vibrations and elastic stability of thin circular plates with variable profile abstract circular plates of thicknesses varying to the functional relationho 1 r a n wherenis a positive integer are studied in the present paper uniform elastic boundary restraints are considered and the first two natural frequency coefficients corresponding to axisymmetric modes and the buckling in plane pressure are determined using simple polynomial co ordinate functions which identically satisfy the boundary conditions and the classical rayleigh ritz method an independent solution is also obtained by means of the differential quadrature technique

three dimensional vibration analysis of circular and annular plates via the chebyshev ritz method abstract a three dimensional free vibration analysis of circular and annular plates is presented via the chebyshev ritz method the solution procedure is based on the linear small strain three dimensional elasticity theory selecting chebyshev polynomials which can be expressed in terms of cosine functions as the admissible functions a convenient governing eigenvalue equation can be derived through the ritz method according to the geometric properties of circular and annular plates the vibration is divided into three distinct categories axisymmetric vibration torsional vibration and circumferential vibration each vibration category can be further subdivided into the antisymmetric and symmetric ones in the thickness direction convergence and comparison study demonstrated the high accuracy and efficiency of the present method the present approach shows a distinct advantage over some other ritz solutions in that stable numerical operation can be guaranteed even when a large number of admissible functions is employed therefore not only lower order but also higher order eigenfrequencies can be obtained by using sufficient terms of the chebyshev polynomials finally some valuable results for annular plates with one or both edges clamped are given and discussed in detail

transverse vibrations and stability of polar orthotropic circular plates high level relationships abstract a set of approximate high level relationships are presented for the determination of frequency and buckling parameters of polar orthotropic circular plates with edges elastically restrained against rotation the relationships are abstracted from results obtained by the ritz method and are applicable for both design and sensitivity analysis the series solution for the governing differential equations are extended to include elastically restrained boundary conditions and are used to establish the accuracy of these relationships and to resolve the inconsistencies in previously published results

on the rediscovery of halley s iterative method for computing the zero of an analytic function abstract we show that halley s basic sequence resulting from accelerating the order of convergence of newton s method is the most efficient way of doing so in terms of usage of certain derivatives this fact could explain why this process of accelerating the convergence of newton s method is so frequently rediscovered then we present an algorithmic way of recognizing halley s family and we apply this algorithm to examples of rediscoveries

on a family of symmetric polynomials abstract in this paper we provide two simple approaches to the explicit expression of a family of symmetric polynomials introduced and studied in milovanovic and cvetkovic j math anal appl 311 2005 191 thereby improving on their observations

hydroelastic vibration of two identical rectangular plates a theoretical study is presented on the hydroelastic vibration of two identical rectangular plates coupled with a bounded fluid it is assumed that the plates are clamped along the plate edges and an ideal fluid is surrounded by the two rectangular plates and a rigid rectangular container the velocity potential satisfying the fluid boundary conditions is expanded in terms of a finite fourier series and the modal displacements of the plates are also expanded by the finite fourier series for the compatibility requirement along the contacting surface between the plates and the fluid two transverse vibration modes in phase and out of phase are observed in the symmetric fluid coupled structure each in phase mode is assumed as a combination of the beam modes in air but every out of phase mode is assumed as a combination of polynomials satisfying the plate boundary condition and fluid volume conservation the coupled natural frequencies are obtained from the relationship between the reference kinetic energy of the structure including the fluid and the maximum strain energy of the two plates the proposed analytical method was found to be in good agreement with the results of a three dimensional finite element analysis

accelerated convergence in newton s method newton s method is based on a linear approximation of the function whose roots are to be determined taken at the current point and the resulting algorithm is known to converge quadratically in a procedure to increase the rate of convergence the author modifies the target function in such a way that newton s method applied to the modified function will yield a faster rate of convergence

n th root extraction double iteration process and newton s method a double iteration process already used to find the nth root of a positive real number is analysed and showed to be equivalent to the newton s method these methods are of order two and three higher order methods for finding the nth root are also mentioned

inverse shifted inverse and rayleigh quotient iteration as newton s method the l 2 normalized inverse shifted inverse and rayleigh quotient iterations are classic algorithms for approximating an eigenvector of a symmetric matrix this work establishes rigorously that each iterate produced by one of these three algorithms can be viewed as a newton s method iterate followed by a normalization the equivalences given here are not meant to suggest changes to the implementations of the classic eigenvalue algorithms however they add further understanding to the formal structure of these iterations and they provide an explanation for their good behavior despite the possible need to solve systems with nearly singular coefficient matrices a historical development of these eigenvalue algorithms is presented using our equivalences and traditional newton s method theory helps to gain understanding as to why normalized newton s method inverse iteration and shifted inverse iteration are only linearly convergent and not quadratically convergent as would be expected and why a new l

variants of newton s method using fifth order quadrature formulas abstract some variants of newton s method are developed in this work in order to solve nonlinear equations depending on one or several variables based in rules of quadrature of fifth order we prove the third or fifth order of convergence of these methods for dimension one and the second or third order in several variables depending on the behaviour of the second derivative moreover different numeric tests confirm or improve theoretic results and allow us to compare these variants with newton s classical method

newton s method and high order algorithms for the nth root computation two modifications of newton s method to accelerate the convergence of the nth root computation of a strictly positive real number are revisited both modifications lead to methods with prefixed order of convergence p n p 2 we consider affine combinations of the two modified pth order methods which lead to a family of methods of order p with arbitrarily small asymptotic constants moreover the methods are of order p 1 for some specific values of a parameter then we consider affine combinations of the three methods of order p 1 to get methods of order p 1 again with arbitrarily small asymptotic constants the methods can be of order p 2 with arbitrarily small asymptotic constants and also of order p 3 for some specific values of the parameters of the affine combination it is shown that infinitely many pth order methods exist for the nth root computation of a strictly positive real number for any p 3

on rediscovered methods for solving equations abstract in the recent papers of gerlach siam rev 36 1994 272 276 and ford and pennline siam rev 38 1996 658 659 a class of iterative methods for solving a single equation f x 0 with arbitrary rate of convergence has been presented in this paper we show that this class is equivalent to five other classes of iterative methods derived earlier in various ways and expressed in different forms the proofs of equivalence of all considered iteration formulas are given

historical development of the newton raphson method this expository paper traces the development of the newton raphson method for solving nonlinear algebraic equations through the extant notes letters and publications of isaac newton joseph raphson and thomas simpson it is shown how newton s formulation differed from the iterative process of raphson and that simpson was the first to give a general formulation in terms of fluxional calculus applicable to nonpolynomial equations simpson s extension of the method to systems of equations is exhibited

newton s method s basins of attraction revisited in this paper we revisit the chaotic number of iterations needed by newton s method to converge to a root here we consider a simple modified newton method depending on a parameter it is demonstrated using polynomiography that even in the simple algorithm the presence and the position of the convergent regions i e regions where the method converges nicely to a root can be complicatedly a function of the parameter

on the use of orthogonal polynomials in the rayleigh ritz method for the study of the flexural vibration and buckling of isotropic and orthotropic rectangular plates series composed of recently proposed orthogonal polynomial functions are used in the rayleigh ritz method to generate results for a number of flexural vibration and buckling problems for rectangular isotropic and orthotropic plates it is demonstrated that the approach yields excellent results not only for natural frequencies and buckling loads but also for the more sensitive mode shape bending moment and shear force distribution problem

relationships between order and efficiency of a class of methods for multiple zeros of polynomials the behavior of a class of high order methods for solving polynomial equations is examined it is shown that the number of iterations for local convergence to a multiple zero to the limits of attainable accuracy is bounded independent of the multiplicity of the zero and decreases as the order of the method increases for the higher order methods the number of iterations decreases as the multiplicity increases computational efficiency as a function of degree order and multiplicity is investigated and an effective choice of order is recommended numerical examples are provided

hydroelastic vibration of a circular plate submerged in a bounded compressible fluid abstract an analytical method for the linear free vibration of a circular plate submerged in a fluid was developed by the rayleigh ritz method based on the fourier bessel series expansion it is assumed that the plate is clamped at an offcenter location of a rigid cylindrical container and the fluid bisected by the plate in the container is non viscous and compressible since a combination of the dry modal functions of the circular plate can approximate the wet vibration modes the functions were used to form a set of linearly independent functions of the rayleigh ritz method it was found that the theoretical results could predict well the fluid coupled natural frequencies with excellent accuracy compared with the finite element analysis results the effects of the fluid compressibility and the offcenter distance on the natural frequencies were also observed

three dimensional vibration analysis of rectangular thick plates on pasternak foundation the free vibration characteristics of rectangular thick plates resting on elastic foundations have been studied based on the three dimensional linear and small strain elasticity theory the foundation is described by the pasternak two parameter model the ritz method is used to derive the eigenvalue equation of the rectangular plate by augmenting the strain energy of the plate with the potential energy of the elastic foundation the chebyshev polynomials multiplied by a boundary function are selected as the admissible functions of the displacement functions in each direction the approach is suitable for rectangular plates with arbitrary boundary conditions convergence and comparison studies have been performed on square plates on elastic foundations with different boundary conditions it is shown that the present method has a rapid convergent rate stable numerical operation and very high accuracy parametric investigations on the dynamic behaviour of clamped square thick plates on elastic foundations have been carried out in detail with respect to different thickness span ratios and foundation parameters some results found for the first time have been given and some important conclusions have been drawn

three dimensional vibration analysis of thick rectangular plates using chebyshev polynomial and ritz method this paper describes a method for free vibration analysis of rectangular plates with any thicknesses which range from thin moderately thick to very thick plates it utilises admissible functions comprising the chebyshev polynomials multiplied by a boundary function the analysis is based on a linear small strain three dimensional elasticity theory the proposed technique yields very accurate natural frequencies and mode shapes of rectangular plates with arbitrary boundary conditions a very simple and general programme has been compiled for the purpose for a plate with geometric symmetry the vibration modes can be classified into symmetric and antisymmetric ones in that direction in such a case the computational cost can be greatly reduced while maintaining the same level of accuracy convergence studies and comparison have been carried out taking square plates with four simply supported edges as examples it is shown that the present method enables rapid convergence stable numerical operation and very high computational accuracy parametric investigations on the vibration behaviour of rectangular plates with four clamped edges have also been performed in detail with respect to different thickness side ratios aspect ratios and poissons ratios these results may serve as benchmark solutions for validating approximate two dimensional theories and new computational techniques in future 2002 elsevier science ltd all rights reserved

on the inversion of infinite moment matrices abstract motivated by 8 we study the existence of the inverse of an infinite hermitian positive definite matrix in short hpd matrix from the point of view of the asymptotic behaviour of the smallest eigenvalues of the finite sections we prove a sufficient condition to assure the inversion of an hpd matrix with square summable rows for infinite toeplitz matrices we introduce the notion of asymptotic toeplitz matrix and we show that under certain assumptions the inverse of an infinite toeplitz positive definite matrix is asymptotic toeplitz such inverses are computed in terms of the limits of the coefficients of the associated orthogonal polynomials we apply these results in the context of the theory of orthogonal polynomials in particular we show that for measures on the unit circle t verifying that the smallest eigenvalue of the finite sections of the corresponding moment matrix are away from zero in the limit we may assure the existence of all the limits of the coefficients of the orthonormal polynomials with respect to such measures

fast switch and spline scheme for accurate inversion of nonlinear functions the new first choice solution to kepler s equation abstract numerically obtaining the inverse of a function is a common task for many scientific problems often solved using a newton iteration method here we describe an alternative scheme based on switching variables followed by spline interpolation which can be applied to monotonic functions under very general conditions to optimize the algorithm we designed a specific ultra fast spline routine we also derive analytically the theoretical errors of the method and test it on examples that are of interest in physics in particular we compute the real branch of lambert s w y function which is defined as the inverse of xexp x and we solve kepler s equation in all cases our predictions for the theoretical errors are in excellent agreement with our numerical results and are smaller than what could be expected from the general error analysis of spline interpolation by many orders of magnitude namely by an astonishing 3 10 22 factor for the computation of w in the range w y 0 10 and by a factor 2 10 4 for kepler s problem in our tests this scheme is much faster than newton raphson method by a factor in the range 10 4 10 3 for the execution time in the examples when the values of the inverse function over an entire interval or for a large number of points are requested for kepler s equation and tolerance 10 6 rad the algorithm outperforms newton s method for all values of the number of points n 2

rotation invariant moment problems an important theorem of marcel riesz cf 14 states that the polynomials are dense in l2 x when x is a determinate measure on the real line in the indeterminate case riesz also characterized the measures x for which the polynomials are dense in l2 they are the so called nevanlinna extremal measures introduced in nevanlinna 11 it does not seem to be known whether the polynomials are dense in l2 g when x is a determinate measure on r d d 1 cf the expository paper by fuglede 7 as well as the research problems book 8 p 529 where devinatz poses the problem as question 1 and ascribes it to the physicist john challifour 1978 in this paper we shall settle the question in the negative there exist rotation invariant measures on r d d 1 which are determinate but for which the polynomials are not dense in l2 p such measures x are necessarily of the following very special form

kepler equation and accelerated newton method we study the efficiency of the accelerated newton method garlach siam rev 36 1994 272 276 for several orders of convergence versus danby s method for the resolution of kepler s equation we find that the cited method of order three is competitive with danby s method and the classical newton s method we also generalize the accelerated newton method for the resolution of system of algebraic equations obtaining a formula of order three and a proof of its convergence its application to several examples shows that its efficiency is greater than newton s method

the index of determinacy for measures and the norm of orthonormal polynomials for determinate measures having moments of every order we define and study an index of determinacy which checks the stability of determinacy under multiplication by even powers of it z for z a complex number using this index of determinacy we solve the problem of determining for which z e c the sequence p m z n m e n belongs to t2 where pn n is the sequence of orthonormal polynomials associated with the measure u

dynamics of a rectangular plate resting on an elastic foundation and partially in contact with a quiescent fluid abstract in this study a method of analysis is presented for investigating the effects of elastic foundation and fluid on the dynamic response characteristics natural frequencies and associated mode shapes of rectangular kirchhoff plates for the interaction of the kirchhoff plate pasternak foundation a mixed type finite element formulation is employed by using the g teaux differential the plate finite element adopted in this study is quadrilateral and isoparametric having four corner nodes and at each node four degrees of freedom are present one transverse displacement two bending moments and one torsional moment therefore a total number of 16 degrees of freedom are assigned to each element a consistent mass formulation is used for the eigenvalue solution in the mixed finite element analysis the plate structure considered is assumed clamped or simply supported along its edges and resting on a pasternak foundation furthermore the plate is fully or partially in contact with fresh water on its one side for the calculation of the fluid structure interaction effects generalized fluid structure interaction forces a boundary element method is adopted together with the method of images in order to impose an appropriate boundary condition on the fluid s free surface it is assumed that the fluid is ideal i e inviscid incompressible and its motion is irrotational it is also assumed that the plate elastic foundation system vibrates in its in vacuo eigenmodes when it is in contact with fluid and that each mode gives rise to a corresponding surface pressure distribution on the wetted surface of the structure at the fluid structure interface continuity considerations require that the normal velocity of the fluid is equal to that of the structure the normal velocities on the wetted surface of the structure are expressed in terms of the modal structural displacements obtained from the finite element analysis by using the boundary integral equation method the fluid pressure is eliminated from the problem and the fluid structure interaction forces are calculated in terms of the generalized hydrodynamic added mass coefficients due to the inertial effect of fluid to asses the influences of the elastic foundation and fluid on the dynamic behavior of the plate structure the natural frequencies and associated mode shapes are presented furthermore the influence of the submerging depth on the dynamic behavior is also investigated

empirical versus asymptotic rate of convergence of a class of methods for solving a polynomial equation abstract given alternative methods with identical order of convergence for solving the polynomial equation z 0 the method with the smaller asymptotic error constant might be assumed to be superior in terms of the number of iterations required for convergence we present empirical evidence for a parameterized class of methods of second order showing that a parameter choice which does not correspond to the minimal asymptotic error constant may nevertheless be superior in practice

dynamic responses of reissner mindlin plates with free edges resting on tensionless elastic foundations abstract dynamic response analysis is presented for a reissner mindlin plate with four free edges resting on a tensionless elastic foundation of the winkler type and pasternak type the mechanical loads consist of transverse partially distributed impulsive loads and in plane static edge loads while the temperature field is assumed to exhibit a linear variation through the thickness of the plate the material properties are assumed to be independent of temperature the two cases of initially compressed plates and of initially heated plates are considered the formulations are based on reissner mindlin first order shear deformation plate theory and include the plate foundation interaction and thermal effects a set of admissible functions is developed for the dynamic response analysis of moderately thick plates with four free edges the galerkin method the gauss legendre quadrature procedure and the runge kutta technique are employed in conjunction with this set of admissible functions to determine the deflection time and bending moment time curves as well as shape mode curves an iterative scheme is developed to obtain numerical results without using any assumption on the shape of the contact region the numerical illustrations concern moderately thick plates with four free edges resting on tensionless elastic foundations of the winkler type and pasternak type from which results for conventional elastic foundations are obtained as comparators the results confirm that the plate will have stronger dynamic behavior than its counterpart when it is supported by a tensionless elastic foundation

finite element analysis of three dimensional vibrations of thick circular and annular plates abstract three dimensional vibrations of thick circular and annular plates are analyzed by a finite element method which with a properly assumed set of displacement field is different from the traditional 3 d finite element analysis and is reduced to a sequence of 2 d analyses one for each circumferential wave number the present approach has several unique features 1 it can obtain vibration frequencies which are comparable to or as accurate as those by other three dimensional approaches whenever comparisons between them are possible and yet the formulation of the present method is simpler and its application is straightforward 2 different simply supported boundary conditions can be exactly imposed with the present approach the latter feature may be quite difficult if not impossible for other methods in the present analysis vibration frequencies of a circular and an annular plate under different combinations of boundary conditions wave numbers and finite element meshes are calculated vibration characteristics are examined and some typical mode shapes are shown graphically then the results of some example problems by the present method are compared to those available in the literature to show its validity and accuracy

a survey on the high convergence orders and computational convergence orders of sequences abstract twenty years after the classical book of ortega and rheinboldt was published five definitions for the q convergence orders of sequences were independently and rigorously studied i e some orders characterized in terms of others by potra 1989 resp beyer ebanks and qualls 1990 the relationship between all the five definitions only partially analyzed in each of the two papers was not subsequently followed and moreover the second paper slept from the readers attention the main aim of this paper is to provide a rigorous selfcontained and as much as possible a comprehensive picture of the theoretical aspects of this topic as the current literature has taken away the credit from authors who obtained important results long ago moreover this paper provides rigorous support for the numerical examples recently presented in an increasing number of papers where the authors check the convergence orders of different iterative methods for solving nonlinear systems of equations tight connections between some asymptotic quantities defined by theoretical and computational elements are shown to hold

the forced vibration of a thin plate floating on an infinite liquid abstract in this paper a solution is presented for the harmonically forced vibration of an arbitrary thin plate floating on the surface of an infinite liquid the full linear potential problem for the liquid is solved by the use of the appropriate green s function a variational equation which the plate liquid system must satisfy is derived and a solution by the rayleigh ritz method is presented examples of possible calculations are given for a square and a rectangular plate

orthogonal polynomials and analytic functions associated to positive definite matrices abstract for a positive definite infinite matrix a we study the relationship between its associated sequence of orthonormal polynomials and the asymptotic behaviour of the smallest eigenvalue of its truncation a n of size n n for the particular case of a being a hankel or a hankel block matrix our results lead to a characterization of positive measures with finite index of determinacy and of completely indeterminate matrix moment problems respectively

a class of iterative methods with third order convergence to solve nonlinear equations algebraic and differential equations generally co build mathematical models either lack or intractability of their analytical solution often forces workers to resort to an iterative method and face the likely challenges of slow convergence non convergence or even divergence this manuscript presents a novel class of third order iterative techniques in the form of x k 1 g u x k x k f x k u x k to solve a nonlinear equation f with the aid of a weight function u the class currently contains an invert and average g k i a an average and invert g k a i and an invert and exponentiate g k e branch each branch has several members some of which embed second order newton s g n third order chebychev s g c or halley s g h solvers class members surpassed stand alone applications of these three well known methods other methods are also permitted as auxiliaries provided they are at least of second order asymptotic convergence constants are calculated assignment of class parameters to non members carries them to a common basis for comparison this research also generated a one step solver that is usable for post priori analysis trouble shooting and comparison

a termination criterion for iterative methods used to find the zeros of polynomials a new criterion for terminating iterations when searching for polynomial zeros is described this method does not depend on the number of digits in the mantissa moreover it can be used to determine the accuracy of the resulting zeros examples are included

pseudocomposition a technique to design predictor corrector methods for systems of nonlinear equations abstract a new technique for designing iterative methods for solving nonlinear systems is presented this procedure called pseudocomposition uses a known method as a predictor and the gaussian quadrature as a corrector the order of convergence of the resulting scheme depends among other factors on the order of the last two steps of the predictor we also introduce a new iterative algorithm of order six and apply the mentioned technique to generate a new method of order ten finally some numerical test are shown

a basic family of iteration functions for polynomial root finding and its characterizations abstract let p x be a polynomial of degree n 2 with coefficients in a subfield k of the complex numbers for each natural number m 2 let lm x be the m m lower triangular matrix whose diagonal entries are p x and for each j 1 m 1 its jth subdiagonal entries are p j x j for i 1 2 let lmi x be the matrix obtained from lm x by deleting its first i rows and its last i columns l1 1 x 1 then the function bm x x p x det l m 1 1 x det l m 1 x defines a fixed point iteration function having mth order convergence rate for simple roots of p x for m 2 and 3 bm x coincides with newton s and halley s respectively the function bm x is a member of s m m n 2 where for any m m s m m is the set of all rational iteration functions g x k x such that for all roots of p x then g x i mm i x x i with i x k x and well defined at any simple root given g s m m and a simple root of p x gi 0 i 1 m 1 and the asymptotic constant of convergence of the corresponding fixed point iteration is m 1 g m m for bm x we obtain m 1 m det l m 1 2 det l m 1 if all roots of p x are simple bm x is the unique member of s m m n 2 by making use of the identity 0 i 0 n p i x i x i we arrive at two recursive formulas for constructing iteration functions within the s m m family in particular the family of bm x can be generated using one of these formulas moreover the other formula gives a simple scheme for constructing a family of iteration functions credited to euler as well as schroder whose mth order member belongs to s m mn m 2 the iteration functions within s m m can be extended to any arbitrary smooth function f with the uniform replacement of p j with f j in g as well as in m

a one parameter family of locally quartically convergent zero finding methods a one parameter family of iteration functions for finding simple and multiple zeros of analytic functions is derived the family includes as a special case traub s quartic square root method and as limiting cases the kiss method of order 4 the halley and the newton methods all the methods of the family are locally quartically convergent for a simple or multiple zero with known multiplicity the asymptotic error constants for the methods of the family are given the decreasing ratio at infinity of iteration functions is discussed the optimum parameter of the family for polynomials is given

on the closed representation for the inverses of hessenberg matrices the general representation for the elements of the inverse of any hessenberg matrix of finite order is here extended to the reduced case with a new proof those entries are given with proper hessenbergians from the original matrix it justifies both the use of linear recurrences of unbounded order for such computations on matrices of intermediate order and some elementary properties of the inverse these results are applied on the resolvent matrix associated to a finite hessenberg matrix in standard form two examples on the unit disk are given

fixed point and newton s methods for solving a nonlinear equation from linear to high order convergence in this paper we revisit the necessary and sufficient conditions for linear and high order convergence of fixed point and newton s methods based on these conditions we extend schroder s process of the first kind to increase the order of convergence of the fixed point method we also obtain two processes to increase the order of convergence of newton s method one of which is schroder s process of the second kind for which several forms are also presented a link between schroder s two processes is given examples and numerical experiments are included

third order methods from quadrature formulae for solving systems of nonlinear equations we extend to p dimensional problems a modification of the newton method based on quadrature formulas of order at least one which produces iterative methods with order of convergence three a general error analysis providing the higher order of convergence is given these new methods may be more efficient then other third order methods as they do not require the use of the second order frechet derivative

accurate computation of weights in classical gauss christoffel quadrature rules for many classical gauss christoffel quadrature rules there does not exist a method which guarantees a uniform level of accuracy for the gaussian quadrature weights at all quadrature nodes unless the nodes are known exactly more disturbing some algebraic expressions for these weights exhibit an excessive sensitivity to even the smallest perturbations in the node location this sensitivity rapidly increases with high order quadrature rules current uses of very high order quadratures are common with the advent of more powerful computers and a loss of accuracy in the weights has become a problem and must be addressed a simple but efficient and general method for improving the accuracy of the computation of the quadrature weights is proposed it ensures a high level of accuracy for these weights even though the nodes may carry a significant large error in addition a highly efficient root finding iterative technique with superlinear converging rates for computing the nodes is developed it uses solely the quadrature polynomials and their first derivatives a comparison of this method with the eigenvalue method of golub and welsh implemented in most standard software libraries is made the proposed method outperforms the latter from the point of view of both accuracy and efficiency the legendre lobatto radau hermite and laguerre quadrature rules are examined

free vibration of polar orthotropic circular plates of variable thickness with elastically restrained edge abstract asymmetric vibrations of polar orthotropic circular plates of linearly varying thickness with elastic rigid support are discussed on the basis of the classical plate theory an approximate solution of the problem is obtained by the rayleigh ritz method using functions based on static deflection of polar orthotropic circular plates this type of approximating functions has a faster rate of convergence as compared to the polynomial co ordinate functions frequency parameters of the plate have been obtained for different values of taper parameter flexibility parameter and rigidity ratio in the first three modes of vibration a comparison of results for special cases of isotropy uniform thickness and of axisymmetric vibrations obtained by other techniques such as the finite element method and the receptance method shows excellent agreement

hydroelastic analysis of fluid storage tanks by using a boundary integral equation method abstract this paper presents a boundary integral equation method in conjunction with the method of images in order to investigate the dynamic behaviour wet frequencies and associated mode shapes of fluid containing structures in the analysis of the linear fluid structure system it is assumed that the fluid is ideal and fluid forces are associated with inertial effects of the contained fluid this implies that the fluid pressure on the wetted surface of the structure is in phase with the structural acceleration the in vacuo dynamic properties of the dry structure are obtained by using standard finite element software in the wet part of the analysis the fluid structure interaction effects are calculated in terms of the generalized added mass coefficients by use of the boundary integral equation method together with the method of images in order to impose the 0 boundary condition on the free surface in this study three different test cases are considered i a clamped free cylindrical tank with a rigid bottom ii a flexible circular plate in a rigid cylindrical tank and iii a flexible cylindrical shell with flexible end plates hermetic can the fluid storage tanks in this investigation are assumed partially or completely filled with water to assess the influence of the contained fluid on the dynamic characteristics the wet natural frequencies and associated mode shapes are calculated the predictions compare well with available experimental and numerical data

a comparison of methods for terminating polynomial iterations abstract three methods of terminating polynomial root finding iterations are compared one based on explicit calculation of rounding errors one based on differences in the iterates and one based on different methods of calculating the polynomial in extensive experiments with randomly generated polynomials it was found that the simplest method based on differences in the iterates usually gives the lowest actual error in the root

symmetries of the julia sets of konig s methods for polynomials abstract let r be the group of symmetries of the julia set of a rational map r and k f n be the konig s method for polynomial f of order n 2 for any given integer n 2 we prove that if f is in normal form then f is a subgroup of k f n we also obtain a necessary and sufficient condition for the julia set of k f n to be a line

free vibration of a partially liquid filled and submerged horizontal cylindrical shell abstract the dynamic characteristics i e natural frequencies and mode shapes of a partially filled and or submerged horizontal cylindrical shell are examined in this investigation it is assumed that the fluid is ideal and fluid forces are associated with inertial effects only namely the fluid pressure on the wetted surface of the structure is in phase with the structural acceleration the in vacuo dynamic characteristics of the cylindrical shell are obtained using standard finite element software in the wet part of the analysis it is assumed that the shell structure preserves its in vacuo mode shapes when in contact with the contained and or surrounding fluid and that each mode shape gives rise to a corresponding surface pressure distribution of the shell the fluid structure interaction effects are calculated in terms of generalized added masses using a boundary integral equation method together with the method of images in order to impose an appropriate boundary condition on the free surface to assess the influence of the contained and or surrounding fluid on the dynamic behaviour of the shell structure the wet natural frequencies and associated mode shapes were calculated and compared with available experimental measurements

newton s method and generation of a determinantal family of iteration functions abstract it is well known that halley s method can be obtained by applying newton s method to the function f f gerlach siam rev 36 1994 272 276 gives a generalization of this approach and for each m 2 recursively defines an iteration function gm x having order m kalantari et al j comput appl math 80 1997 209 226 and kalantari technical report dcs tr 328 department of computer science rutgers university new brunswick nj 1997 derive and characterize a determinantal family of iteration functions called the basic family bm x m 2 in this paper we prove gm x bm x on the one hand this implies that gm x enjoys the previously derived properties of bm x i e the closed formula its efficient computation an expansion formula which gives its precise asymptotic constant as well as its multipoint versions on the other this gives a new insight on the basic family and newton s method

effect of fluid depth on the hydroelastic vibration of free edge circular plate abstract this paper is concerned with the effect of water depth on the free vibration of free edge circular plates resting on a free fluid surface the problem addressed is formulated by using the hankel transformation method which leads to dual integral equations the solution of dual integral equations is solved numerically by using fourier bessel series the fluid is assumed to be inviscid and incompressible the kirchhoff theory of plates is used to model the elastic thin plate numerical results are given in non dimensional form for free edge circular plates in order to be ready to use in applications to validate the theoretical results experiments were carried out experimental results are in good agreement with theoretical results it is found that the effect of the fluid depth can be neglected when the fluid depth is greater than the diameter of the circular plates but becomes significant as the water depth decreases it is also observed in the experimental results that the fluid damping increases as the water depth decreases

buckling and vibrations of polar orthotropic annular plates on elastic foundation subjected to hydrostatic peripheral loading an analysis is presented of the effect of hydrostatic peripheral loading and that of an elastic foundation on the natural frequencies of free axisymmetric vibrations and critical buckling parameter of linearly varying thickness annular plates exhibiting polar orthotropic characteristics on the basis of classical plate theory the numerical solution of the governing differential equation derived by using hamilton s energy principle is obtained by the spline technique method for three different boundary conditions inner and outer both clamped inner clamped and outer simply supported inner free and outer clamped transverse displacements moments in normalized form for a particular set of plate parameters and buckling loads in compression for different values of taper constant foundation modulus and rigidity ratios have been computed for the first two modes of vibration the numerical results are compared with exact values and with the results of other authors the close agreement of the values illustrates the versatility of this simple and elegant method provided the number of nodal points is taken appropriately

on general convergence in extracting radicals via a fundamental family of iteration functions newton s method is well known to be generally convergent for solving x n c 0 in this paper we first extend this result to the next two members of an infinite family of high order methods referred to here as the basic family which starts with newton s method while computing roots of unity numerically is a trivial task studying the general convergence of the basic family in this simple case is an important first step toward the understanding of the global behavior of this fundamental family with the aid of polynomiography techniques for the visualization of polynomial root finding we further conjecture the general convergence of all members of the basic family when extracting radicals using the computer algebra system maple we obtain some partial results toward the proof of our conjecture

small eigenvalues of large hermitian moment matrices abstract we consider an infinite hermitian positive definite matrix m which is the moment matrix associated with a measure with infinite and compact support on the complex plane we prove that if the polynomials are dense in l 2 then the smallest eigenvalue n of the truncated matrix m n of m of size n 1 n 1 tends to zero when n tends to infinity in the case of measures in the closed unit disk we obtain some related results

generalization of taylor s theorem and newton s method via a new family of determinantal interpolation formulas and its applications abstract the general form of taylor s theorem for a function f k k where k is the real line or the complex plane gives the formula f pn rn where pn is the newton interpolating polynomial computed with respect to a confluent vector of nodes and rn is the remainder whenever f 0 for each m 2 n 1 we describe a determinantal interpolation formula f pm n rm n where pm n is a rational function in x and f itself these formulas play a dual role in the approximation of f or its inverse for m 2 the formula is taylor s and for m 3 is a new expansion formula and a pade approximant by applying the formulas to pn for each m 2 p m m 1 p m m n 2 is a set of n rational approximations that includes pn and may provide a better approximation to f than pn hence each taylor polynomial unfolds into an infinite spectrum of rational approximations the formulas also give an infinite spectrum of rational inverse approximations as well as a fundamental k point iteration function bm k for each k m defined as the ratio of two determinants that depend on the first m k derivatives application of our formulas have motivated several new results obtained in sequel papers i theoretical analysis of the order of b m k k 1 m proving that it ranges from m to the limiting ratio of generalized fibonacci numbers of order m ii computational results with the first few members of bm k indicating that they outperform traditional root finding methods e g newton s iii a novel polynomial rootfinding method requiring only a single input and the evaluation of the sequence of iteration functions bm 1 at that input this amounts to the evaluation of special toeplitz determinants that are also computable via a recursive formula iv a new strategy for general root finding v new formulas for approximation of e and other special numbers

two applications of the subnormality of the hessenberg matrix related to general orthogonal polynomials abstract in this paper we prove two consequences of the subnormal character of the hessenberg matrix d when the hermitian matrix m of an inner product is a moment matrix if this inner product is defined by a measure supported on an algebraic curve in the complex plane then d satisfies the equation of the curve in a noncommutative sense we also prove an extension of the krein theorem for discrete measures on the complex plane based on properties of subnormal operators

application of two dimensional orthogonal plate function to flexural vibration of skew plates abstract a set of two dimensional orthogonal plate functions is used as an admissible deflection function for the study of the flexural vibration of skew plates by the rayleigh ritz method from a prescribed starting function satisfying the boundary conditions the higher terms in the two dimensional orthogonal plate functions are generated by using the gram schmidt orthogonalization process the method is applied to obtain results for a few particular rhombic plates with different support conditions at various skew angles where possible the numerical results presented are compared with the values obtained by other numerical methods in the literature the results should be of considerable value and practical significance in design applications

primal dual affine scaling interior point methods for linear complementarity problems a first order affine scaling method and two m th order affine scaling methods for solving monotone linear complementarity problems lcps are presented all three methods produce iterates in a wide neighborhood of the central path the first order method has o nl 2 lognl 2 log lognl 2 iteration complexity if the lcp admits a strict complementary solution then both the duality gap and the iteration sequence converge superlinearly with q order two if m omega log sqrt n l then both higher order methods have o sqrt n l iteration complexity the q order of convergence of one of the methods is m 1 for problems that admit a strict complementarity solution while the q order of convergence of the other method is m 1 2 for general monotone lcps

on the roles of minimization and linearization in least squares finite element models of nonlinear boundary value problems in this paper we examine the roles of minimization and linearization in the least squares finite element formulations of nonlinear boundary values problems the least squares principle is based upon the minimization of the least squares functional constructed via the sum of the squares of appropriate norms of the residuals of the partial differential equations in the present case we consider l 2 norms since the least squares method is independent of the discretization procedure and the solution scheme the least squares principle suggests that minimization should be performed prior to linearization where linearization is employed in the context of either the picard or newton iterative solution procedures however in the least squares finite element analysis of nonlinear boundary value problems it has become common practice in the literature to exchange the sequence of application of the minimization and linearization operations the main purpose of this study is to provide a detailed assessment on how the finite element solution is affected when the order of application of these operators is interchanged the assessment is performed mathematically through an examination of the variational setting for the least squares formulation of an abstract nonlinear boundary value problem and also computationally through the numerical simulation of the least squares finite element solutions of both a nonlinear form of the poisson equation and also the incompressible navier stokes equations the assessment suggests that although the least squares principle indicates that minimization should be performed prior to linearization such an approach is often impractical and not necessary

semi analytical shape functions in the finite element analysis of rectangular plates this paper presents a method for the dynamic analysis of a thin elastic isotropic rectangular plate the method is a hybrid of finite element theory and classical thin plate theory the displacement functions are derived from sanders thin shell equations and are expanded in power series expressions for mass and stiffness are determined by precise analytical integration the free vibrations of rectangular plates with various boundary conditions are studied by following this method the results obtained reveal that the frequencies calculated in this way are in good agreement with those obtained by others

buckling and vibration of polar orthotropic circular plate resting on winkler foundation buckling and vibrational behavior of polar orthotropic circular plates of linearly varying thickness is presented on the basis of classical plate theory the plate is resting on winkler type foundation an approximate solution has been obtained by ritz method which employs basis functions based upon static deflection of polar orthotropic plates the effect of elastic foundation and that of orthotropy on the natural frequency of plate has been illustrated for different values of taper parameter flexibility parameter in plane force and nodal diameter the critical buckling load for clamped and simply supported plates have been obtained a comparison of results with those available in literature has been presented

hessenberg matrix for sums of hermitian positive definite matrices and weighted shifts abstract in this work we introduce an algebraic operation between bounded hessenberg matrices and we analyze some of its properties we call this operation m sum and we obtain an expression for it that involves the cholesky factorization of the corresponding hermitian positive definite matrices associated with the hessenberg components this work extends a method to obtain the hessenberg matrix of the sum of measures from the hessenberg matrices of the individual measures introduced recently by the authors for subnormal matrices to matrices which are not necessarily subnormal moreover we give some examples and we obtain the explicit formula for the m sum of a weighted shift in particular we construct an interesting example a subnormal hessenberg matrix obtained as the m sum of two not subnormal hessenberg matrices

free vibrations of a solid circular plate of linearly varying thickness and attached to a winkler foundation abstract the title problem is solved by using linear analysis and the rayleigh schmidt method frequency coefficients corresponding to the lower symmetric and antisymmetric modes of transverse vibration are tabulated since it is assumed that the thin isotropic plate is elastically restrained against translation and rotation the free edge case turns out to be a particular situation of the mechanical system under study a comparison with results obtained by means of the finite element method is also presented very good agreement is achieved for the situations considered

on the moment problem in the bounded case abstract through the matrix treatment of the theory of orthogonal polynomials on curves or domains of the complex plane we extend to arbitrary bounded regions the results of atzmon 1975 for the unit disc

sixteenth order method for nonlinear equations modification of newton s method with higher order convergence is presented the modification of newton s method is based on king s fourth order method the new method requires three step per iteration analysis of convergence demonstrates that the order of convergence is 16 some numerical examples illustrate that the algorithm is more efficient and performs better than classical newton s method and other methods

letter to the editor accelerated convergence in newton s method for approximating square roots in this paper we construct a modification of newton s method to accelerate the convergence of this method to the approximation of the positive square root of a positive real number from this modification we can define a new iterative process with prefixed order q n q 2

a new modified king werner method for solving nonlinear equations in this paper a new method for solving nonlinear equations f x 0 is presented analysis of the convergence shows that the asymptotic convergence order of this method is 1 3 some numerical results are given to demonstrate its efficiency

choosing the forcing terms in an inexact newton method an inexact newton method is a generalization of newton s method for solving f x 0 f mathbb r n to mathbb r n in which at the kth iteration the step s k from the current approximate solution x k is required to satisfy a condition f x k f x k s k leqslant eta k f x k for a forcing term eta k in 0 1 in typical applications the choice of the forcing terms is critical to the efficiency of the method and can affect robustness as well promising choices of the forcing terms are given their local convergence properties are analyzed and their practical performance is shown on a representative set of test problems

orthogonal matrix polynomials and higher order recurrence relations abstract it is well known that orthogonal polynomials on the real line satisfy a three term recurrence relation and conversely every system of polynomials satisfying a three term recurrence relation is orthogonal with respect to some positive borel measure on the real line we extend this result and show that every system of polynomials satisfying some 2 n 1 term recurrence relation can be expressed in terms of orthonormal matrix polynomials for which the coefficients are n n matrices we apply this result to polynomials orthogonal with respect to a discrete sobolev inner product and other inner products in the linear space of polynomials as an application we give a short proof of krein s characterization of orthogonal polynomials with a spectrum having a finite number of accumulation points

on schroder s families of root finding methods schroder s methods of the first and second kind for solving a nonlinear equation f x 0 originally derived in 1870 are of great importance in the theory and practice of iteration processes they were rediscovered several times and expressed in different forms during the last 130 years it was proved in the paper of petkovic and herceg 1999 7 that even seven families of iteration methods for solving nonlinear equations are mutually equivalent in this paper we show that these families are also equivalent to another four families of iteration methods and find that all of them have the origin in schroder s generalized method of the second kind presented in 1870 in the continuation we consider smale s open problem from 1994 about possible link between schroder s methods of the first and second kind and state the link in a simple way

effect of elastic foundation on axisymmetric vibrations of polar orthotropic circular plates of variable thickness abstract axisymmetric vibrations of a polar orthotropic circular plate of variable thickness linear as well as parabolic resting on an elastic foundation of winkler type are discussed on the basis of the classical theory of plates an approximate solution of the problem is obtained by the ritz method frequency parameters of plates with different edge conditions for the first three modes of vibration and for various values of the taper parameter the foundation modulus and the rigidity ratio are presented

asymmetric vibrations and elastic stability of polar orthotropic circular plates of linearly varying profile abstract asymmetric vibration of polar orthotropic circular plates of linearly varying thickness subjected to hydrostatic in plane force are discussed on the basis of classical plate theory an approximate solution of the problem has been obtained by the ritz method which employs functions based upon the static deflection of polar orthotropic plates this method has a faster rate of convergence as compared to the polynomial co ordinate functions frequency parameters of the plate with elastically restrained edge conditions are presented for the three modes of vibrations for various values of taper parameter rigidity ratio flexibility parameter and buckling load parameter the critical buckling loads for elastically restrained edge conditions have been obtained a comparison of results with those available in the literature shows an excellent agreement

flexural vibration of polygonal plates using characteristic orthogonal polynomials in two variables a set of characteristic orthogonal polynomials in two variables is used as deflection functions to obtain the natural frequencies and mode shapes of polygonal plates by the rayleigh ritz method the set of orthogonal polynomials is constructed by employing the gram schmidt orthogonalization procedure the first six natural frequencies are numerically evaluated for triangular plates of different configurations by using these orthogonal functions and the corresponding mode shapes are plotted results are compared with those obtained previously by other methods