on sensitivity of gauss christoffel quadrature in numerical computations the question how much does a function change under perturbations of its arguments is of central importance in this work we investigate sensitivity of gauss christoffel quadrature with respect to small perturbations of the distribution function in numerical quadrature a definite integral is approximated by a finite sum of functional values evaluated at given quadrature nodes and multiplied by given weights consider a sufficiently smooth integrated function uncorrelated with the perturbation of the distribution function then it seems natural that given the same number of function evaluations the difference between the quadrature approximations is of the same order as the difference between the original and perturbed approximated integrals that is perhaps one of the reasons why to our knowledge the sensitivity question has not been formulated and addressed in the literature though several other sensitivity problems motivated in particular by computation of the quadrature nodes and weights from moments have been thoroughly studied by many authors we survey existing particular results and show that even a small perturbation of a distribution function can cause large differences in gauss christoffel quadrature estimates we then discuss conditions under which the gauss christoffel quadrature is insensitive under perturbation of the distribution function present illustrative examples and relate our observations to known conjectures on some sensitivity problems

comment on solutions to quasi relativistic multi configurative hartree fock equations in quantum chemistry by c argaez and m melgaard in a recent paper published in nonlinear analysis theory methods applications argaez and melgaard have studied excited states for pseudo relativistic multi configuration methods their paper follows a previous work of mine in the non relativistic case m lewin solutions of the multiconfiguration equations in quantum chemistry arch ration mech anal 171 1 2004 83 114 doi 10 1007 s00205003 0281 6 the main results of the paper of argaez and melgaard are correct but the proofs are both wrong and incomplete

morse index of some min max critical points i application to multiplicity results on donne un resultat general sur des bornes inferieures pour des indices de morse de points critiques obtenus par des principes de min max en combinant cette information avec une inegalite semiclassique on obtient des estimations pointues sur la croissance de certaines valeurs critiques a partir desquelles on deduit de nouveaux resultats de multiplicite pour des solutions d equations elliptiques semi lineaires d ordre 2

the condition number of real vandermonde krylov and positive definite hankel matrices we show that the euclidean condition number of any positive definite hankel matrix of order n geq 3 may be bounded from below by gamma n 1 16n with gamma exp 4 cdot it catalan pi approx 3 210 and that this bound may be improved at most by a factor 8 gamma n similar estimates are given for the class of real vandermonde matrices the class of row scaled real vandermonde matrices and the class of krylov matrices with hermitian argument improved bounds are derived for the case where the abscissae or eigenvalues are included in a given real interval our findings confirm that all such matrices including for instance the famous hilbert matrix are ill conditioned already for moderate order as application we describe implications of our results for the numerical condition of various tasks in numerical analysis such as polynomial and rational i nterpolation at real nodes determination of real roots of polynomials computation of coefficients of orthogonal polynomials or the iterative solution of linear systems of equations

spectral properties of hankel matrices and numerical solutions of finite moment problems abstract after proving that any hankel matrix generated by moments of positive functions is conditioned essentially the same as the hilbert matrix of the same size we show a preconditioning technique i e a congruence transform of the original hankel matrix that drastically reduces its ill conditioning applications of this result to classical orthogonal polynomial sequences and to modified moment problems are given also we outline an efficient algorithm for the computation of the function f x w x exp p x where w x is positive and p x is a polynomial of degree n 1 from the knowledge of its first n moments

on the calculation of jacobi matrices abstract given a jacobi matrix the problem in question is to find the jacobi matrix corresponding to the weight function modified by a polynomial r galant and gautschi derived algorithms based on the generalized christoffel theorem of uvarov applicable when the roots of r are known in this paper we present two methods not requiring the explicit knowledge of the roots of r we also obtain various properties of the similarity transformations between jacobi matrices which we prove by simple matrix calculus without using the generalized christoffel theorem

a reliable method for computing m pade approximants on arbitrary staircases abstract the m pade approximation problem is defined which contains as a special case the hermite pade approximation as well as the newton pade approximation different well known methods for computing m pade approximations are studied they are based on a normal solution table it is shown that the concept of one of them computing bases of sets induced by interpolation conditions can be generalized to the singular case a characterization of special bases leads to a representation of the m pade solution set like in beckermann this journal 1990 a simple efficient and reliable method for computing such bases by so called transfer matrices is derived finally examples of hermite pade and newton pade approximation are given the connection to other well known reliable algorithms for rational interpolation is discussed

the structure of the singular solution table of the m pade approximation problem abstract a general interpolation problem is defined which contains as special cases the m pade approximation problem as well as classical interpolation problems e g newton pade approximation to study the structure of the solution table a new concept the minimal solution is introduced statements on the local and global structure of the solution table are obtained as a special case the well known block structure of the pade table is derived finally a characterization of the solution set is given

the methods of vorobyev and lanczos abstract this paper deals with the method of moments of vorobyev which is a particular case of galerkin s method this method is generalized in different ways and some of its properties are studied in detail as an application it is showed that the lanczos method for solving systems of linear equations can be derived from it and that some generalizations could be obtained

on the theory and computation of nonperfect pade hermite approximants abstract for a vector of k 1 power series we introduce two new types of rational approximations the weak pade hermite form and the weak pade hermite fraction a recurrence relation is then presented which computes pade hermite forms along with their weak counterparts along a sequence of perfect points in the pade hermite table the recurrence relation results in a fast algorithm for calculating a pade hermite approximant of any given type when the vector of power series is perfect the algorithm is shown to calculate a pade hermite form of type n0 nk in o kn 2 operations where n n0 nk this complexity is the same as that of other fast algorithms the new algorithm also succeeds in the nonperfect case usually with only a moderate increase in cost

higher energy solutions in the theory of phase transitions a variational approach abstract we establish the existence of a higher energy solution to the vector ginzburg landau equation with a triple well potential on a bounded and smooth domain on the plane this solution is obtained by a linking argument in implementing this variational approach we make several considerations on the dynamics of the negative gradient flow in particular we use the conley index to contruct a suitable one dimensional invariant set this solution has morse index two in the nondegenerate case we discuss its structure in connection with the so called triple junction configurations

how bad are hankel matrices considered are hankel vandermonde and krylov basis matrices it is proved that for any real positive definite hankel matrix of order n its spectral condition number is bounded from below by 3 cdot 2 n 6 also proved is that the spectral condition number of a krylov basis matrix is bounded from below by 3 frac 1 2 cdot 2 frac n 2 3 for v v x 1 ldots x n a vandermonde matrix with arbitrary but pairwise distinct nodes x 1 ldots x n we show that mbox rm cond 2 v geq 2 n 2 n frac 1 2 if either x j leq 1 or x j geq 1 for all j then mbox rm cond 2 v geq 2 n 2

algebraic description of the finite stieltjes moment problem abstract the stieltjes problem of moments seeks for a nondecreasing positive distribution function on the semi axis 0 so that its moments match a given infinite sequence of positive real numbers m 0 m 1 in his seminal paper investigations on continued fractions published in 1894 stieltjes gave a complete solution including the conditions for the existence and uniqueness in relation to his main goal the convergence theory of continued fractions one can also reformulate the stieltjes problem of moments as looking for a sequence of positive distribution functions 1 2 where the nth distribution function has n points of increase and m 0 m 1 m 2 n 1 represent its first 2n moments i e as the sequence of the finite stieltjes moment problems this view can be linked to iterative solution of large linear algebraic systems providing that m 0 m 1 are moments of some linear self adjoint and coercive operator a on a hilbert space with respect to a given vector f the finite stieltjes moment problems determine the iterations of the conjugate gradient method applied for solving a u f and vice versa here the existence and uniqueness is guaranteed by the properties of the operator a reformulation for finite sequences matrices and finite vectors is obvious this fundamental link raises a question on how the solution of the finite stieltjes moment problem can be described purely algebraically this has motivated the presented exposition built upon ideas published previously by several authors since the description uses matrices of moments it is not intended for numerical computations

a smooth variational principle with applications to subdifferentiability and to differentiability of convex functions we show that typically lower semicontinuous functions on a banach space densely inherit lower subderivatives of the same degree of smoothness as the norm in particular every continuous convex function on a space with a g teaux weak hadamard frechet smooth renorm is densely g teaux weak hadamard frechet differentiable our technique relies on a more powerful analogue of ekeland s variational principle in which the function is perturbed by a quadratic like function this smooth variational principle has very broad applicability in problems of nonsmooth analysis

continued fractions with applications i introductory examples ii more basics iii convergence criteria iv continued fractions and three term recurrence relations v correspondence of continued fractions vi hypergeometric functions vii moments and orthogonality viii pade approximants ix some applications in number theory x zero free regions xi digital filters and continued fractions xii applications to some differential equations appendix some continued fraction expansions references index

vector orthogonal polynomials abstract the following well known framework will be applied to hermite pade approximants first the hermite pade type approximants will be defined from the error of approximation some supplementary conditions will be fixed in order to obtain hermite pade forms vector orthogonal polynomials will be deduced from these conditions their existence and uniqueness are studied in the normal case these results are ensured then some properties of vector orthogonal polynomials will be proved

approximants de pade hermite 2eme partie programmation in this second paper we propose an algorithm for the construction of the tables and of the pade hermite tables 3 then we present some numerical applications

abstract criteria for multiple solutions to nonlinear coupled equations involving magnetic schrodinger operators we consider a system of nonlinear coupled equations involving magnetic schrodinger operators and general potentials we provide the criteria for the existence of multiple solutions to these equations as special cases we get the classical results on existence of infinitely many distinct solutions within hartree and hartree fock theory of atoms and molecules subject to an external magnetic fields we also extend recent results within this theory including coulomb system with a constant magnetic field a decreasing magnetic field and a physically measurable magnetic field

the impact of stieltjes work on continued fractions and orthogonal polynomials additional material abstract in the recent new edition of the collected works of t j stieltjes one of us gave an impression of the impact of stieltjes work a century after his death 43 in this paper we give an update and mention some observations which were missing from 43 and some results which appeared during the last two years and which are directly related to stieltjes work in particular there is a large section on special polynomials which were already considered by stieltjes but which were rediscovered later often without the knowledge of stieltjes work

approximants de pade hermite 1ere partie theorie we present in this first paper a generalization of pade approximants which gives us as particular cases shafer s and baker sd log approximants first we define these approximants following an old idea of hermite then we prove some fundamental properties for their constructions

hermite pade approximation basic notions and theorems abstract our purpose is to give a brief exposition of basic notions and facts on hermite pade approximation theory a hermite pade form hpf is defined here in the same way as a pade approximant is as a consequence in some cases hpf does not exist even in the modified version hpf need not be unique therefore it was worthwhile to choose some hpf as the optimal one all criteria of existence unicity etc are phrased in a unified manner they base on few quantities depending on the power series under consideration and are usable in the theory rather than in the numerical practice

vector orthogonal polynomials we give the relation between hermite pade approximants and vector orthogonal polynomials an algorithm for calculating vector orthogonal polynomials near the diagonal is described an exact multiple integral formula for vector orthogonal polynomials is proved an example is given showing how the recurrence relations of pade may be used to calculate hermite pade approximants of degrees m mu mu with mu increasing

asymptotic behavior of radial solutions for a semilinear elliptic problem on an annulus through morse index abstract we study the asymptotic behavior of radial solutions for a singularly perturbed semilinear elliptic dirichlet problem on an annulus we show that morse index informations on such solutions provide a complete description of the blow up behavior as a by product we exhibit some sufficient conditions to guarantee that radial ground state solutions blow up and concentrate at the inner outer boundary of the annulus

b splines and hermite pade approximants to the exponential function this paper is the continuation of a work initiated in p sablonniere an algorithm for the computation of hermite pade approximations to the exponential function divided differences and hermite pade forms numer algorithms 33 2003 443 452 about the computation of hermite pade forms hpf and associated hermite pade approximants hpa to the exponential function we present an alternative algorithm for their computation based on the representation of hpf in terms of integral remainders with b splines as peano kernels using the good properties of discrete b splines this algorithm gives rise to a great variety of representations of hpf of higher orders in terms of hpf of lower orders and in particular of classical pade forms we give some examples illustrating this algorithm in particular another way of constructing quadratic hpf already described by different authors finally we briefly study a family of cubic hpf

morse type information on palais smale sequences obtained by min max principles in the context of the min max approach to critical point theory but without the usual compactness assumptions a la palais smale and the nondegeneracy conditions a la fredholm we construct almost critical points with two sided estimates on their approximate morse indices which are arbitrarily close to certain a priori given dual sets this additional topological and analytical information about almost critical sequences can sometimes be crucial in the proof of their convergence and therefore in solving the corresponding variational problem 1994 john wiley sons inc

how to choose modified moments modified moments of a nonnegative measure determine uniquely the recurrence coefficients of the polynomials orthogonal with respect to the given nonnegative measure the sensitivity of the underlying nonlinear maps with regard to perturbations in the modified moments has been studied by gautschi and fischer and is further analyzed in this paper in particular we discuss the question of how not to choose the support of the underlying two measures