on newton s method and halley s method for the principal pth root of a matrix if a is a matrix with no negative real eigenvalues and all zero eigenvalues of a are semisimple the principal pth root of a can be computed by newton s method or halley s method with a preprocessing procedure if necessary we prove a new convergence result for newton s method and discover an interesting property of newton s method and halley s method in terms of series expansions we explain how the convergence of newton s method and halley s method can be improved when the eigenvalues of a are known or when a is a singular matrix we also prove new results on pth roots of m matrices and h matrices and consider the application of newton s method and halley s method to find the principal pth roots of these special matrices ams classification 65f30 41a58 15a48

computing the frechet derivative of the matrix logarithm and estimating the condition number the most popular method for computing the matrix logarithm is the inverse scaling and squaring method which is the basis of the recent algorithm of al mohy and higham siam j sci comput 34 2012 pp c152 c169 for real matrices we develop a version of the latter algorithm that works entirely in real arithmetic and is twice as fast as and more accurate than the original algorithm we show that by differentiating the algorithms we obtain backward stable algorithms for computing the frechet derivative we demonstrate experimentally that our two algorithms are more accurate and efficient than existing algorithms for computing the frechet derivative and we also show how the algorithms can be used to produce reliable estimates of the condition number of the matrix logarithm

an algebraic approximation for the matrix exponential in singularly perturbed boundary value problems a symmetric difference scheme for linear stiff or singularly perturbed boundary value problems of first order with constant coefficients is constructed being based on a stability function containing a matrix square root its essential feature is the unconditional stability in the absence of purely imaginary eigenvalues of the coefficient matrix local damping of errors uniform stability and uniform second order convergence are proved the computation of the specific matrix square root by a well known stable variant of newton s method is discussed a numerical example confirming the results is given

on solution of the periodic riccati equations results presented by mori in the article named above ibid vol ac 30 pp 162 164 feb 1985 which give some bounds for the trace and the determinant of the positive definite solution to the algebraic riccati matrix equation are shown to be erroneous and suggestions to remedy these errors are made

non removable singularities of a fourth order nonlinear partial differential equation abstract the aim of this paper is to give a classification of the solutions x y of the fourth order nonlinear equation y y x x 2 x y x y x x y y 0 det 2 3 4 in a punctured domain which are c 2 at the singularity we prove that this kind of solutions either have a removable singularity or are asymptotic to rotationally symmetric solutions near the singularity a geometric description of solutions which are not c 1 at the singularity is also given

perturbation bounds for matrix square roots and pythagorean sums abstract perturbation estimates for the square root r a 1 2 and pythagorean sum p i a 2 1 2 of complex matrices are proved we present bounds in the spectral norm for the cases that r and p are accretive i e have positive definite hermitian parts with special attention to the case of a large condition of a the results are based on bounds for the solution of the matrix sylvester equation and for the separation of two matrices

computing the square roots of matrices with central symmetry for computing square roots of a nonsingular matrix a which are functions of a two well known fast and stable algorithms which are based on the schur decomposition of a were proposed by bjork and hammarling a bjork s hammarling a schur method for the square root of a matrix linear alg appl 52 53 1983 127 140 for square roots of general complex matrices and by higham n j higham computing real square roots of a real matrix linear alg appl 88 89 1987 405 430 for real square roots of real matrices in this paper we further consider the computation of the square roots of matrices with central symmetry we first investigate the structure of the square roots of these matrices and then develop several algorithms for computing the square roots we show that our algorithms ensure significant savings in computational costs as compared to the use of standard algorithms for arbitrary matrices

qualitative properties of solutions to some nonlinear elliptic equations in r 2 o introduction elliptic equations in this paper we investigate properties of the solutions to the au r x e u x x r 2 for functions r x which are positive near infinity equations of this kind arise from a variety of situations such as from prescribing gaussian curvature in geometry i 2 and from combustion theory in physics 3 recently a series ofworks have been sought to understand the existence and the qualitative properties of the solutions of ni i 5 and ni cheng 4 1 considered the case where r x is nonpositive mcowen 6 1 and aviles 7 investigated the situation where r x 0 in some order as ixl in our previous paper 1 we consider a special case where r is a constant we proved that the solutions are radially symmetric and hence classified all the solutions in this paper we consider more general functions r x first we obtain the asymptotic behavior of the solution near infinity consequently we prove that all the solutions satisfy an identity which is somewhat of a generalization of the well known kazdan warner condition finally using the asymptotic behavior together with the further development of the method employed in our previous paper i 1 we show that all the solutions are radially symmetric provided r is radially symmetric and nonincreasing this part can be viewed as the completion of 1 throughout this paper we assume that the function r x is positive near infinity in 1 we study the asymptotic behavior of the solution u x of let fl 1 2n r2 r x e dx under some appropriate conditions we show that the solutions approach

on some integral inequalities similar to hardy s inequality abstract the aim of the present paper is to establish some new integral inequalities which claim their origin in the well known hardy s integral inequality the method by which we obtain our results is quite elementary and based on the idea used by levinson to obtain the generalizations of hardy s integral inequality

krylov approximations for matrix square roots in stiff boundary value problems recently we proposed an algebraic difference scheme with extended stability properties for linear boundary value problems involving stiff differential equations of first order here an efficient approximation scheme is presented for matrix square roots which provides the stabilization of that scheme in case of stiffness it combines the use of low rank matrix approximations from projections onto krylov subspaces with an accelerated sign iteration for the matrix square root the krylov approximation being accurate in eigenspaces with large eigenvalues preserves the stability of the scheme and the 0 n3 square root computation need be performed only in lower dimension operation counts and numerical results show that the effort for the numerical scheme is essentially proportional to the number of stiff components but not to the norm of the coefficient matrix approximation properties of low rank krylov matrices which may be of independent interest are analyzed 1 the sqrt one step difference scheme a standard approximation for the differential equation in the linear boundary value problem bvp u x a x u x g x x e 0 1 u x er bou o b u l 8 is the trapezoidal rule on a suitable grid 0 xo 1 this restriction leads to unacceptable small stepsizes in regions where the solution is smooth this situation is usually referred to as stiffness in certain cases however a relaxation received march 27 1990 revised december 5 1990 1991 mathematics subject classification primary 65l10 65l20 65f30

conformal deformation of metrics on s 2 soit d un domaine lisse de s 2 et k une fonction lisse definie sur d alors pour que le probleme aux limites de neumann u k exp 2u 1 sur d u n 0 sur d ait une solution u on a a quand aire d max x d k x 0

periodic solutions of riccati equations applied to multirate sampling from some basic relations in estimation theory a comprehensive set of formulae are obtained for the periodic solution of riccati equations to obtain numerically stable algorithms square root formulations are introduced both the discrete time and continuous time cases are considered closed loop properties for periodic systems are also investigated the results are applied to multirate measurement sampling the control of a continuous stirred tank reactor serves as an illustrative example the temperature measurements are then complemented by concentration analysis available at a slower rate than the basic sampling and control rate the results give some guidelines about the benefits of multirate sampling

newton s method for the matrix square root one approach to computing a square root of a matrix a is to apply newton s method to the quadratic matrix equation f x x2 a 0 two widely quoted matrix square root iterations obtained by rewriting this newton iteration are shown to have excellent mathematical convergence properties however by means of a perturbation analysis and supportive numerical examples it is shown that these simplified iterations are numerically unstable a further variant of newton s method for the matrix square root recently proposed in the literature is shown to be for practical purposes numerically stable

a structured condition number for self adjoint polynomial matrix equations with applications in linear control abstract based on the classic definition of condition number a structured condition number is proposed for a class of self adjoint polynomial matrix equations the explicit formula of the structured condition number is derived with applying newly defined linear operators the structured condition number can be applied to some important polynomial matrix equations including the continuous time algebraic riccati equation care the discrete time algebraic lyapunov equation dale etc compared with the state of the art condition numbers for care and dale the newly proposed structured condition number can measure the sensitivity of the solution better which is validated by numerical examples

residual methods for the large scale matrix pth root and some related problems abstract the problem of finding the p th root of a matrix has received special attention in the last few years standard approaches for this problem include and combine some variations of newton s method which in turn involve matrix factorizations that in general are not suitable for large scale problems motivated by some recently developed low cost iterative schemes for nonlinear problems we consider and analyze specialized residual methods that only require a few matrix matrix products per iteration and hence are suitable for the large scale case as a by product we also discuss the advantages of residual methods for general nonlinear problems whose variables separate preliminary and encouraging numerical results are presented for computing p th roots of large scale symmetric and positive definite matrices for different values of p

computing real square roots of a real matrix bjorck and hammarling 1 describe a fast stable schur method for computing a square root x of a matrix a x2 a we present an extension of their method which enables real arithmetic to be used throughout when computing a real square root of a real matrix for a nonsingular real matrix a conditions are given for the existence of a real square root and for the existence of a real square root which is a polynomial in a the number of square roots of the latter type is determined the conditioning of matrix square roots is investigated and an algorithm is given for the computation of a well conditioned square root

a schur method for the square root of a matrix abstract a fast and stable method for computing the square root x of a given matrix a x 2 a is developed the method is based on the schur factorization a qsq h and uses a fast recursion to compute the upper triangular square root of s it is shown that if x 2 a is not large then the computed square root is the exact square root of a matrix close to a the method is extended for computing the cube root of a matrices exist for which the square root computed by the schur method is ill conditioned but which nonetheless have well conditioned square roots an optimization approach is suggested for computing the well conditioned square roots in these cases

a parallel algorithm for principal n th roots of matrices an iterative algorithm for computing the principal nth root of a positive definite matrix is presented the algorithm is based on the gauss legendre approximation of a definite integral we present a parallelization in which we use as many processors as the order of the approximation an analysis of the error introduced at each step of the iteration indicates that the algorithm converges more rapidly as the order of the approximation and thus the number of processors increases we describe the results of our implementation of an eight processor meiko cs 2 comparing the parallel algorithm to the fastest sequential algorithm which is the hoskins walton method

an efficient algorithm for the discrete time algebraic riccati equation the authors develop an algorithm to solve the standard discrete time algebraic riccati equation by using a skew hamiltonian transformation and the square root method the algorithm is structure preserving and efficient because the hamiltonian structure is fully exploited and only orthogonal transformations are used the efficiency and stability of the algorithm are analyzed numerical examples are included

hessenberg eigenvalue eigenmatrix relations abstract explicit relations between eigenvalues eigenmatrix entries and matrix elements of unreduced hessenberg matrices are derived the main result is based on the taylor expansion of the adjugate of zi h on the one hand and inherent properties of hessenberg matrix structure on the other hand this result is utilized to construct computable relations between eigenvalues eigenvector components eigenvalues of principal submatrices and products of lower diagonal elements generalizing similar identities for jacobi matrices

on square roots of m matrices abstract the question of the existence and uniqueness of an m matrix which is a square root of an m matrix is discussed the results are then used to derive some new necessary and sufficient conditions for a real matrix with nonpositive off diagonal elements to be an m matrix

a formula for the general solution of riccati type matrix differential equations abstract in this paper a closed formula for the general solution of a class of non symmetric riccati matrix differential equations in given in terms of exponentials of matrices related to the data

the affine cauchy problem abstract the aim of this paper is to solve the cauchy problem for locally strongly convex surfaces which are extremal for the equiaffine area functional these surfaces are called affine maximal surfaces and here we give a new complex representation which let us describe the solution to the corresponding cauchy problem as applications we obtain a generalized symmetry principle characterize when a curve in r 3 can be a geodesic or pre geodesic of a such surface and study the helicoidal affine maximal surfaces finally we investigate the existence and uniqueness of affine maximal surfaces with a given analytic curve in its singular set

boundary problems and periodic riccati equations we consider the resolution problem of a periodic riccati equation from the point of view of a generalized boundary problem that lets us reduce the differential problem to an algebraic equation of riccati type sufficient conditions for the existence and calculus of special types of solutions are given

extremal metrics of zeta function determinants on 4 manifolds in conformal geometry the sobolev inequality at a critical exponent has received much attention in particular the determination of the best constants has played a crucial role in the yamabe problem in dimension two the analogous problem deals with the moser trudinger inequality on a compact riemann surface m2 there exists a constant c c m so that f e47w2 c m if f ivw12 1 and f w 0 the connection of this inequality with geometry comes through the zeta functional determinant of the laplacian as defined by ray singer for a riemannian metric g let 0 a1 a2 be the spectrum

improved inverse scaling and squaring algorithms for the matrix logarithm a popular method for computing the matrix logarithm is the inverse scaling and squaring method which essentially carries out the steps of the scaling and squaring method for the matrix exponential in reverse order here we make several improvements to the method putting its development on a par with our recent version emph siam j matrix anal appl 31 2009 pp 970 989 of the scaling and squaring method for the exponential in particular we introduce backward error analysis to replace the previous forward error analysis obtain backward error bounds in terms of the quantities a p 1 p for several small integer p instead of a and use special techniques to compute the argument of the pad e approximant more accurately we derive one algorithm that employs a schur decomposition and thereby works with triangular matrices and another that requires only matrix multiplications and the solution of multiple right hand side linear systems numerical experiments show the new algorithms to be generally faster and more accurate than their existing counterparts and suggest that the schur based method is the method of choice for computing the matrix logarithm

a power method for computing square roots of complex matrices abstract in this paper higher order convergent methods for computing square roots of nonsingular complex matrices are derived these methods are globally convergent and are based on eigenvalue shifting and powering specifically it is shown for each positive integerr 2 a convergent method of orderrcan be developed these algorithms can be used to compute square roots of general nonsingular complex matrices such as computing square roots of matrices with negative eigenvalues

commutators of hardy operators in this paper we establish hardy s integral inequalities for commutators generating hardy operators with one sided dyadic cmo functions

variation of cone metrics on riemann surfaces we discuss the variational properties of the unique conical metric of constant curvature 1 associated to a compact riemann surface together with a weighted divisor 2005 elsevier inc all rights reserved

on some generalizations of hardy s integral inequality in this paper we establish some new generalizations of the hardy s inequality by using a fairly elementary analysis the inequalities given here contain in the special cases some of the recent generalizations of hardy s integral inequality appearing in the literature

periodic solutions of periodic riccati equations for periodically time varying matrix riccati equations controllability and observability in the usual sense are shown to be sufficient for the existence of a unique positive definite periodic solution

computing primary solutions of equations involving primary matrix functions abstract the matrix equation f x a where f is an analytic function and a is a square matrix is considered some results on the classification of solutions are provided when f is rational a numerical algorithm is proposed to compute all solutions that can be written as a polynomial of a for real data the algorithm yields the real solutions using only real arithmetic numerical experiments show that the algorithm performs in a stable fashion when run in finite precision arithmetic

prescribing curvature on compact surfaces with conical singularities we study the berger nirenberg problem on surfaces with conical singularities i e we discuss conditions under which a function on a riemann surface is the gaussian curvature of some conformal metric with a prescribed set of singularities of conical types

solving algebraic and differential riccati operator equations abstract by application of the riesz dunford functional calculus a method for solving generalized algebraic riccati operator equations in hilbert space is given results are applied to the problem of finding explicit solutions for cauchy problems and two point boundary value problems for generalized riccati operator differential equations

numerical approximation of the product of the square root of a matrix with a vector abstract given an n n symmetric positive definite matrix a and a vector c two numerical methods for approximating a 1 2 c are developed analyzed and computationally tested the first method applies a newton iteration to a specific nonlinear system to approximate a 1 2 c while the second method applies a step control method to numerically solve a specific initial value problem to approximate a 1 2 c assuming that a is first reduced to tridiagonal form the first method requires o n 2 operations per iteration while the second method requires o n operations per iteration in contrast numerical methods that first approximate a 1 2 and then compute a 1 2 c generally require o n 3 operations per iteration

on some variants of hardy s inequality abstract the present paper deals with some new generalizations and extensions of a certain variant of hardy s inequality given by izumi and izumi the method employed in our analysis is quite elementary and the results established in this paper provide new estimates on this type of integral inequalities