perron type theorems for functional differential equations with infinite delay in a banach space for functional differential equations in a banach space we obtain some results on the asymptotic behavior of solutions which correspond to perron type theorems for differential equations without delay or with finite delay

accurate calculation of the solutions to the thomas fermi equations abstract we obtain highly accurate solutions to the thomas fermi equations for atoms and atoms in very strong magnetic fields we apply the pade hankel method numerical integration power series with pade and hermite pade approximants and chebyshev polynomials both the slope at origin and the location of the right boundary in the magnetic field case are given with unprecedented accuracy

on thermal stability of a reactive third grade fluid in a channel with convective cooling the walls in this paper the thermal stability of a reactive third grade liquid flowing steadily between two parallel plates with symmetrical convective cooling at the walls is investigated the system is assumed to exchange heat with the ambient following newton s cooling law and the reaction is exothermic under arrhenius kinetics neglecting the consumption of the material approximate solutions are constructed for the governing nonlinear boundary value problem using regular perturbation techniques together with a special type of hermite pade approximants and important properties of the velocity and temperature fields including bifurcations and thermal criticality conditions are discussed it is observed that a combined increase in non newtonian parameter and convective cooling enhances the thermal stability of the material

a perron type theorem for functional differential equations a perron type theorem about the existence of the strict lyapunov exponents of the solutions of retarded functional differential equations is established

on the divergence of gradient expansions for kinetic energy functionals in the potential functional theory we consider the density of a fermionic system as a functional of the potential in one dimensional case where it is approximated by the thomas fermi term plus semiclassical corrections through the gradient expansion we compare this asymptotic series with the exact answer for the case of the harmonic oscillator and the morse potential it is found that the leading thomas fermi term is in agreement with the exact density but the subdominant term does not agree in terms of the asymptotic behavior because of the presence of oscillations in the exact density but their absence in the gradient expansion however after regularization of the density by convolution with a gaussian the agreement can be established even in the subdominant term moreover it is found that the expansion is always divergent and its terms grow proportionally to the factorial function of the order similar to the well known divergence of perturbation series in field theory and the quantum anharmonic oscillator pade hermite approximants allow summation of the series and one of the branches of the approximants agrees with the density

method of summation of some slowly convergent series a new method of summation of slowly convergent series is proposed it may be successfully applied to the summation of generalized and basic hypergeometric series as well as some classical orthogonal polynomial series expansions in some special cases our algorithm is equivalent to wynn s epsilon algorithm weniger transformation e j weniger nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series computer physics reports 10 1989 189 371 or the technique recently introduced by cizek et al j cizek j zamastil l skala new summation technique for rapidly divergent perturbation series hydrogen atom in magnetic field journal of mathematical physics 44 3 2003 962 968 in the case of trigonometric series our method is very similar to the homeier s h transformation while in the case of orthogonal series to the k transformation two iterated methods related to the proposed method are considered some theoretical results and several illustrative numerical examples are given

a look ahead levinson algorithm for general toeplitz systems the authors present an extension of levinson s algorithm that is guaranteed to be weakly stable for a large class of general toeplitz matrices namely those that do not have many consecutive ill conditioned leading principal submatrices the new algorithm adapts itself to the given toeplitz matrix by skipping over all the ill conditioned leading principal submatrices encountered during the solution process this is done by a look ahead strategy that monitors the condition of the leading principal submatrices and if necessary switches to a block step of suitable size the overhead of the look ahead algorithm is typically small compared to the classical levinson algorithm and in addition a reliable condition number estimate is produced

the summation of the ordinary and renormalized perturbation series for the ground state energy of the quartic sextic and octic anharmonic oscillators using nonlinear sequence transformations the strongly divergent rayleigh schrodinger perturbation expansion for the ground state energy of the quartic sextic and octic anharmonic oscillator and a corresponding renormalized perturbation expansion f vinette and j ci ek j math phys 32 3392 1991 are summed by pade approximants by levin s sequence transformation d levin int j comput math b 3 371 1973 and by a closely related sequence transformation which was suggested recently e j weniger comput phys rep 10 189 1989 it is shown that the renormalized perturbation expansion can be summed much more easily than the original perturbation expansion from which it was derived and that levin s sequence transformation diverges and is not able to sum the perturbation expansions the pade summation of the renormalized perturbation expansions gives relatively good results in the quartic and sextic case in the case of the octic anharmonic oscillator even the renormalized perturbation expansion is not pade summable the best results are clearly obtained by the new sequence transformation which for instance is able to sum the renormalized perturbation expansions for the infinite coupling limits of the quartic sextic and octic anharmonic oscillator and which produces at least in the quartic and sextic case extremely accurate results

sextic and octic anharmonic oscillators connection between strong coupling and weak coupling expansions for the eigenvalues of the hamiltonians with m 3 4 we propose representations reproducing the well known weak coupling asymptotic perturbation and strong coupling expansions as limiting cases some analytical relations which the strong coupling coefficients must satisfy are presented

high order analysis of the limit cycle of the van der pol oscillator we have applied the lindstedt poincare method to study the limit cycle of the van der pol oscillator obtaining the numerical coefficients of the series for the period and for the amplitude to order 859 hermite pade approximants have been used to extract the location of the branch cut of the series with unprecedented accuracy 100 digits both series have then been resummed using an approach based on pade approximants where the exact asymptotic behaviors of the period and the amplitude are taken into account our results improve drastically all previous results obtained on this subject

high order differential approximants we introduce a new form of differential approximant for the summation of power series the method is a special type of pade hermite approximant it consists of a high order linear differential equation with polynomial coefficients that is satisfied approximately by the partial sum of the power series this method is able to reproduce the polylogarithmic functions exactly numerical evidence suggests that this is currently one of the best methods of singularity analysis for many problems

algebraic approximants a new method of series analysis the authors propose a new method of series analysis in which the available series coefficients are fitted to an algebraic equation such a proposal restricts solutions to have algebraic singularities with rational critical exponents the advantage of the method over the method of differential approximants is that in those cases where the two methods are exact the algebraic approximant method usually though not invariably requires far fewer series coefficients for its exact specification and hence discovery the method is therefore potentially valuable in the study of two dimensional critical systems in particular

application of the quadratic approximants to simple cubic lattice trees recent work has drawn attention to the utility of algebraic approximants for series expansion analysis the special case of quadratic approximants are applied to lattice trees weakly embedded on the simple cubic lattice the resulting estimates of xc the reciprocal of the growth constant are compared with those obtained from other methods of analysis

an efficient recursive total least squares algorithm for fir adaptive filtering an algorithm for recursively computing the total least squares tls solution to the adaptive filtering problem is described this algorithm requires o n multiplications per iteration to effectively track the n dimensional eigenvector associated with the minimum eigenvalue of an augmented sample covariance matrix it is shown that the recursive least squares rls algorithm generates biased adaptive filter coefficients when the filter input vector contains additive noise the tls solution on the other hand is seen to produce unbiased solutions examples of standard adaptive filtering applications that result in noise being added to the adaptive filter input vector are cited computer simulations comparing the relative performance of rls and recursive tls are described

bender wu branch points in the cubic oscillator the analytic continuation of the resonances of the cubic anharmonic oscillator to complex values of the coupling constant is studied with semiclassical and numerical methods bender wu branch points at which level crossing occurs are calculated and labelled by a process of analytic continuation the different resonances are the values that a single analytic function takes on different sheets of a riemann surface whose topology is described

new summation technique for rapidly divergent perturbation series hydrogen atom in magnetic field the perturbation series for the ground state energy of the hydrogen atom in the external magnetic field is summed via the sequence transformations the formula for the large order behavior of the partial sums of the series is derived from this formula a new general sequence transformation is suggested this transformation contains free parameters that can be further optimized it is shown that if the renormalization approach is used the optimal choice of these parameters leads to the previously suggested weniger transformation

non hermitian matrix description of the calpcalt symmetric anharmonic oscillators the symmetric differential schrodinger equation h e with the operator h h x p2 ax4 i x3 cx2 i x h x on l2 is studied at a 0 it is rearranged as a linear algebraic diagonalization with rigorous proof our non variational construction of bound states offers an infinite dimensional analogue to the recent finite dimensional quasi exact solution available at the less common a 0

chebyshev expansion on intervals with branch points with application to the root of kepler s equation a chebyshev hermite pade method when two or more branches of a function merge the chebyshev series of u l will converge very poorly with coefficients a n of t n l falling as o 1 n a for some small positive exponent a however as shown in j p boyd chebyshev polynomial expansions for simultaneous approximation of two branches of a function with application to the one dimensional bratu equation appl math comput 143 2002 189 200 it is possible to obtain approximations that converge exponentially fast in n if the roots that merge are denoted as u 1 l and u 2 l then both branches can be written without approximation as the roots of u u 1 l u u 2 l u email protected l email protected l by expanding the nonsingular coefficients of the quadratic b l and c l as chebyshev series and then applying the usual roots of a quadratic formula we can approximate both branches simultaneously with error that decreases proportional to exp email protected for some constant s 0 where n is the truncation of the chebyshev series this is dubbed the chebyshev shafer or chebyshev hermite pade method because it substitutes chebyshev series for power series in the generalized pade approximants known variously as shafer or hermite pade approximants here we extend these ideas first we explore square roots with branches that are both real valued and complex valued in the domain of interest illustrated by meteorological baroclinic instability second we illustrate triply branched functions via roots of the kepler equation f u l e email protected u email protected 0 only one of the merging roots is real valued and the root depends on two parameters l e rather than one nonetheless the chebyshev hermite pade scheme is successful over the whole two dimensional parameter plane we also discuss how to cope with poles and logarithmic singularities that arise in our examples at the extremes of the expansion domain

summation of the eigenvalue perturbation series by multi valued pade approximants application to resonance problems and double wells quadratic pade approximants are used to obtain energy levels both for the anharmonic oscillator x2 2 lambda x4 and for the double well x2 2 lambda x4 in the first case the complex valued energy of the resonances is reproduced by summation of the real terms of the perturbation series the second case is treated formally as an anharmonic oscillator with a purely imaginary frequency we use the expansion around the central maximum of the potential to obtain a complex perturbation series on the unphysical sheet of the energy function then we perform an analytical continuation of this solution to the neighbouring physical sheet taking into account the supplementary branch of quadratic approximants in this way we can reconstruct the real energy by summation of the complex series such an unusual approach eliminates the double degeneracy of states that makes ordinary perturbation theory around the minima of the double well potential incorrect

determination of f from the asymptotic series for f x about x 0 a difficult and long standing problem in mathematical physics concerns the determination of the value of f from the asymptotic series for f x about x 0 in the past the approach has been to convert the asymptotic series to a sequence of pade approximants pnn x and then to evaluate these approximants at x unfortunately for most physical applications the sequence pnn is slowly converging and does not usually give very accurate results in this paper the results of extensive numerical studies for a large class of functions f x associated with strong coupling lattice approximations are reported it is conjectured that for large n pnn f b ln n a numerical fit to this asymptotic behavior gives an accurate extrapolation to the value of f

the anharmonic oscillator perturbation series for cubic and quartic energy distortion twenty five terms of the perturbation series are calculated for the first five energy levels of an oscillator with lambda x4 energy distortion and a similar calculation is carried out for the first six levels for lambda x3 distortion for the positive quartic the alternating series is summed using the aitken delta 2 transformation for lambda from 0 1 to 100 an exponential integral model and a truncated binomial model are used to transform the series for negative distortion energies and the complex energy eigenvalues are calculated for lambda from 0 01 to 1

on the existence of compressive solitary waves in compacting media necessary conditions for the existence of compressive solitary wave solutions of a partial differential equation derived by scott and stevenson 1986 which describes the two phase fluid flow in a medium compacting under gravity are derived it is shown that for compressive solitary wave solutions to exist which satisfy certain boundary conditions it is necessary that n m 1 where n and m are the exponents in power laws relating the permeability of the medium and the viscosity of the solid matrix respectively to the voidage the effect of the value of the exponents n and m on the shape of the solitary wave is investigated by using existing analytical solutions and new numerical solutions

self similar factor approximants the problem of reconstructing functions from their asymptotic expansions in powers of a small variable is addressed by deriving an improved type of approximants the derivation is based on the self similar approximation theory which presents the passage from one approximant to another as the motion realized by a dynamical system with the property of group self similarity the derived approximants because of their form are called self similar factor approximants these complement the obtained earlier self similar exponential approximants and self similar root approximants the specific feature of self similar factor approximants is that their control functions providing convergence of the computational algorithm are completely defined from the accuracy through order conditions these approximants contain the pade approximants as a particular case and in some limit they can be reduced to the self similar exponential approximants previously introduced by two of us it is proved that the self similar factor approximants are able to reproduce exactly a wide class of functions which include a variety of nonalgebraic functions for other functions not pertaining to this exactly reproducible class the factor approximants provide very accurate approximations whose accuracy surpasses significantly that of the most accurate pade approximants this is illustrated by a number of examples showing the generality and accuracy of the factor approximants even when conventional techniques meet serious difficulties

thermal stability of a reactive third grade fluid in a cylindrical pipe an exploitation of hermite pade approximation technique abstract this study is devoted to investigate the thermal stability of a reactive third grade fluid flowing steadily through a cylindrical pipe with isothermal wall it is assumed that the reaction is exothermic under arrhenius kinetics neglecting the consumption of the material approximate solutions are constructed for the governing nonlinear boundary value problem using regular perturbation techniques together with a special type of hermite pade approximants and important properties of the velocity and temperature fields including bifurcations and thermal criticality conditions are discussed

improved methods of estimating critical indices via fractional calculus efficiencies of certain methods for the determination of critical indices from power series expansions are shown to be considerably improved by a suitable implementation of fractional differentiation in the context of the ratio method rm kinship of the modified strategy with the ad hoc shifted rm is established and the advantages are demonstrated further in the course of the estimation of critical points significant betterment of convergence properties of diagonal pade approximants is observed on several occasions by invoking this concept test calculations are performed on i various ising spin 1 2 lattice models for susceptibility series attended with a ferromagnetic phase transition ii complex model situations involving confluent and antiferromagnetic singularities and iii the chain generating functions for self avoiding walks on triangular square and simple cubic lattices

addition theorem of slater type orbitals a numerical evaluation of barnett coulson lowdin functions when using the one centre two range expansion method to evaluate multicentre integrals over slater type orbitals stos it may become necessary to compute numerical values of the corresponding fourier coefficients also known as barnett coulson lowdin functions bclfs bouferguene and jones 1998 j chem phys 109 5718 to carry out this task it is crucial to not only have a stable numerical procedure but also a fast algorithm in previous work bouferguene and rinaldi 1994 int j quantum chem 50 21 bclfs were represented by a double integral which led to a numerically stable algorithm but this turned out to be disappointingly time consuming the present work aims at exploring another path in which bclfs are represented either by an infinite series involving modified bessel functions or by an integral whose integrand is a smooth function both of these representations have the advantage of being symmetrical with respect to the cusp parameter a and the radial variable r as a consequence it is no longer necessary to split the integrals over r 0 into several components with a different analytical form in each of these a numerical study is also carried out to help select the most appropriate method to be used in practice

extraction of an eigenvector oriented spectrum from the mesa coefficients in this communication an efficient algorithm is proposed for extracting an orthogonal eigenvector oriented spectrum directly from maximum entropy spectral analysis mesa coefficients in effect the algorithm estimates the eigenvector corresponding to the smallest eigenvalue of the data covariance matrix simulations show that in a narrow band array processing application the multisource resolution of mesa can be significantly improved with the use of this algorithm

anharmonic oscillator constructing the strong coupling expansions two novel approaches to construction of the strong coupling expansion for the anharmonic oscillator with the potential v x 1 2 x2 g 4 x4 are proposed the first one is simply a straightforward solution of the schrodinger equation via the nonlinearization technique resulting in the rapidly convergent perturbation series the second one is a version of the path integral perturbation theory but with an unconventional choice of the zeroth approximation action nine leading coefficients of the strong coupling expansion are computed they decrease rapidly the ninth one being of the order of 10 9 three leading corrections of the nonlinearization approach provide the ground state energy within a relative accuracy of 10 7 10 9 at an arbitrary coupling g the explicit formulas for corrections enable one to study the analyticity properties of the energy as a function of the coupling g in the second approach the strong coupling expansion coefficients are computed as an infinite linear series of the weak

the asymptotic behaviour of algebraic approximants we study the convergence of algebraic approximants to a function represented by a power series we consider for an arbitrary but fixed degree approximant sequences which can be generated recursively by use of sergeyev s algorithm for the exponential function a logarithmic function and a power of a binomial we find explicit formulae for the coefficients that appear in a resulting linear recurrence relation we assume that the error equation may be linearized for small errors analysis then yields the generic dominant term in the asymptotic behaviour of the error when a large number of terms of the series are used extensive numerical results confirm the behaviour finally we compare this behaviour with that for the closely related method of drazin tourigny in which the degree of the approximants grows without bound

hyperasymptotics and the linear boundary layer problem why asymptotic series diverge the simplest problem with boundary layers epsilon 2 u xx u f x is used to illustrate i why the perturbation series in powers of epsilon is asymptotic but divergent ii why the optimally truncated expansion is superasymptotic in the sense that that error is proportional to exp mbox constant epsilon and iii how to obtain an improved hyperasymptotic approximation

joint use of the weniger transformation and hyperasymptotics for accurate asymptotic evaluations of a class of saddle point integrals when the weniger transformation is systematically used in resumming the asymptotic series arising from the application of the steepest descent method can display dramatic numerical instabilities that prevent it from improving the accuracy achievable via superasymptotics in the present paper an explanation of such pathologies through the concept of resurgence introduced by berry and howls several years ago within the context of hyperasymptotics is proposed in particular the way the topology of the whole set of the saddles influences the resummation capabilities of the weniger transformation is here investigated for the integrals defining the lowest order cuspoid diffraction catastrophes eventually a powerful and easily implementable resummation scheme based on a joint use of the weniger transformation and hyperasymptotics is proposed for taking care of the above pathologies such a joint action seems to encompass the main virtues of both approaches and the preliminary numerical results obtained from its application show that relative errors several orders of magnitude smaller than those achievable via superasymptotics can be achieved with modest implementation efforts

asymptotic behavior and oscillation of functional differential equations asymptotic relations between the solutions of a linear autonomous functional differential equation and the solutions of the corresponding perturbed equation are established in the scalar case it is shown that the existence of a nonoscillatory solution of the perturbed equation often implies the existence of a real eigenvalue of the limiting equation the proofs are based on a recent perron type theorem for functional differential equations

nonautonomous equations with arbitrary growth rates a perron type theorem abstract for a linear equation x a t x we show that the asymptotic behavior of its solutions is reproduced by the solutions of the nonlinear equation x a t x f t x for any sufficiently small perturbation f more precisely we show that if the lyapunov exponents of the linear equation are limits even for general exponential rates e c t for an arbitrary function then the same happens with the lyapunov exponents of the solutions of the nonlinear equations without introducing new values our approach is based on lyapunov s theory of regularity

mathematical properties of a new levin type sequence transformation introduced by cižek zamastil and skala i algebraic theory ci ek zamastil and skala j math phys 44 962 968 2003 introduced in connection with the summation of the divergent perturbation expansion of the hydrogen atom in an external magnetic field a new sequence transformation which uses as input data not only the elements of a sequence sn n 0 of partial sums but also explicit estimates n n 0 for the truncation errors the explicit incorporation of the information contained in the truncation error estimates makes this and related transformations potentially much more powerful than for instance pade approximants special cases of the new transformation are sequence transformations introduced by levin int j comput math b 3 371 388 1973 and weniger comput phys rep 10 189 371 1989 secs 7 9 numer algor 3 477 486 1992 and also a variant of richardson extrapolation philos trans r soc london ser a 226 299 349 1927 the algebraic theory of these transformations explicit expressions recurrence formulas explicit expres

summation of asymptotic expansions of multiple valued functions using algebraic approximants application to anharmonic oscillators the divergent rayleigh schrodinger perturbation expansions for energy eigenvalues of cubic quartic sextic and octic oscillators are summed using algebraic approximants these approximants are generalized pade approximants that are obtained from an algebraic equation of arbitrary degree numerical results indicate that given enough terms in the asymptotic expansion the rate of convergence of the diagonal staircase approximant sequence increases with the degree different branches of the approximants converge to different branches of the function the success of the high degree approximants is attributed to their ability to model the function on multiple sheets of the riemann surface and to reproduce the correct singularity structure in the limit of large perturbation parameter an efficient recursive algorithm for computing the diagonal approximant sequence is presented

complex variable hvpt complex variable hypervirial perturbation theory is applied to the case of oscillator and coulomb potentials perturbed by a single term potential of the form vxn or vrn respectively the trial calculations reported show that this approach can produce accurate complex energies for resonant states via a simple and speedy calculation and can also be useful in studies of pt symmetry and tunnelling resonance effects

applications of hermite pade approximants to water waves and the harmonic oscillator on a lattice three types of hermite pade approximants are considered known respectively as quadratic integral and differential pade approximants the singularity structure of each type of approximant is described it is more complicated than that of the standard pade approximant and this property may often be used to estimate the types of singularity of a function from its power series expansion as well as evaluate it on its branch cuts two applications in different fields are described which illustrate these properties of the above types of hermite pade approximants the first concerns the characteristic values of mathieu s equation which are related to the energy eigenvalues of the harmonic oscillator on a lattice the second concerns the investigation of the singularity structure and values of various physical quantities associated with periodic and solitary water waves

rayleigh schrodinger perturbation theory with a strong perturbation anharmonic oscillators the bound state solutions of schrodinger s equation for the anharmonic oscillator potentials v x2 lambda x2k k 2 3 have been investigated using elementary techniques of low order variational perturbation theory for the quartic oscillator k 2 a scaled harmonic potential provides a remarkably accurate model for all lambda although this model is slightly less satisfactory for higher order anharmonicities k or 3 the perturbation procedures remain effective and can be applied successfully provided that higher order terms are calculated

asymptotic response of observables from divergent weak coupling expansions a fractional calculus assisted pade technique appropriate constructions of pade approximants are believed to provide reasonable estimates of the asymptotic large coupling amplitude and exponent of an observable given its weak coupling expansion to some desired order in many instances however sequences of such approximants are seen to converge very poorly we outline here a strategy that exploits the idea of fractional calculus to considerably improve the convergence behavior pilot calculations on the ground state perturbative energy series of quartic sextic and octic anharmonic oscillators reveal clearly the worth of our endeavor

hydrogenic atoms in the external potential v r gr λr2 exact solutions and ground state eigenvalue bounds using moment methods an infinite number of exact solutions for the ground and excited states of an s wave hydrogenic atom with the perturbation v r gr lambda r2 are constructed subject to special relations between g lambda and the nuclear charge z a powerful method of stieltjes moments is employed to generate rapidly converging upper and lower bounds to the ground state energy e0 z g lambda for arbitrary g z and positive lambda the accuracies of these bounds are relatively insensitive to the strength of the perturbation the moment method also generates the exact s wave ground state solutions mentioned above

link between the strong coupling and the weak coupling asymptotic perturbation expansions for the quartic anharmonic oscillator for eigenvalues of the hamiltonian of the quartic anharmonic oscillator we give a representation which reproduces two well known limiting cases from the theory of the anharmonic oscillator expansions the weak coupling asymptotic perturbation and the convergent strong coupling we give an estimation of the radius of convergence of the strong coupling expansion for the ground state eigenvalue

toeplitz eigensystem solver in this paper a novel algorithm for computing the minimum eigenvalue and associated eigenvector is presented this algorithm is a derivative of the inverse iteration method which requires a linear system to be solved in each iteration taking advantage of the toeplitz structure a pipelined toeplitz system solver can be applied to solve the toeplitz system in o n time units with o n processors this brings the total computing time for solving the toeplitz minimum eigenvalue down to o kn time units where k is the number of iterations this algorithm makes it possible to perform real time processing of the pisarenko s high resolution spectrum estimate

hypervirial perturbation theory revisited a wide range of topics in the area of hypervirial perturbation theory hvpt is discussed it is shown that with the use of a few simple procedures hvpt is capable of high accuracy for many problems results from many previous works in the literature are found to be improvable by the careful use of hvpt with appropriate choice of the unperturbed potential and of the origin at which the energy expansion is carried out two multi well problems from the literature are analysed in detail to show the value of a combination of hvpt and finite difference methods

upper and lower bounds of the ground state energy of anharmonic oscillators using renormalized inner projection renormalization is a method often used to approximate the eigenvalues of a hamiltonian that cannot be solved exactly it consists of splitting the hamiltonian into a solvable part and a remainder which is then minimized the inner projection technique first introduced by lowdin in the sixties was developed to bracket the eigenvalues between lower and upper bounds combining renormalization and lowdin s inner projection yielded the so called renormalized inner projection technique in this study this method will be applied to the quartic sextic and octic anharmonic oscillators lower and upper energy bounds are obtained for finite values of the coupling constant as well as for the infinite case the relation between the renormalized inner projection and perturbation theory will also be discussed another feature of this study is the importance of symbolic computation in allowing us to manipulate expressions with unevaluated parameters and to perform calculations in rational arithmetics or high decimal precision thus lowdin s rational approximants can be expressed explicitly as rational fractions in terms of the coupling constant and values for the limit constant can be obtained with amazing high accuracy namely 62 33 and 21 decimal places for the quartic sextic and octic oscillator respectively