notes on tqr algorithms historically the algorithm for completely solving the symmetric tridiagonal eigenvalue problem is the tqr algorithm rational variants of tqr were used when only the complete spectrum of eigenvalues was desired since they have the advantage of avoiding square roots several historical variants of tqr use fewer operations than the one used in lapack however these variants have known accuracy problems in this paper a new variant of tqr is developed which is similar to parts of the variant proposed by sack 1972 this variant has fewer operations than the one in lapack which results in faster timings and it avoids the accuracy problems of the other historical variants in addition it is shown how the eigenvectors can be recovered from the rational algorithms

on the formulation of high frequency dissipative time stepping algorithms for nonlinear dynamics part ii second order methods we present in this paper the formulation of a new high frequency dissipative time stepping algorithm for nonlinear elastodynamics that is second order accurate in time the new scheme exhibits unconditional energy dissipation and momentum conservation and thus the given name of energy dissipative momentum conserving second order scheme edmc 2 leading also to the conservation of the relative equilibria of the underlying physical system the unconditional character of these properties applies not only with respect to the time step size but equally important with respect to the considered elastic potential moreover the dissipation properties are fully controlled through an algorithmic parameter reducing to existing fully conserving schemes if desired the design of the new algorithm is described in detail including a complete analysis of its dissipation conservation properties in the fully nonlinear range of finite elasticity to motivate the different constructions that lead to the dissipative properties of the final scheme the same arguments are used first in the construction of new linear time stepping algorithms for the system of linear elastodynamics including first and second order schemes the new schemes exhibit a rigorous decay of the physical energy with the second order schemes formulated in a general two stage formula accommodating the aforementioned control parameter in the dissipation of the energy a complete spectral analysis of the new schemes is presented in this linear range to evaluate their different numerical properties in particular the dissipative character of the proposed schemes in the high frequency range is fully demonstrated in fact it is shown that the new second order scheme is l stable most remarkably the extension of these ideas to the nonlinear range is accomplished avoiding the use of multi stage formulae given the freedom gained in using general nonlinear relations while preserving the conservation laws of the momenta and the corresponding relative equilibria several representative numerical simulations are presented in the context of nonlinear elastodynamics to evaluate the performance of the newly proposed schemes

conservative rigid body dynamics by convected base vectors with implicit constraints abstract a conservative time integration formulation is developed for rigid bodies based on a convected set of orthonormal base vectors the base vectors are represented in terms of their absolute coordinates and thus the formulation makes use of three translation components plus nine components of the base vectors orthogonality and unit length of the base vectors are imposed by constraining the equivalent green strain components and the kinetic energy is represented corresponding to rigid body motion the equations of motion are obtained via hamilton s equations including the zero strain conditions as well as external constraints via lagrange multipliers subsequently the lagrange multipliers associated with the internal zero strain constraints are eliminated by use of a set of orthogonality conditions between the generalized displacements and the momentum vector leaving a set of differential equations without additional algebraic constraints on the base vectors a discretized form of the equations of motion is obtained by starting from a finite time increment of the hamiltonian and retracing the steps of the continuous formulation in discrete form in terms of increments and mean values over each integration time increment in this discrete form the lagrange multipliers are given in terms of a representative value within the integration time interval and the equations of motion are recast into a conservative mean value and finite difference format the lagrange multipliers are eliminated explicitly within each integration interval leaving a projection operator expressed in terms of displacement component mean values hereby the number of variables is reduced by six for each rigid body in the problem and the difference equations lead to conservation of the orthonormality conditions for the local base vectors examples demonstrate the efficiency and accuracy of the procedure

accurate recovery of recursion coefficients from gaussian quadrature formulas abstract we present an algorithm for the accurate recovery of recursion coefficients from quadrature formulas with positive weights based on the differential form of the quotient difference algorithm the process is slightly faster than the gragg harrod algorithm and is forward stable in the sense of componentwise relative error when the gaps between the nodes are available as data this result shows constructively that the converse problem is well posed when the data are required to be accurate floating point numbers this is not a contradiction of the examples given by gragg and harrod which are ill posed in a vector norm because in that case very small numbers are not required to have any significant digits

generalized anti gauss quadrature rules gauss quadrature is a popular approach to approximate the value of a desired integral determined by a measure with support on the real axis laurie proposed an n 1 point quadrature rule that gives an error of the same magnitude and of opposite sign as the associated n point gauss quadrature rule for all polynomials of degree up to 2 n 1 this rule is referred to as an anti gauss rule it is useful for the estimation of the error in the approximation of the desired integral furnished by the n point gauss rule this paper describes a modification of the n 1 point anti gauss rule that has n k nodes and gives an error of the same magnitude and of opposite sign as the associated n point gauss quadrature rule for all polynomials of degree up to 2 n 2 k 1 for some k 1 we refer to this rule as a generalized anti gauss rule an application to error estimation of matrix functionals is presented

integrating finite rotations abstract we study the integration of problems of evolution in the rotation group instead of attacking the problem in the nonlinear differential manifold so 3 pure rotational dynamics as is usually done we derive equations for the complete problem of motion translational and rotational dynamics on an extended manifold we develop a generalization of runge kutta methods that by design ensures that the solution will remain on the manifold for any choice of the tableau this is obtained through configuration updates performed via the exponential map we show how certain terms can be approximated while retaining the order of accuracy of the scheme and how the method conserves the total momentum of the system within this framework we develop two nonlinearly unconditionally stable time integration schemes that are associated with discrete laws of conservation dissipation of the total energy the dissipating algorithm generalizes to the nonlinear case the high frequency damping characteristics provided by some well known conventional methods we present numerical results to support our analysis and we develop a complete application of this methodology to the nonlinear dynamics of three dimensional rods undergoing large displacements and finite rotations under the assumption of small strains

calculation of gauss quadratures with multiple free and fixed knots algorithms are derived for the evaluation of gauss knots in the presence of fixed knots by modification of the jacobi matrix for the weight function of the integral simple gauss knots are obtained as eigenvalues of symmetric tridiagonal matrices and a rapidly converging simple iterative process based on the merging of free and fixed knots of quadratic convergence is presented for multiple gauss knots the procedures also allow for the evaluation of the weights of the quadrature corresponding to the simple gauss knots a new characterization of simple gauss knots as a solution of a partial inverse eigenvalue problem is derived

the numerically stable reconstruction of jacobi matrices from spectral data we present an expose of the elementary theory of jacobi matrices and in particular their reconstruction from the gaussian weights and abscissas many recent works propose use of the diagonal hermitian lanczos process for this purpose we show that this process is numerically unstable we recall rutishauser s elegant and stable algorithm of 1963 based on plane rotations implement it efficiently and discuss our numerical experience we also apply rutishauser s algorithm to reconstruct a persymmetric jacobi matrix from its spectrum in an efficient and stable manner

anti gaussian quadrature formulas an anti gaussian quadrature formula is an n 1 point formula of degree 2n 1 which integrates polynomials of degree up to 2n 1 with an error equal in magnitude but of opposite sign to that of the n point gaussian formula its intended application is to estimate the error incurred in gaussian integration by halving the difference between the results obtained from the two formulas we show that an anti gaussian formula has positive weights and that its nodes are in the integration interval and are interlaced by those of the corresponding gaussian formula similar results for gaussian formulas with respect to a positive weight are given except that for some weight functions at most two of the nodes may be outside the integration interval the anti gaussian formula has only interior nodes in many cases when the kronrod extension does not and is as easy to compute as the n 1 point gaussian formula

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

bdf like methods for nonlinear dynamic analysis we present several time integration algorithms of second order accuracy that are numerically simple and effective for nonlinear elastodynamic problems these algorithms are based on a general four step scheme that has a resemblance to the backward differentiation formulas we also present an extension to the composite strategy of the bathe method appropriate values for the algorithmic parameters are determined based on considerations of stability and dissipativity and less dissipative members of each algorithm have been identified we demonstrate the convergence characteristics of the proposed algorithms with a nonlinear dynamic problem having analytic solutions and test these algorithms with several three dimensional nonlinear elastodynamic problems involving large deformations and rotations employing st venant kirchhoff and compressible neo hookean hyperelastic material models these tests show that stable computations are obtained with the proposed algorithms in nonlinear situations where the trapezoidal rule encounters a well known instability

numerical integration of non linear elastic multi body systems this paper is concerned with the modelling of nonlinear elastic multi body systems discretized using the finite element method the formulation uses cartesian co ordinates to represent the position of each elastic body with respect to a single inertial frame the kinematic constraints among the various bodies of the system are enforced via the lagrange multiplier technique the resulting equations of motion are stiff non linear differential algebraic equations the integration of these equations presents a real challenge as most available techniques are either numerically unstable or present undesirable high frequency oscillations of a purely numerical origin an approach is proposed in which the equations of motion are discretized so that they imply conservation of the total energy for the elastic components of the system whereas the forces of constraint are discretized so that the work they perform vanishes exactly the combination of these two features of the discretization guarantees the stability of the numerical integration process for non linear elastic multi body systems examples of the procedure are presented

on the sensitivity of orthogonal polynomials to perturbations in the moments modified moments of a positive weight distribution determine uniquely the recursion coefficients of the polynomials orthogonal with respect to the given distribution we analyze the sensitivity of the underlying nonlinear maps with regard to perturbations in the modified moments and introduce pertinent descriptive concepts weight distributions of chebyshev and jacobi type are studied in particular

accurate singular values and differential qd algorithms in september 1991 j w demmel and w m kahan were awarded the second siam prize in numerical linear algebra for their paper accurate singular values of bidiagonal matrices 1 referred to as dk hereafter among several valuable results was the observation that the standard bidiagonal qr algorithm used in linpack 2 and in many other svd programs can be simplified when the shift is zero and of greater importance no subtractions occur the last feature permits very small singular values to be found with almost all the accuracy permitted by the data and at no extra cost

a self stabilized algorithm for enforcing constraints in multibody systems a new algorithm is developed for the enforcement of constraints within the framework of nonlinear flexible multibody system modeled with the finite element approach the proposed algorithm exactly satisfies the constraints at the displacement and velocity levels and furthermore it achieves nonlinear unconditional stability by imposing the vanishing of the work done by the constraint forces when combined with specific discretizations of the inertial and elastic forces identical convergence rates are observed for the displacements velocities and lagrange multipliers the proposed algorithm is closely related to the stabilized index 2 or ggl method although no additional multipliers are introduced in the present approach these desirable characteristics are obtained without resorting to numerically dissipative algorithms if high frequencies are present in the system i e the system is physically stiff dissipative schemes become necessary the proposed algorithm is extended to deal with this situation

robust integration schemes for flexible multibody systems a framework for the development of robust time integration schemes for nonlinear flexible multibody systems is presented the proposed schemes are designed to meet four specific requirements nonlinear unconditional stability of the scheme a rigorous treatment of both geometric and material nonlinearities exact satisfaction of the constraints and the presence of high frequency numerical dissipation specific algorithms are presented for rigid bodies cables beams shells and elasto dynamics the kinematic nonlinearities are treated in a rigorous manner for all elements and the material nonlinearities can be handled when the constitutive laws stem from the existence of a strain energy density function the treatment of the constraint equations associated with the six lower pair joints are presented as well the efficiency and robustness of the proposed approach is demonstrated with specific numerical examples

interpolation of rotational variables in nonlinear dynamics of 3d beams the formulation of dynamic procedures for three dimensional 3 d beams requires extensive use of the algebra pertaining to the non linear character of the rotation group in space the corresponding extraction procedure to obtain the rotations that span a time step has certain limitations which can have a detrimental eect on the overall stability of a time integration scheme the paper describes two algorithms for the dynamics of 3 d beams which dier in their manifestation of the above limitation the rst has already been described in the literature and involves the interpolation of iterative rotations while an alternative formulation which eliminates the above eect by design requires interpolation of incremental rotations theoretical arguments are backed by numerical results similarities between the conventional and new formulation are pointed out and are shown to be big enough to enable easy transformation of the conventional formulation into the new one 1998 john wiley sons ltd

conservation properties of a time fe method part iii mechanical systems with holonomic constraints a galerkin based discretization method for index 3 differential algebraic equations pertaining to finite dimensional mechanical systems with holonomic constraints is proposed in particular the mixed galerkin mg methodis introduced which leads in a natural way to time stepping schemes that inherit major conservation properties of the underlying constrained hamiltonian system namely total energy and angular momentum in addition to that the constraints on the configuration level and on the velocity momentum level are fulfilled exactly the application of the mg method to specific mechanical systems such as the pendulum rigid body dynamics and the coupled motion of rigid and flexible bodies is presented related numerical examples are investigated to evaluate the numerical performance of the mg 1 and mg 2 method

numerical integration as a finite matrix approximation to multiplication operator abstract in this article numerical integration is formulated as evaluation of a matrix function of a matrix that is obtained as a projection of the multiplication operator on a finite dimensional basis the idea is to approximate the continuous spectral representation of a multiplication operator on a hilbert space with a discrete spectral representation of a hermitian matrix the gaussian quadrature is shown to be a special case of the new method the placement of the nodes of numerical integration and convergence of the new method are studied

energy decaying scheme for non linear beam models abstract this paper is concerned with the time integration of the equations of motion describing the non linear dynamic response of beams desirable characteristics of integration schemes for structural dynamics problems include unconditional stability accuracy and high frequency numerical dissipation several schemes exist that present all these features when applied to linear problems though the application of those schemes to non linear problems is often successful proofs of unconditional stability are rarely derived a finite difference integration scheme is derived in this paper for the non linear dynamic response of beams though of a finite difference nature the proposed scheme mimics the integration scheme obtained by applying the time discontinuous galerkin methodology to a single degree of freedom linear oscillator a formal proof of unconditional stability for the non linear problem is derived based on an energy decay argument numerical examples using the proposed scheme are given and the results are compared with the predictions of other available schemes the accuracy of the proposed scheme and its high frequency numerical dissipation characteristics are demonstrated in these examples

computation of gauss type quadrature formulas gaussian formulas for a linear functional l such as a weighted integral are best computed from the recursion coefficients relating the monic polynomials orthogonal with respect to l in gauss type formulas one or more extraneous conditions such as pre assigning certain nodes replace some of the equations expressing exactness when applied to high order polynomials these extraneous conditions may be applied by modifying the same number of recursion coefficients we survey the methods of computing formulas from recursion coefficients methods of obtaining recursion coefficients and modifying them for gauss type formulas and questions of existence and numerical accuracy associated with those computations

fernando s solution to wilkinson s problem an application of double factorization abstract suppose that one knows a very accurate approximation to an eigenvalue of a symmetric tridiagonal matrix t a good way to approximate the eigenvector x is to discard an appropriate equation say the rth from the system t i x 0 and then to solve the resulting underdetermined system in any of several stable ways however the output x can be completely inaccurate if r is chosen poorly and in the absence of a quick and reliable way to choose r this method has lain neglected for over 35 years experts in boundary value problems have known about the special structure of the inverse of a tridiagonal matrix since the 1960s and their double triangular factorization technique down and up gives directly the redundancy of each equation and so reveals the set of good choices for r the relation of double factorization to the eigenvector algorithm of godunov and his collaborators is described the results extend to band matrices and to zero entries in eigenvectors and have uses beyond eigenvector computation

vibrations of rectangular plates of bilinearly varying thickness and with general boundary conditions abstract results are presented for the fundamental frequency of a rectangular plate having a thickness which varies in a bilinear fashion in the x direction translational and rotational flexibilities are taken into account at all edges a simple algorithm which allows one to evaluate the fundamental frequency of vibration is derived by making use of the ritz method and expressing the fundamental displacement function in terms of a polynomial co ordinate function which satisfies approximately the natural boundary conditions

numerical solution of the orr sommerfeld equation using the viscous green function and split gaussian quadrature we continue our study of the construction of numerical methods for solving two point boundary value problems using green functions building on the successful use of split gauss type quadrature schemes here we adapt the method for eigenvalue problems in particular the orr sommerfeld equation of hydrodynamic stability theory use of the green function for the viscous part of the problem reduces the fourth order ordinary differential equation to an integro differential equation which we then discretize using the split gaussian quadrature and product integration approach of our earlier work along with pseudospectral differentiation matrices for the remaining differential operators as the latter are only second order the resulting discrete equations are much more stable than those obtained from the original differential equation this permits us to obtain results for the standard test problem plane poiseuille flow at unit streamwise wavenumber and reynolds number 10000 that we believe are the most accurate to date

matrix methods for quadrature formulas on the unit circle a survey in this paper we give a survey of some results concerning the computation of quadrature formulas on the unit circle like nodes and weights of gauss quadrature formulas for the estimation of integrals with respect to measures on the real line can be computed from the eigenvalue decomposition of the jacobi matrix szeg quadrature formulas for the approximation of integrals with respect to measures on the unit circle can be obtained from certain unitary five diagonal or unitary hessenberg matrices that characterize the recurrence for an orthogonal laurent polynomial basis these quadratures are exact in a maximal space of laurent polynomials orthogonal polynomials are a particular case of orthogonal rational functions with prescribed poles more general szeg quadrature formulas can be obtained that are exact in certain spaces of rational functions in this context the nodes and the weights of these rules are computed from the eigenvalue decomposition of an operator mobius transform of the same five diagonal or hessenberg matrices

computation of gauss kronrod of quadrature rules recently laurie presented a new algorithm for the computation of 2n 1 point gauss kronrod quadrature rules with real nodes and positive weights this algorithm first determines a symmetric tridiagonal matrix of order 2n 1 from certain mixed moments and then computes a partial spectral factorization we describe a new algorithm that does not require the entries of the tridiagonal matrix to be determined and thereby avoids computations that can be sensitive to perturbations our algorithm uses the consolidation phase of a divide and conquer algorithm for the symmetric tridiagonal eigenproblem we also discuss how the algorithm can be applied to compute kronrod extensions of gauss radau and gauss lobatto quadrature rules throughout the paper we emphasize how the structure of the algorithm makes efficient implementation on parallel computers possible numerical examples illustrate the performance of the algorithm

computational schemes for non linear elasto dynamics this paper deals with the development of computational schemes for the dynamic analysis of non linear elastic systems the focus of the investigation is on the derivation of unconditionally stable time integration schemes presenting high frequency numerical dissipation for these types of problem at first schemes based on galerkin and time discontinuous galerkin approximations applied to the equations of motion written in the symmetric hyperbolic form are proposed though useful these schemes require casting the equations of motion in the symmetric hyperbolic form which is not always possible furthermore this approaches to unacceptably high computational costs next unconditionally stable schemes are proposed that do not rely on the symmetric hyperbolic form both energy preserving and energy decaying schemes are derived numerical examples are presented to demonstrate the accuracy and efficiency of the proposed schemes copyright 1999 john wiley sons ltd

energy conserving time integration scheme for geometrically exact beam an energy conserving finite element formulation for the dynamic analysis of geometrically non linear beam like structures undergoing large overall motions has been developed the formulation uses classical displacement based planar beam finite elements described in an inertial frame it takes into account finite axial bending and shear strains a theoretically consistent approach is used to derive a novel and simple energy conserving scheme using the unconventional incremental strain update rather than the standard strong form numerical examples demonstrate perfect energy and momenta conservation stability and robustness of the scheme and good convergence properties in terms of both the newton raphson method and time step size c 2006 elsevier b v all rights reserved

the existence and construction of rational gauss type quadrature rules abstract consider a hermitian positive definite linear functional f and assume we have m distinct nodes fixed in advance anywhere on the real line in this paper we then study the existence and construction of n th rational gauss radau m 1 and gauss lobatto m 2 quadrature formulas that approximate f f these are quadrature formulas with n positive weights and with the n m remaining nodes real and distinct so that the quadrature is exact in a 2 n m dimensional space of rational functions further we also consider the case in which the functional is defined by a positive bounded borel measure on an interval for which it is required in addition that the nodes are all in the support of the measure

numerical integration of the stiff dynamics of geometrically exact shells an energy dissipative momentum conserving scheme this paper presents a new family of time stepping algorithms for the integration of the dynamics of non linear shells we consider the geometrically exact shell theory involving an inextensible director field the so called five parameter shell model the main characteristic of this model is the presence of the group of finite rotations in the configuration manifold describing the deformation of the solid in this context we develop time stepping algorithms whose discrete solutions exhibit the same conservation laws of linear and angular momenta as the underlying physical system and allow the introduction of a controllable non negative energy dissipation to handle the high numerical stiffness characteristic of these problems a series of algorithmic parameters for the different components of the deformation of the shell i e membrane bending and transverse shear fully control this numerical dissipation recovering existing energy momentum schemes as a particular choice of these algorithmic parameters we present rigorous proofs of the numerical properties of the resulting algorithms in the full non linear range furthermore it is argued that the numerical dissipation is introduced in the high frequency range by considering the proposed algorithm in the context of a linear problem the finite element implementation of the resulting methods is described in detail as well as considered in the final arguments proving the aforementioned conservation dissipation properties we present several representative numerical simulations illustrating the performance of the newly proposed methods the robustness gained over existing methods in these stiff problems is confirmed in particular copyright 2002 john wiley sons ltd

energy preserving decaying schemes for non linear beam dynamics using the helicoidal approximation abstract we develop two unconditionally stable displacement based time stepping schemes for the non linear dynamic response of beams the first algorithm guarantees the exact discrete conservation of energy and momentum the second is associated with an energy decay inequality that achieves control of the unresolved frequencies by means of a numerical dissipation mechanism both schemes emanate from a weak form of the equations of dynamic equilibrium referred to a fixed pole space and time discretizations are based on the exponential parameterization of motion this implies that the beam reference line and the trajectories of the beam nodes are helicoids in space the exponential mapping approach allows a unified treatment of translations and rotations greatly simplifying the derivation of the algorithms and their analysis the capabilities and performance of the new schemes are demonstrated and discussed with the aid of numerical simulations

on practical integration of semi discretized nonlinear equations of motion part 1 reasons for probable instability and improper convergence time integration is the most versatile method for analyzing the general case of nonlinear semi discretized equations of motion however the approximate responses of such analyses generally do not converge properly and might even display numerical instability this is a very significant shortcoming especially in practical time integration herein after illustrating that this phenomenon is viable even for very simple nonlinear dynamic models sources of the shortcoming are discussed in detail the conclusion is that in time integration of nonlinear dynamic mathematical models of physically stable structural systems responses may converge improperly for three major reasons these reasons are 1 inadequate number of iterations before terminating nonlinearity solutions 2 deficiencies in the formulation of some time integration methods and 3 the inherent behaviour of the models of some special dynamic systems in addition limitations on computational facilities and improper consideration of these limitations may impair the numerical stability and convergence of the computed responses the differences between static and dynamic analyses are also discussed from the viewpoint of the numerical errors induced by nonlinearity

solution of second kind fredholm integral equations by means of gauss and anti gauss quadrature rules this paper is concerned with the numerical approximation of fredholm integral equations of the second kind a nystrom method based on the anti gauss quadrature formula is developed and investigated in terms of stability and convergence in appropriate weighted spaces the nystrom interpolants corresponding to the gauss and the anti gauss quadrature rules are proved to furnish upper and lower bounds for the solution of the equation under suitable assumptions which are easily verified for a particular weight function hence an error estimate is available and the accuracy of the solution can be improved by approximating it by an averaged nystrom interpolant the effectiveness of the proposed approach is illustrated through different numerical tests

gauss quadratures and jacobi matrices for weight functions not of one sign construction of gauss quadratures with prescribed knots via jacobi matrices is extended to the case where not all orthogonal polynomials exist due to the weight function changing sign an algorithm is described and is demonstrated by calculating the knots of kronrod schemes and other gauss quadratures with prescribed knots

on the numerical evaluation of fredholm determinants some significant quantities in mathematics and physics are most naturally expressed as the fredholm determinant of an integral operator most notably many of the distribution functions in random matrix theory though their numerical values are of interest there is no systematic numerical treatment of fredholm determinants to be found in the literature instead the few numerical evaluations that are available rely on eigenfunction expansions of the operator if expressible in terms of special functions or on alternative numerically more straightforwardly accessible analytic expressions e g in terms of painleve transcendents that have masterfully been derived in some cases in this paper we close the gap in the literature by studying projection methods and above all a simple easily implementable general method for the numerical evaluation of fredholm determinants that is derived from the classical nystrom method for the solution of fredholm equations of the second kind using gauss legendre or clenshaw curtis as the underlying quadrature rule we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel in particular we get exponential convergence for analytic kernels which are typical in random matrix theory the application of the method to the distribution functions of the gaussian unitary ensemble gue in the bulk scaling limit and the edge scaling limit is discussed in detail after extending the method to systems of integral operators we evaluate the two point correlation functions of the more recently studied airy and airy 1 processes

generalized block anti gauss quadrature rules golub and meurant describe how pairs of gauss and gauss radau quadrature rules can be applied to determine inexpensively computable upper and lower bounds for certain real valued matrix functionals defined by a symmetric matrix however there are many matrix functionals for which their technique is not guaranteed to furnish upper and lower bounds in this situation it may be possible to determine upper and lower bounds by evaluating pairs of gauss and anti gauss rules unfortunately it is difficult to ascertain whether the values determined by gauss and anti gauss rules bracket the value of the given real valued matrix functional therefore generalizations of anti gauss rules have recently been described such that pairs of gauss and generalized anti gauss rules may determine upper and lower bounds for real valued matrix functionals also when pairs of gauss and standard anti gauss rules do not the available generalization requires the matrix that defines the functional to be real and symmetric the present paper reviews available anti gauss and generalized anti gauss rules and extends them in several ways that allow applications in new situations in particular the genarlized anti gauss rules for a real valued non negative measure described in pranic and reichel j comput appl math 284 235 243 2015 are extended to allow the estimation of the error in matrix functionals defined by a non symmetric matrix as well as to matrix valued matrix functions modifications that give simpler formulas and thereby make the application of the rules both easier and applicable to a larger class of problems also are described

generalized gaussian quadrature rules for systems of arbitrary functions a numerical algorithm is presented for the construction of generalized gaussian quadrature rules originally introduced by s karlin and w studden over three decades ago the quadrature rules to be discussed possess most of the desirable properties of the classical gaussian integration formulae such as positivity of the weights rapid convergence mathematical elegance etc the algorithm is applicable to a wide class of functions including smooth functions not necessarily polynomials as well as functions with end point singularities such as those encountered in the solution of integral equations complex analysis potential theory and several other areas the performance of the algorithm is illustrated with several numerical examples

calculation of gauss kronrod quadrature rules the jacobi matrix of the 2n 1 point gauss kronrod quadrature rule for a given measure is calculated efficiently by a five term recurrence relation the algorithm uses only rational operations and is therefore also useful for obtaining the jacobi kronrod matrix analytically the nodes and weights can then be computed directly by standard software for gaussian quadrature formulas

upper and lower bounds for frequencies of trapezoidal and triangular plates upper and lower bounds are given for the two lowest frequencies of vibration of clamped trapezoidal and triangular plates it is believed that this is the first time that rigorous bounds have been calculated for the vibrational frequencies of plates of these shapes

integration of elastic multibody systems by invariant conserving dissipating algorithms i formulation abstract this work presents a novel methodology for the dynamic analysis of general non linear flexible multibody systems in part i we develop the 6 d compact representation of motion for those body models which motion may be described by a displacement field plus an independent rotation field this approach explores the fundamental properties of rigid body motion and in particular the coupled nature of linear and angular quantities in both kinematics and dynamics inspiring a novel parameterization technique based on the exponential map using the proposed approach we derive the governing equations for the case of multibody systems composed by rigid bodies and geometrically non linear beams connected by holonomic constraints these equations provide the starting point for the derivation of a class of numerical algorithms characterized by non conventional conservation properties in part ii of this work we develop the algorithms and illustrate their properties with the aid of some numerical applications

vibrational frequencies of clamped plates of variable thickness abstract a method is presented for computing lower and upper bounds for vibrational frequencies of clamped plates with general thickness variations the method is illustrated for plates with linear tapers

spectral methods for orthogonal rational functions abstract we present an operator theoretic approach to orthogonal rational functions based on the identification of a suitable matrix representation of the multiplication operator associated with the corresponding orthogonality measure two alternatives are discussed leading to representations which are linear fractional transformations with matrix coefficients acting on infinite hessenberg or five diagonal unitary matrices this approach permits us to recover the orthogonality measure throughout the spectral analysis of an infinite matrix depending uniquely on the poles and the parameters of the recurrence relation for the orthogonal rational functions besides the zeros of the orthogonal and para orthogonal rational functions are identified as the eigenvalues of matrix linear fractional transformations of finite hessenberg or five diagonal matrices as an application we use operator perturbation theory results to obtain new relations between the support of the orthogonality measure and the location of the poles and parameters of the recurrence relation for the orthogonal rational functions

on the design of energy preserving and decaying schemes for flexible nonlinear multi body systems abstract energy preserving schemes achieve unconditional stability for nonlinear systems by establishing discrete energy preservation statements several schemes have been presented by various authors within drastically different frameworks finite difference schemes based on a mid point approximation galerkin and time discontinuous galerkin approximations of the equations of motion written in the symmetric hyperbolic form finite elements in time and 2 stage fsal runge kutta methods furthermore different types of parameterization of finite rotations were used in the various formulations this paper presents a unified finite difference framework which readily allows comparing the various schemes and their respective properties numerical examples are presented and show that the predictions of two of these schemes are in very close agreement with each other

spectra of multiplication operators as a numerical tool we introduce a numerical procedure for the construction of interpolation and quadrature formulae on bounded convex regions in the plane the construction is based on the behavior of spectra of certain multiplication operators and leads to nodes which are inside a prescribed convex region in mathbb r 2 the resulting interpolation schemes are numerically stable and the quadrature formulae have positive weights and almost but not quite optimal numbers of nodes the performance of the algorithm is illustrated by several numerical examples

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

unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum we show that for rigid body dynamics the mid point rule formulated in body co ordinates exactly conserves energy and the norm of the angular momentum for incremental force free motions but fails to conserve the direction of the angular momentum vector further we show that the mid point rule formulated in the spatial representation is in general physically and geometrically meaningless an alternative algorithm is developed which exactly preserves energy and the total spatial angular momentum in incremental force free motions the implicit version of this algorithm is unconditionally stable and second order accurate the explicit version conserves exactly angular momentum in incremental force free motions numerical simulations are presented which illustrate the excellent performance of the proposed procedure even for incremental rotations over 65 degrees the procedure is directly applicable to transient dynamic calculations of geometrically exact rods and shells