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

a residual recurrence for halley s method for the matrix pth root abstract in this note we prove a residual relation for halley s method for finding the principal p th root of a matrix

halley s method for the matrix sector function the matrix n sector function is a generalization of the matrix sign function it can be used to determine the number of eigenvalues of a matrix in a specific sector of the complex plane and to extract the eigenpairs belonging to this sector without explicitly computing the eigenvalues it is known that newton s method which can be used for computing the matrix sign function is not globally convergent for the matrix sector function the only existing algorithm for computing the matrix sector function is based on the continued fraction expansion approximation to the principal nth root of an arbitrary complex matrix in this paper we introduce a new algorithm based on halley s generalized iteration formula for solving nonlinear equations it is shown that the iteration has good error propagation properties and high accuracy finally we give two application examples and summarize the results of our numerical experiments comparing newton s the continued fraction and halley s method

the matrix sign function a survey of the matrix sign function is presented including some historical background definitions and properties approximation theory and computational methods and condition theory and estimation procedures applications to areas such as control theory eigendecompositions and roots of matrices are outlined and some new theoretical results are also given

ramanujan s theories of elliptic functions to alternative bases in his famous paper 3 10 pp 23 39 ramanujan offers several beautiful series representations for 1 pi he first states three formulas one of which is frac 4 pi sum limits mathbf n 0 infty frac left 6 mathbf n 1 right left frac 1 2 right 3 n left mathbf n right 3 4 n where a o 1 and for each positive integer n left mathbf a right n mathbf a left mathbf a 1 right left mathbf a 2 right left mathbf a mathbf n 1 right

monotonicity properties of generalized elliptic integrals with respect to the parameter abstract for a 0 1 2 and r 0 1 let k a r and e a r k r and e r denote the generalized elliptic integrals the complete elliptic integrals respectively of the first and second kinds respectively in this paper the authors present the necessary and sufficient conditions under which certain familiar combinations defined in terms of the generalized elliptic integrals and elementary functions are monotone in a 0 1 2 by these results sharp bounds expressed in terms of k r e r and elementary functions are obtained for k a r and e a r thus showing the dependence on the parameter a of k a r and e a r and substantially improving the known related results

fast and stable algorithms for computing the principal nth root of a complex matrix and the matrix sector function this paper presents rapidly convergent and more stable recursive algorithms for finding the principal nth root of a complex matrix and the associated matrix sector function the developed algorithms significantly improve the computational aspects of finding the principal nth root of a matrix and the matrix sector function thus the developed algorithms will enhance the capabilities of the existing computational algorithms such as the principal nth root algorithm the matrix sign algorithm and the matrix sector algorithm for developing applications to control system problems

propagation of singularities and maximal functions in the plane in this work we mainly generalize bourgain s circular maximal function to include variable coefficient averages our techniques involve a combination of bourgain s basic ideas plus microlocal analysis in particular to see the role of curvature we rely heavily on methods used in studying propogation of singularities for hyperbolic differential equations we also show that forp 2 there is local smoothing inl p for solutions to the wave equation

from harmonic analysis to arithmetic combinatorics arithmetic combinatorics or additive combinatorics is a fast developing area of research combining elements of number theory combinatorics harmonic analysis and ergodic theory its arguably best known result and the one that brought it to global prominence is the proof by ben green and terence tao of the long standing conjecture that primes contain arbitrarily long arithmetic progressions there are many accounts and expositions of the green tao theorem including the articles by kra 119 and tao 182 in the bulletin the purpose of the present article is to survey a broader highly interconnected network of questions and results built over the decades and spanning several areas of mathematics of which the green tao theorem is a famous descendant an old geometric problem lies at the heart of key conjectures in harmonic analysis a major result in partial differential equations invokes combinatorial theorems on intersecting lines and circles an unexpected argument points harmonic analysts towards additive number theory with consequences that could have hardly been anticipated we will not try to give a comprehensive survey of harmonic analysis combinatorics or additive number theory we will not even be able to do full justice to our specific areas of focus instead referring the reader to the more complete expositions and surveys listed in section 7 our goal here is to emphasize the connections between these areas we will thus concentrate on relatively few problems chosen as much for their importance to their fields as for their links to each other the article is written from the point of view of an analyst who in the course of her work was gradually introduced to the questions discussed here and found them fascinating we hope that the reader will enjoy a taste of this experience

determining continuous time state equations from discrete time state equations via the principal q th root method fast computational methods are developed for finding the equivalent continuous time state equations from discrete time state equations the computational methods utilize the direct truncation method the matrix continued fraction method and the geometric series method in conjunction with the principal q th root of the discrete time system matrix for quick determination of the approximants of a matrix logarithm function it is shown that the use of the principal q th root of a matrix enables us to enlarge the convergence region of the expansion of a matrix logarithm function and to improve the accuracy of the approximants of the matrix logarithm function

roots of real matrices abstract an algorithm for computing the roots of a matrix with real elements and the real part of the eigenvalues not zero will be described the algorithm is based on the sign matrix given in an earlier paper and the projectors of the matrix

on hypergeometric functions and function spaces the aim of this paper is to discuss the role of hypergeometric functions in function spaces and to prove some new results for these functions the first part of this paper proves results such as monotone convexity and concavity properties of sums of products of hypergeometric functions the second part of our results deals with the space a of all normalized analytic functions f f 0 0 f 0 1 in the unit disk and the subspace r f a a n r such that re ei f z 0 z for f a we consider integral transforms of the type v f 01 t f tz t dt where t is a real valued nonnegative weight function normalized so that 01 t 1 we obtain conditions on and the function such that v f takes each member of r into a starlike function of order 0 1 2 these results extend and improve the earlier known results in these directions we end the paper with an open problem

functional inequalities for the quotients of hypergeometric functions letf a b c x be the gaussian hypergeometric series and for 0 r 1 let formula g d anderson m k vamanamurthy and m vuorinen raised recently the following problem for whicha b 0 1 does formula hold for allr s 0 1 they also proved this inequality fora b 1 2 the main purpose of this paper is to give an answer to this problem fora b 1 and to find a lower bound for the summ r m s fora 0 2 andb 0 2 a

a fast method for computing the principal nth roots of complex matrices abstract based on the generalized continued fraction method for finding the nth roots of real numbers this paper presents a fast computation method for finding the principal nth roots of complex matrices computation algorithms with high convergence rates are developed and their global convergence properties are investigated from the viewpoint of systems theory

a kakeya type problem for circles we prove full hausdorff dimension in a variant of the kakeya problem involving circles in the plane and also sharp estimates for the relevant maximal function these results can also be formulated in terms of the wave equation in two space variables a novelty in our approach is the use of ideas from computational geometry

computation of the principal nth roots of complex matrices a new computational algorithm for finding the principal n th roots and associated roots of complex matrices is developed via a recursive solution of a block discrete state equation

block power method for computing solvents and spectral factors of matrix polynomials abstract this paper is concerned with the extension of the power method used for finding the largest eigenvalue and associated eigenvector of a matrix to its block from for computing the largest block eigenvalue and associated block eigenvector of a non symmetric matrix based on the developed block power method several algorithms are developed for solving the complete set of solvents and spectral factors of a matrix polynomial without prior knowledge of the latent roots of the matrix polynomial moreover when any right left solvent of a matrix polynomial is given the proposed method can be used to determine the corresponding left right solvent such that both right and left solvents have the same eigenspectra the matrix polynomial of interest must have distinct block solvents and a corresponding non singular polynomial matrix the established algorithms can be applied in the analysis and or design of systems described by high degree vector differential equations and or matrix fraction descriptions

pade error estimates for the logarithm of a matrix the error of pade approximations to the logarithm of a matrix and related hypergeometric functions is analysed by obtaining an exact error expansion with positive coefficients it is shown that the error in the matrix approximation at x is always less than the scalar approximation error at x when x x a more detailed analysis involving the interlacing properties of the zeros of the pade denominator polynomials shows that for a given order of approximation the diagonal pade approximants are the most accurate similarly knowing that the denominator zeros must lie in the interval 1 leads to a simple upper bound on the condition number of the matrix denominator polynomial which is a crucial indicator of how accurately the matrix pade approximants can be evaluated numerically in this respect the pade approximants to the logarithm are very well conditioned for x 0 25 this latter condition can be ensured by using the inverse scaling and squaring procedure for evaluating the logarithm

on iterative algorithms for the polar decomposition of a matrix and the matrix sign function in this paper we consider relations between the principal iterations from the pade family of kenney and laub for computing the matrix sign function and the principal iterations from the reciprocal pade family of greco iannazzo and poloni and the dual pade family of zi tak we show global convergence of the principal reciprocal pade iterations and the principal dual pade iterations we adopt the dual pade family of iterations to which the newton method belongs for computing the unitary polar factor of a nonsingular matrix we present numerical experiments with the scaled newton method of higham for computing the unitary polar factor which show how the quality of the computed inverses of matrices in the scaled newton method affects the accuracy of the computed polar factorization of a nonsingular matrix it indicates that assumptions under which the backward stability of the scaled newton method has been proved in the literature cannot be weaker

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

wave front sets local smoothing and bourgain s circular maximal theorem the purpose of this paper is to improve certain known regularity results for the wave equation and to give a simple proof of bourgain s circular maximal theorem 1 we use easy wave front analysis along with techniques previously used in proofs of the carleson sj6lin theorem see 3 5 7 and in the proof of sharp regularity properties of fourier integral operators 13 the circular means operators are defined by

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

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 generalization of bourgain s circular maximal theorem where d r is the normalized surface measure on r s1 it is easy to see that m is not bounded on l2 see example 1 1 below a well known result of bourgain 1 asserts that m is bounded on lp for 2 p we will consider the question of boundedness of m and m from lp to lq unless stated to the contrary we will be dealing only with functions defined on r2 absolute constants will be denoted by c and the notation will mean up to a constant

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

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

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

monotonicity and positivity of coefficients of power series expansions associated with newton and halley methods for the matrix pth root abstract if a is a matrix with all eigenvalues in the disk z 1 1 the principal pth root of a can be computed by newton s method or halley s method the study of newton s method and halley s method for the matrix pth root can be done through a study of power series expansions of some sequences of scalar functions in this paper we prove monotonicity results for the coefficients in the power series expansions for both newton s method and halley s method and for all integer p 2 the sign patterns of these coefficients can be seen directly from the monotonicity results we then use these monotonicity results and their consequences to obtain some new error estimates in the matrix case to obtain a monotonic convergence result when a is a nonsingular m matrix and to obtain a structure preserving result when a is a nonsingular m matrix or a real nonsingular h matrix with positive diagonal entries

on pth roots of stochastic matrices in markov chain models in finance and healthcare a transition matrix over a certain time interval is needed but only a transition matrix over a longer time interval may be available the problem arises of determining a stochastic pth root of a stochastic matrix the given transition matrix by exploiting the theory of functions of matrices we develop results on the existence and characterization of matrix pth roots and in particular on the existence of stochastic pth roots of stochastic matrices our contributions include characterization of when a real matrix has a real pth root a classification of pth roots of a possibly singular matrix a sufficient condition for a pth root of a stochastic matrix to have unit row sums and the identification of two classes of stochastic matrices that have stochastic pth roots for all p we also delineate a wide variety of possible configurations as regards existence nature primary or nonprimary and number of stochastic roots and develop a necessary condition for existence of a stochastic root in terms of the spectrum of the given matrix

maximal operators and differentiation theorems for sparse sets we study maximal averages associated with singular measures on r our main result is a construction of singular cantor type measures supported on sets of hausdorff dimension 1 e with 0 e 1 e 1 e as a consequence we are able to answer a question of aversa and preiss on density and differentiation theorems for singular measures in one dimension our proof combines probabilistic techniques with the methods developed in multidimensional euclidean harmonic analysis in particular there are strong similarities to bourgain s proof of the circular maximal theorem in two dimensions

on hypergeometric functions and generalizations of legendre s relation we prove some convexity properties for a sum of hypergeometric functions and obtain a generalization of legendre s relation for complete elliptic integrals we apply these results to prove some inequalities for hypergeometric functions incomplete beta functions and legendre functions

a bilinear approach to the restriction and kakeya conjectures the purpose of this paper is to investigate bilinear variants of the restriction and kakeya conjectures to relate them to the standard formulations of these conjectures and to give applications of this bilinear approach to existing conjectures the methods used are based on several observations and results of bourgain see 2 6 together with some refinements by moyua vargas and vega 17 18 this paper is organized as follows in the first section we discuss bilinear restriction estimates and show how one can pass back and forth between these estimates and the standard restriction estimates we also generalize the 12 7 bilinear restriction estimate of 18 to higher dimensions in the second section we give analogues of the above results for the kakeya operator in particular we give a bilinear improvement to wolff s kakeya theorem in arbitrary dimension in the third section we give applications of these bilinear estimates in three dimensions for example we are able to improve the 42 11 exponent in wolff s restriction theorem to 34 9 we are also able to prove a sharp lp lq restriction theorem which improves on the classical l2 l4 tomas stein theorem and also give some concrete progress on a bilinear restriction conjecture of klainerman and machedon we also give a non bilinear approach to these estimates which gives weaker results but is more direct and probably has a wider range of application

on prescribing geodesic curvature on d2 abstract some new results are obtained for the problem of prescribing geodesic curvature k on d when k possesses some kinds of symmetries

on the convergence of schroder iteration functions for pth roots of complex numbers in this work a condition on the starting values that guarantees the convergence of the schroder iteration functions of any order to a pth root of a complex number is given convergence results are derived from the properties of the taylor series coefficients of a certain function the theory is illustrated by some computer generated plots of the basins of attraction

regions of convergence of a pade family of iterations for the matrix sector function and the matrix pth root in this paper we prove a conjecture on a common region of a convergence of pade iterations for the matrix sector function for this purpose we show that all pade approximants to a special case of hypergeometric function have a power series expansion with positive coefficients using a sharpened version of schwarz s lemma we also demonstrate a better estimate of the convergence speed our results are also applicable to a family of rational iterations for computing the matrix pth root

explicit convergence regions of newton s method and chebyshev s method for the matrix pth root abstract for a square matrix with all eigenvalues in a suitable region in the complex plane its principal pth root exists and is the limit of the convergent sequence generated by newton s method or chebyshev s method starting from the identity matrix such a region is called a convergence region for newton s method or chebyshev s method in this paper we obtain explicit convergence regions that significantly expand existing ones

elliott s identity and hypergeometric functions elliott s identity involving the gaussian hypergeometric series contains as a special case the classical legendre identity for complete elliptic integrals the aim of this paper is to derive a differentiation formula for an expression involving the gaussian hypergeometric series which for appropriate values of the parameters implies elliott s identity and which also leads to concavity convexity properties of certain related functions we also show that elliott s identity is equivalent to a formula of ramanujan on the differentiation of quotients of hypergeometric functions applying these results we obtain a number of identities associated with the legendre functions of the first and the second kinds respectively

inverse m matrices abstract this is an attempt at a comprehensive expository study of those nonnegative matrices which happen to be inverses of m matrices and is aimed at an audience conversant with basic ideas of matrix theory a theme is the parallels anddifferences between the class of m matrices and the class of inverse m matrices among the primary tools used are diagonal multiplications the neumann expansion and the form of the inverse of a partitioned matrix some of the results seem not to have appeared before and several unsolved problems are mentioned

problemes de neumann non lineaires sur les varietes riemanniennes nonlinear neumann problems on riemannian manifolds let m g be a c compact riemannian manifold of dimension n 2 whose boundary b is an n 1 dimensional submanifold and let m m b be the interior of m study of neumann problems of the form x 0 in m d dn g y 0 on b where for every t x y r m b t x and g t y are bounded by c 1 t a or c exp t a application to the determination of a conformal metric for which the scalar curvature of m and the mean curvature of b take prescribed values

determination of a matrix function using the divided difference method of newton and the interpolation technique of hermite computing a function f a of an n by n matrix a is a frequently occurring problem in control theory and other applications in this paper we introduce an effective approach for the determination of matrix function f a we propose a new technique which is based on the extension of newton divided difference and the interpolation technique of hermite and using the eigenvalues of the given matrix a the new algorithm is tested on several problems to show the efficiency of the presented method finally the application of this method in control theory is highlighted

computation of roots of real and complex matrices an algorithm is presented in this paper by which the rth root of real or complex matrices can be found without the computation of the eigenvalues and eigenvectors of the matrix all required computations are in the real domain the method is based on the newton raphson algorithm and is capable of finding roots even when the matrix is defective computing the root of a matrix from eigenvalues and eigenvectors would be the preferred method if these data were available

on continuum incidence problems related to harmonic analysis abstract we consider averaging operators over curves and surfaces satisfying the rotational curvature condition of phong and stein using combinatorial arguments restricted weak type versions of the standard estimates are obtained under weaker assumptions on the regularity of these surfaces we also consider maximal functions over circles in the plane and obtain a simpler proof of wolff s endpoint result

a geometric proof of the circular maximal theorem is bounded on l r for p 2 simple examples show that this fails for p 2 in this note we derive bourgain s result by geometric and combinatorial methods in particular we do not use the fourier transform in any way our proof is based on a combinatorial argument from 6 which in turn uses marstrand s three circle lemma 7 and a lemma involving two circles that seems to originate in 10 the three circle lemma was used in 7 to prove the following result which is a simple consequence of bourgain s theorem suppose a planar set e has the property for every point in the plane e contains some circle with that point as center then e has positive measure thus we show here that marstrand s lemma in combination with other ideas does indeed allow one to establish the stronger maximal function estimate furthermore we demonstrate in section 4 how to obtain the entire known range of l l estimates for the circular maximal function which is optimal possibly up to endpoints see 9 and 11 by using the methods from sections 2 and 3 one perhaps significant distinction from the techniques developed in 1 8 11 and 12 which involve the fourier transform is the fact that the methods presented here do not seem to yield estimates for the global maximal function mf x sup 0 t

a double mean field equation related to a curvature prescription problem abstract we study a double mean field type pde related to a prescribed curvature problem on compacts surfaces with boundary 0 1 u 2 k e u k e u 1 in u 2 h e u 2 h e u 2 1 on here and are real parameters k h are smooth positive functions on and respectively and is the outward unit normal vector to we provide a general blow up analysis then a moser trudinger inequality which gives energy minimizing solutions for some range of parameters finally we provide existence of min max solutions for a wider range of parameters which is dense in the plane if is not simply connected

fast and stable algorithms for computing the principal square root of a complex matrix this note presents recursive algorithms that are rapidly convergent and more stable for finding the principal square root of a complex matrix also the developed algorithms are utilized to derive the fast and stable matrix sign algorithms which are useful in developing applications to control system problems

a small note on symmetric geodesic curvature on d2 some new results are obtained for the problem of prescribing geodesic curvature k on d when k possesses some kinds of symmetries

a computer aided method for solvents and spectral factors of matrix polynomials a new approach for determining the complete sets of solvents and spectral factors of a monic matrix polynomial is proposed a systematic method for determining the initial guess for the extended multidimensional newton raphson method is first proposed such that the eigenspectrum corresponding to each solvent of the matrix polynomial can be determined with the evaluated eigenspectra complete sets of solvents and spectral factors of a monic matrix polynomial can be obtained by utilizing the applications and the advantages of the principal nth root method the matrix sign function and the block power method the established algorithms can be applied in the analysis and or design of systems described by high degree vector differential equations and or matrix fractions

the dual pade families of iterations for the matrix pth root and the matrix p sector function in this paper we consider the pade family of iterations b laszkiewicz k email protected tak a pade family of iterations for the matrix sector function and the matrix pth root numer linear algebra appl 16 2009 951 970 and a new dual pade family of iterations for computing the principal pth root of a matrix including the newton and halley methods as particular cases we prove convergence of iterations of these families in certain regions we also propose a new dual pade family of iterations for computing the matrix p sector function and we determine a certain region of convergence for this purpose we study properties of the pade approximants to the function 1 z 1 p we show a connection of the series expansion with respect to b of the iterates generated by iterations of the dual pade family for computing the matrix pth root i b 1 p with binomial scalar expansion of 1 b 1 p