg appl 428 2008 1009 1030 14 that the schur complement of a strictly diagonally dominant matrix is strictly diagonally dominant as well as its diagonal schur complement also if a matrix is an h matrix then its schur complement and diagonal schur complement are h matrices too see j liu y huang some properties on schur complements of h matrices and diagonally dominant matrices linear alg appl 389 2004 365 380 13 recent research see j liu y huang f zhang the schur complements of generalized doubly diagonally dominant matrices linear alg appl 378 2004 231 244 12 j liu j li z huang x kong some properties of schur complements and diagonal schur complements of diagonally dominant matrices linear alg appl 428 2008 1009 1030 14 showed that the similar statements hold for some special subclasses of h matrices the aim of this paper is to give more invariance results of this type and simplified proofs for some already known results by using scaling approach

some properties of schur complements and diagonal schur complements of diagonally dominant matrices in this paper we prove that the diagonal schur complement of a strictly doubly diagonally dominant matrix is strictly doubly diagonally dominant matrix the same holds for the diagonal schur complement of a strictly generalized doubly diagonally dominant matrix and a nonsingular h matrix we point out that under certain assumptions the diagonal schur complement of a strictly doubly doubly product diagonally dominant matrix is also strictly doubly doubly product diagonally dominant further we provide the distribution of the real parts of eigenvalues of a diagonal schur complement of h matrix we also show that the schur complement of a diagonally dominant matrix is not always diagonally dominant by a numerical example and then obtain a sufficient condition to ensure that the schur complement of a diagonally dominant matrix is diagonally dominant

matrix models with hard walls geometry and solutions we discuss various aspects of most general multisupport solutions to matrix models in the presence of hard walls i e in the case where the eigenvalue support is confined to subdomains of the real axis the structure of the solution at the leading order is described by semiclassical or generalized whitham krichever hierarchies as in the unrestricted case derivatives of tau functions for these solutions are associated with families of riemann surfaces with possible double points and satisfy the witten dijkgraaf verlinde verlinde equations we then develop the diagrammatic technique for finding free energy of this model in all orders of the t hooft expansion in the reciprocal matrix size generalizing the feynman diagrammatic technique for the hermitian one matrix model due to eynard

complexity of bezout s theorem v polynomial time we show that there are algorithms which find an approximate zero of a system of polynomial equations and which function in polynomial time on the average the number of arithmetic operations is cn 4s where n is the input size and c a universal constant

a newton method for systems of m equations in n variables the classical newton kantorovich method for solving systems of equations f x 0 uses the inverse of the jacobian of f at each iteration if the number of equations is dierent than the number of variables or if the jacobian cannot be assumed nonsingular a generalized inverse of the jacobian can be used in a newton method whose limit points are stationary points of kf x k 2 we study conditions for local convergence of this method prove quadratic convergence and implement an adaptive version this iterative method allowing a controlled increase of the ranks of the 2 inverses used in the iterations

on schur complement of block diagonally dominant matrices abstract it is well known that the schur complements of strictly diagonally dominant matrices are strictly diagonally dominant d carlson t markham schur complements of diagonally dominant matrices czech math j 29 104 1979 246 251 the same is true of generalized strictly diagonally dominant matrices jianzhou liu yungqing huang some properties on schur complements of h matrix and diagonally dominant matrices linear algebra appl 389 2004 365 380 in this paper this result is extended to the block strictly diagonally dominant matrices and the generalized block strictly diagonally dominant matrices that is it is shown that the schur complement of a block strictly diagonally dominant matrix is a block strictly diagonally dominant matrix and so is the schur complement of a generalized block strictly diagonally dominant matrix

a new class of nonsymmetric algebraic riccati equations abstract motivated by the study of linear quadratic differential games we introduce a new class of nonsymmetric algebraic riccati equations it is shown that every equation in this class has a unique stabilizing solution which is the solution required to find the open loop nash equilibrium for the differential game we show that the doubling algorithm can be used to find this solution efficiently the solution may also be found by the schur method and under further assumptions by newton s method and a basic fixed point iteration

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 matrices a truth is found that the class of conjugate generalized h matrices is a subclass of the class of generalized 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 matrices furthermore the convergence of the block aor iterative method for linear systems with generalized 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

the dominant degree and disc theorem for the schur complement of matrix the theory of schur complement plays an important role in many fields such as control theory and computational mathematics in this paper we obtain some estimates for the diagonally c diagonally and product c diagonally dominant degree on the schur complement of matrices which improve some relative results as application we present some bounds for the eigenvalues of schur complement by the entries of the original matrix instead of those of the schur complement particularly we obtain that the eigenvalues of the schur complements are located in the gerschgorin circles of the original matrices under certain conditions

the eigenvalue distribution on schur complements of h matrices abstract the paper studies the eigenvalue distribution of some special matrices tong in theorem 1 2 of wen ting tong on the distribution of eigenvalues of some matrices acta math sinica china 20 4 1977 273 275 gives conditions for an n n matrix a sdn idn to have j r a eigenvalues with positive real part and j r a eigenvalues with negative real part a counter example is given in this paper to show that the conditions of the theorem are not true a corrected condition is then proposed under which the conclusion of the theorem holds then the corrected condition is applied to establish some results about the eigenvalue distribution of the schur complements of h matrices with complex diagonal entries several conditions on the n n matrix a and the subset n 1 2 n are presented such that the schur complement matrix a of the matrix a has j r a j r a eigenvalues with positive real part and j r a j r a eigenvalues with negative real part

complexity of bezout s theorem iv probability of success extensions we estimate the probability that a given number of projective newton steps applied to a linear homotopy of a system of n homogeneous polynomial equations in n 1 complex variables of fixed degrees will find all the roots of the system we also extend the framework of our analysis to cover the classical implicit function theorem and revisit the condition number in this context further complexity theory is developed

determining dimension of the solution component that contains a computed zero of a polynomial system abstract when the jacobian of a computed numerical solution of a polynomial system in c n allows very small singular values the solution could be isolated with a multiple multiplicity or may belong to a solution component with positive dimension the algorithm constructed in this article intends to differentiate those cases by determining the dimension of the solution component m in which the solution lies of particular interest is the case when dim m 0 then the solution is of course isolated while the proposed algorithm is experimental it has been tested successfully on the class of problems with the solution in question belonging to a reduced component numerical results are provided to illustrate the accuracy of the algorithm

local convergence of the gauss newton method for injective overdetermined systems of equations under a majorant condition a local convergence analysis of the gauss newton method for solving injective overdetermined systems of nonlinear equations under a majorant condition is provided the convergence as well as results on its rate are established without a convexity hypothesis on the derivative of the majorant function the optimal convergence radius the biggest range for uniqueness of the solution along with some other special cases are also obtained

multihomogeneous newton methods we study multihomogeneous analytic functions and a multihomogeneous newton s method for finding their zeros we give a convergence result for this iteration and we study two examples the evaluation map and the generalized eigenvalue problem

newton s method for overdetermined systems of equations complexity theoretic aspects of continuation methods for the solution of square or underdetermined systems of polynomial equations have been studied by various authors in this paper we consider overdetermined systems where there are more equations than unknowns we study newton s method for such a system

the nekrasov diagonally dominant degree on the schur complement of nekrasov matrices and its applications abstract in this paper we estimate the nekrasov diagonally dominant degree on the schur complement of nekrasov matrices as an application we offer new bounds of the determinant for several special matrices which improve the related results in certain case further we give an estimation on the infinity norm bounds for the inverse of schur complement of nekrasov matrices finally we introduce new methods called schur based super relaxation iteration sssor method and schur based conjugate gradient scg method to solve the linear equation by reducing order the numerical examples illustrate the effectiveness of the derived result

on generalized newton algorithms quadratic convergence path following and error analysis abstract newton iteration is known under some precise conditions to convergence quadratically to zeros of non degenerate systems of polynomials this and other properties may be used to obtain theorems on the global complexity of solving systems of polynomial equations see shub and smale in 6 using a model of computability over the reals however it is not practical and not desirable to actually compute newton iteration exactly in this paper approximate newton iteration is investigated for several generalizations of the newton operator quadratic convergence theorems and a robustness theorem are extended to approximate newton iteration generalizing some of the results in 6 the results here can be used to prove complexity theorems on path following algorithms for solving systems of polynomial equations using a model of computation over the integers malajovich 3

kantorovich s type theorems for systems of equations with constant rank derivatives the famous newton kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of newton s method to a solution of an equation here we present a kantorovich type convergence analysis for the gauss newton s method which improves the result in w m hauszler a kantorovich type convergence analysis for the gauss newton method numer math 48 1986 119 125 and extends the main theorem in i k argyros on the newton kantorovich hypothesis for solving equations j comput appl math 169 2004 315 332 furthermore the radius of convergence ball is also obtained

convergence criterion of newton s method for singular systems with constant rank derivatives abstract the present paper is concerned with the convergence problem of newton s method to solve singular systems of equations with constant rank derivatives under the hypothesis that the derivatives satisfy a type of weak lipschitz condition a convergence criterion based on the information around the initial point is established for newton s method for singular systems of equations with constant rank derivatives applications to two special and important cases the classical lipschitz condition and the smale s assumption are provided the latter in particular extends and improves the corresponding result due to dedieu and kim in j p dedieu m kim newton s method for analytic systems of equations with constant rank derivatives j complexity 18 2002 187 209

on diagonal schur complements of block diagonally dominant matrices it is known that the diagonal schur complements of strictly diagonally dominant matrices are strictly diagonally dominant matrices j z liu y q huang some properties on schur complements of h matrices and diagonally dominant matrices linear algebra appl 389 2004 365 380 and the same is true for nonsingular h matrices j z liu j c li z t huang x kong some properties of schur complements and diagonal schur complements of diagonally dominant matrices linear algebra appl 428 2008 1009 1030 in this paper we research the properties on diagonal schur complements of block diagonally dominant matrices and prove that the diagonal schur complements of block strictly diagonally dominant matrices are block strictly diagonally dominant matrices and the same holds for generalized block strictly diagonally dominant matrices

newton s method with deflation for isolated singularities of polynomial systems we present a modification of newton s method to restore quadratic convergence for isolated singular solutions of polynomial systems our method is symbolic numeric we produce a new polynomial system which has the original multiple solution as a regular root using standard bases a tool for the symbolic computation of multiplicities we show that the number of deflation stages is bounded by the multiplicity of the isolated root our implementation performs well on a large class of applications

the schur complement of ph matrices abstract in this paper we consider the class of p h matrices a subclass of h matrices and using scaling characterization we show that this class is closed under taking the schur complement we show that under certain conditions the perron complement of a p h matrix is a p h matrix we also present a way of constructing a scaling matrix for the given p h matrix and we give eigenvalue localization for the schur complement of a p h matrix using only the entries of the original matrix we illustrate this by numerical examples

some properties on schur complements of h matrices and diagonally dominant matrices in this paper we obtain a theorem on the distribution of eigenvalues for schur complements of h matrices further we give some properties of diagonal schur complements on diagonally dominant matrices and their distribution of eigenvalues

the schur complements of γ diagonally and product γ diagonally dominant matrix and their disc separation in this paper we obtain some estimates for the diagonally and product diagonally dominant degree of the schur complement of matrices as application we present some bounds for the eigenvalues of schur complement by the entries of the original matrix based on these results we give a kind of iteration called the schur based iteration which can solve large scale linear systems though reducing the order by the schur complement and can compute out the results faster

on simple zeros of analytic functions of n variables we give a numerical criterion for a badly conditioned zero of a system of analytic equations to be part of a cluster of two zeros

overdetermined weierstrass iteration and the nearest consistent system we propose a generalization of the weierstrass iteration for overdetermined systems of equations and we prove that the proposed method is the gauss newton iteration to find the nearest system which has at least k common roots and which is obtained via a perturbation of prescribed structure in the univariate case we show the connection of our method to the optimization problem formulated by karmarkar and lakshman for the nearest gcd in the multivariate case we generalize the expressions of karmarkar and lakshman and give explicitly several iteration functions to compute the optimum the arithmetic complexity of the iterations is detailed

computing multiple roots of inexact polynomials we present a combination of two algorithms that accurately calculate multiple roots of general polynomials algorithm i transforms the singular root finding into a regular nonlinear least squares problem on a pejorative manifold and it calculates multiple roots simultaneously from a given multiplicity structure and initial root approximations to fulfill the input requirement of algorithm i we develop a numerical gcd finder containing a successive singular value updating and an iterative gcd refinement as the main engine of algorithm ii that calculates the multiplicity structure and the initial root approximation while limitations exist in certain situations the combined method calculates multiple roots with high accuracy and consistency in practice without using multiprecision arithmetic even if the coefficients are inexact this is perhaps the first blackbox type root finder with such capabilities to measure the sensitivity of the multiple roots a structure preserving condition number is proposed and error bounds are established according to our computational experiments and error analysis a polynomial being ill conditioned in the conventional sense can be well conditioned with the multiplicity structure being preserved and its multiple roots can be computed with high accuracy

theorems on schur complement of block diagonally dominant matrices and their application in reducing the order for the solution of large scale linear systems abstract we firstly consider the block dominant degree for i ii block strictly diagonally dominant matrix and their schur complements showing that the block dominant degree for the schur complement of an i ii block strictly diagonally dominant matrix is greater than that of the original grand block matrix then as application we present some disc theorems and some bounds for the eigenvalues of the schur complement by the elements of the original matrix further by means of matrix partition and the schur complement of block matrix based on the derived disc theorems we give a kind of iteration called the schur based iteration which can solve large scale linear systems though reducing the order by the schur complement and the numerical example illustrates that the iteration can compute out the results faster

a class of complex nonsymmetric algebraic riccati equations associated with h matrix abstract in this paper based on the study of fluid flow models modulated by a markov chain we propose a new class of complex nonsymmetric algebraic riccati equations nares associated with h matrix under the assumption that the diagonal elements of the associated h matrix are on a same ray in the complex plane we generalize the definition of the extremal solution of an h matrix algebraic riccati equation hare whose associated h matrix has positive or negative diagonal elements and show that the extremal solution exists and is unique besides we show that the basic fixed point iterative methods are linearly convergent and the newton s method is quadratically convergent when they are applied to search for the extremal solutions of the hares we also give out criteria for choosing suitable parameters such that the three existing doubling algorithms can deliver the extremal solutions of the hares and show that the doubling algorithms are quadratically convergent more importantly we also propose a preprocessing technique to transform an hare under a certain assumption into a new hare under the aforementioned assumption numerical experiments show that our methods and preprocessing technique are effective

the schur complement of strictly doubly diagonally dominant matrices and its application it is known that the schur complements of doubly diagonally dominant matrices are doubly diagonally dominant in this paper we obtain an estimate for the doubly diagonally dominant degree on the schur complement of strictly doubly diagonally dominant matrices then as an application we obtain that the eigenvalues of the schur complements are located in the brauer ovals of cassini of the original matrices under certain conditions as another application we obtain an upper bound for the infinity norm on the inverse on the schur complement of strictly doubly diagonally dominant matrices further based on the derived results we give a kind of iteration called the schur based iteration which can solve large scale linear systems though reducing the order by the schur complement and can compute out the results faster

eigenvalue localization refinements for the schur complement abstract the theory of schur complement plays an important role in many fields as well as the theory of h matrices in this paper we obtain some bounds for the eigenvalues of schur complement by the entries of the original matrix instead of those of the schur complement for some special h matrices also we show how this result for a wider class can be applied for estimating bounds for the eigenvalues of schur complement of an sdd matrix

further results on h matrices and their schur complements abstract it is well known d carlson t markham schur complements of diagonally dominant matrices czech math j 29 104 1979 246 251 1 that the schur complement of a strictly diagonally dominant matrix is strictly diagonally dominant also if a matrix is an h matrix then its schur complement is an h matrix too j liu y huang some properties on schur complements of h matrices and diagonally dominant matrices linear algebra appl 389 2004 365 380 8 recent research showed that the same type of statement holds for some special subclasses of h matrices see for example liu et al j liu y huang f zhang the schur complements of generalized doubly diagonally dominant matrices linear algebra appl 378 2004 231 244 the aim of this paper is to show that the proof of these results can be significantly simplified by using scaling approach as in zhang et al f zhang et al the schur complement and its applications springer new york 2005 and to give another invariance result of this type

local convergence analysis of inexact gauss newton like methods under majorant condition in this paper we present a local convergence analysis of inexact gauss newton like methods for solving nonlinear least squares problems under the hypothesis that the derivative of the function associated with the least squares problem satisfies a majorant condition we obtain that the method is well defined and converges our analysis provides a clear relationship between the majorant function and the function associated with the least squares problem it also allows us to obtain an estimate of convergence ball for inexact gauss newton like methods and some important special cases