generation of root finding algorithms via perturbation theory and some formulas abstract perturbation theory is systematically used to generate root finding algorithms depending on the number of correction terms in the perturbation expansion and the number of taylor expansion terms different root finding formulas can be generated the way of separating the resulting equations after the perturbation expansion alters the root finding formulas also well known cases such as newton raphson and its second correction namely the householder s iteration are derived as examples moreover higher order algorithms which may or may not be the corrections of well known formulas are derived the formulas are contrasted with each other as well as with some new algorithms obtained by modified adomian decomposition method proposed in ref s abbasbandy improving newton raphson method for nonlinear equations by modified adomian decomposition method applied mathematics and computation 145 2003 887 893

a simple solution of the bratu problem a brief survey of the properties and different treatments of the one dimensional 1d and 2d bratu problems is presented different iterative treatments of the resulting nonlinear system of equations are discussed the finite difference treatment of the problem is considered nonstandard finite difference methods with a simple sinusoidal starting function having an appropriate amplitude are recommended bounds on the amplitude for yielding both lower and upper solutions are given

a rank one reduction formula and its applications to matrix factorizations let a in r m times n denote an arbitrary matrix if x in r n and y in r m are vectors such that omega y t ax ne 0 then the matrix b a omega 1 axy t a a has rank exactly one less than the rank of a this wedderburn rank one reduction formula is easy to prove yet the idea is so powerful that perhaps all matrix factorizations can be derived from it the formula also appears in places such as the positive definite secant updates bfgs and dfp as well as the abs methods by repeatedly applying the formula to reduce ranks a biconjugation process analogous to the gram schmidt process with oblique projections can be developed this process provides a mechanism for constructing factorizations such as text ldm t qr and svd under a common framework of a general biconjugate decomposition v t au omega that is diagonal and nonsingular two characterizations of biconjugation provide new insight into the lanczos method and its breakdown one characterization shows that

an efficient iterative method for solving bratu type equations abstract in this paper an efficient analytical iterative method is introduced for solving the initial value and boundary value bratu problems a quasilinearization method with an optimal parameter is derived and utilized to reduce the highly nonlinear bratu equations to a sequence of linear problems an optimal picard iteration method is used to obtain approximate analytical solution of linearized versions of bratu s equations the numerical results presented the effectiveness and high performance of the proposed method and comparisons are made with many other existing numerical solution techniques the comparisons reveal the efficiency and applicability of the present work

a modified perturbation solution to the one dimensional bratu problem abstract an approximate analytical solution to the one dimensional bratu boundary value problem is introduced in this paper the solution is based on some perturbation expansion methods the first step is taken from the linearized perturbation technique which was developed to solve initial value problem with a nonlinear term and no small parameter an artificial small parameter is embedded and the dependent variable can then be expanded in terms of this parameter however necessary modifications are introduced to implement some techniques to solve a boundary value problem in order to allow for different nonlinearities the current solution showed good convergence at any value of the bratu constant when compared with the exact lower branch of the solution to the one dimensional bratu problem also its recursive nature allows for more iterations and adding more correction terms to the final approximate solution which increases the accuracy

constraint qualifications in quasidifferentiable optimization the classical linearization procedure for differentiable nonlinear programming problems can be naturally generalized to the quasidifferentiable case as in the classical case one has to impose so called constraint qualifications on the constraint functions in order to ensure that optimality of a feasible point implies optimality of the nullvector for the corresponding quasilinearized problem we present various constraint qualifications in a unified setting propose a new one and investigate the relations between these conditions

decay rates for inverses of band matrices spectral theory and classical approximation theory are used to give a new proof of the exponential decay of the entries of the inverse of band matrices the rate of decay of a 1 can be bounded in terms of the essential spectrum of aa for general a and in terms of the essential spectrum of a for positive definite a in the positive definite case the bound can be attained these results are used to establish the exponential decay for a class of generalized eigenvalue problems and to establish exponential decay for certain sparse but nonbanded matrices we also establish decay rates for certain generalized inverses

a transformation approach that makes spai psai and rsai procedures efficient for large double irregular nonsymmetric sparse linear systems abstract a sparse matrix is called double irregular sparse if it has at least one relatively dense column and row and it is double regular sparse if all the columns and rows of it are sparse the sparse approximate inverse preconditioning procedures spai psai t o l and rsai t o l are costly and even impractical to construct preconditioners for a large sparse nonsymmetric linear system with the coefficient matrix being double irregular sparse but they are efficient for double regular sparse problems double irregular sparse linear systems have a wide range of applications and 24 4 of the nonsymmetric matrices in the florida university collection are double irregular sparse for this class of problems we propose a transformation approach which consists of four steps i transform a given double irregular sparse problem into a small number of double regular sparse ones with the same coefficient matrix a ii use spai psai t o l and rsai t o l to construct sparse approximate inverses m of a iii solve the preconditioned double regular sparse linear systems by krylov solvers and iv recover an approximate solution of the original problem with a prescribed accuracy from those of the double regular sparse ones a number of theoretical and practical issues are considered on the transformation approach numerical experiments on a number of real world problems confirm the very sharp superiority of the transformation approach to the standard approach that preconditions the original double irregular sparse problem by spai psai t o l or rsai t o l and solves the resulting preconditioned system by krylov solvers

on optimality conditions in quasidifferentiable optimization a class of quasidifferentiable functions whose directional derivatives are representable as a difference of two sublinear functions is introduced an optimization problem subject to quasidifferentiable equality and inequality constraints is studied optimality conditions for this problem are given in terms of sets inclusion this refines the results of demyanov rubinov and co workers

adomian decomposition method for a reliable treatment of the bratu type equations in this paper we present a framework to determine exact solutions of bratu type equations the algorithm rests mainly on the adomian decomposition method the proposed scheme is illustrated by studying two boundary value problems and an initial value problem of bratu type the first type gives a solution that blows up at the middle of the domain whereas the other equations give bounded solutions

the legendre wavelet method for solving initial value problems of bratu type the aim of this work is to study the legendre wavelets for the solution of initial value problems of bratu type which is widely applicable in fuel ignition of the combustion theory and heat transfer the properties of legendre wavelets together with the gaussian integration method are used to reduce the problem to the solution of nonlinear algebraic equations also a reliable approach for convergence of the legendre wavelet method when applied to a class of nonlinear volterra equations is discussed and an error estimation for the proposed method is also introduced illustrative examples have been discussed to demonstrate the validity and applicability of the technique and the results have been compared with the exact solution we finally show the high accuracy and efficiency of the proposed method

an algorithm for solving boundary value problems in this paper a numerical algorithm based on the decomposition technique is presented for solving a class of nonlinear boundary value problems the method is implemented for well known examples including troesch s and bratu s problems which have been extensively studied the scheme is shown to be highly accurate and only a few terms are required to obtain accurate computable solutions

homotopy perturbation method a new nonlinear analytical technique in this paper a new perturbation method is proposed in contrast to the traditional perturbation methods this technique does not require a small parameter in an equation in this method according to the homotopy technique a homotopy with an imbedding parameter email protected 0 1 is constructed and the imbedding parameter is considered as a small parameter so the method is called the homotopy perturbation method which can take the full advantages of the traditional perturbation methods and homotopy techniques to illustrate its effectiveness and its convenience a duffing equation with high order of nonlinearity is used the result reveals that its first order of approximation obtained by the proposed method is valid uniformly even for very large parameter and is more accurate than the perturbation solutions

comparison between adomian s method and he s homotopy perturbation method in this paper it is revealed that modified form of he s homotopy perturbation method corresponds to adomian s decomposition method for certain nonlinear problems

a classical perturbation technique which is valid for large parameters there are various perturbation techniques for constructing analytical approximations to the oscillatory solutions of second order non linear differential equations 1 2 but many of them apply to weakly non linear cases only to overcome the limitations many novel techniques have been proposed in recent years for example cheung et al 3 proposed a modified lindstedt poincare method and lim et al 4 presented a modified mickens procedure for certain non linear oscillators recently he 5 proposed a perturbation technique which is valid for large parameters but unfortunately o1 and b in ref 5 are the functions of z which is unknown the main purpose of this paper is to point out that there already exists a classical perturbation technique which is valid for large parameters in this paper the following well known duffing equation is considered

reduction of quasidifferentials and minimal representations some criterias for the non minimality of pairs of compact convex sets of a real locally convex topological vector space are proved based on a reduction technique via cutting planes and excision of compact convex subsets following an example of j grzybowski we construct a class of equivalent minimal pairs of compact convex sets which are not connected by translations

an analytic approach to solve multiple solutions of a strongly nonlinear problem based on a new kind of analytic method namely the homotopy analysis method an analytic approach to solve multiple solutions of strongly nonlinear problems is described by using gelfand equation as an example its validity is verified by comparing the approximation series with the known exact solution and different from perturbation techniques this approach is independent upon any small large perturbation quantities so the basic ideas of this approach can be employed to search for multiple solutions of strongly nonlinear problems in science and engineering

iteration method solutions for conservative and limit cycle x1 3 force oscillators abstract an iterative technique is used to calculate a higher order approximation to the periodic solutions of a conservative oscillator for which the elastic force term is proportional to x 1 3 the related van der pol type limit cycle oscillator is also studied

a sparse approximate inverse preconditioner for the conjugate gradient method a method for computing a sparse incomplete factorization of the inverse of a symmetric positive definite matrix a is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient calculations it is proved that in exact arithmetic the preconditioner is well defined if a is an h matrix the results of numerical experiments are presented

chebyshev polynomial expansions for simultaneous approximation of two branches of a function with application to the one dimensional bratu equation we introduce two methods for simultaneously approximating both branches of a two branched function using chebyshev polynomials both schemes remove the pernicious convergence wrecking effects of the square root singularity at the limit point where the two branches meet the chebyshev shafer method gives the approximants as the solution to a quadratic equation the mapping algorithm makes a quadratic change of variable for both the only input information is the set of values of f x at a set of discrete points there is little to choose between the two schemes in accuracy but the single expansion mapping method is more flexible in that it can accommodate different ranges on the two branches the eigenrelation for the one dimensional bratu equation is an interesting example because the upper branch is also singular at another point besides the limit point this too can be removed by subtracting the asymptotic solution which is the lambert w function from the upper branch only when the domain of the variable is infinite the quadratic change of variable can still be applied by substituting rational chebyshev functions which are basis functions for an unbounded interval for chebyshev polynomials we illustrate this by approximating the real valued root of the brill quintic u 5 u l l which was first solved by hermite using elliptic modular functions more than a century ago

a direct projection method for sparse linear systems an oblique projection method is adapted to solve large sparse unstructured systems of linear equations this row projection technique is a direct method which can be interpreted as an oblique kaczmarz type algorithm and is also related to other standard solution methods when a sparsity preserving pivoting strategy is incorporated it is demonstrated that the technique can be superior in terms of both fill in and arithmetic complexity to more standard sparse algorithms based on gaussian elimination this is especially true for systems arising from stiff ordinary differential equations problems in chemical kinetics studies

an efficient method for solving bratu equations in this paper we present a numerical technique for solving bratu equation it is based on the laplace adomain decomposition method which produces an implicit equation in two variables we used the predictor corrector technique to trace the solution curve generated from this equation numerical results and conclusions will be presented

an iterative finite difference method for solving bratu s problem in this paper we propose a new iterative finite difference ifd scheme based on the newton raphson kantorovich approximation method in function space to solve the classical one dimensional bratu s problem this new numerical method produces accurate solutions with low computational cost the effectiveness and accuracy of the ifd method are confirmed through several numerical examples and compared to some existing numerical solvers

a simple perturbation approach to blasius equation in this paper we couple the iteration method with the perturbation method to solve the well known blasius equation the obtained approximate analytic solutions are valid for the whole solution domain comparison with howarth s numerical solution reveals that the proposed method is of high accuracy the first iteration step leads to 6 8 accuracy and the second iteration step yields the 0 73 accuracy of initial slop

new breakdown free variant of ainv method for nonsymmetric positive definite matrices this paper proposes a new breakdown free preconditioning technique called sainv ns of the ainv method of benzi and tuma for nonsymmetric positive definite matrices the resulting preconditioner which is an incomplete factorization of the inverse of a nonsymmetric matrix will be used as an explicit right preconditioner for qmr bicgstab and gmres m methods the preconditoner is reliable pivot breakdown can not occur and effective at reducing the number of iterations some numerical experiments on test matrices are presented to show the efficiency of the new method and comparing to the ainv a algorithm

approximate inverse preconditioners via sparse sparse iterations the standard incomplete lu ilu preconditioners often fail for general sparse indefinite matrices because they give rise to unstable factors l and u in such cases it may be attractive to approximate the inverse of the matrix directly this paper focuses on approximate inverse preconditioners based on minimizing i am f where am is the preconditioned matrix an iterative descent type method is used to approximate each column of the inverse for this approach to be efficient the iteration must be done in sparse mode i e with sparse matrix by sparse vector operations numerical dropping is applied to maintain sparsity compared to previous methods this is a natural way to determine the sparsity pattern of the approximate inverse this paper describes newton global and column oriented algorithms and discusses options for initial guesses self preconditioning and dropping strategies some limited theoretical results on the properties and convergence of approximate inverses are derived numerical tests on problems from the harwell boeing collection and the fidap fluid dynamics analysis package show the strengths and limitations of approximate inverses finally some ideas and experiments with practical variations and applications are presented

on shapiro s results in quasidifferential calculus several necessary minimum conditions for quasidifferentiable programming problems are discussed some statements of shapiro 1986 are shown to be incorrect

linearized perturbation technique and its applications to strongly nonlinear oscillators abstract a new perturbation technique called linearized perturbation method is proposed contrary to the traditional perturbation techniques the unperturbed equations is obtained by linearizing the original nonlinear equation not by setting 0 therefore the obtained results are valid not only for small parameter but also for very large values of the present theory is processed as simple as the straightforward expansion while omits the secular terms completely

preconditioning techniques for nonsymmetric and indefinite linear systems abstract the standard preconditioning techniques for conjugate gradient methods often fail for matrices that are indefinite and or strongly nonsymmetric the most common alterative considered for these cases are either to use expensive direct solvers or to resort to one of many techniques based on the normal equations this paper examines several such alternatives and compares them in particular an incomplete lq factorization is proposed and some of its implementation details are described a number of experiments are reported to compare these methods

a generalized iteration procedure for calculating approximations to periodic solutions of truly nonlinear oscillators abstract an extended iteration method for calculating the periodic solutions of nonlinear oscillator equations is given the procedure is illustrated by applying it to two truly nonlinear oscillator differential equations

a sparse approximate inverse preconditioner for nonsymmetric linear systems this paper is concerned with a new approach to preconditioning for large sparse linear systems a procedure for computing an incomplete factorization of the inverse of a nonsymmetric matrix is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient type methods some theoretical properties of the preconditioner are discussed and numerical experiments on test matrices from the harwell boeing collection and from tim davis s collection are presented our results indicate that the new preconditioner is cheaper to construct than other approximate inverse preconditioners furthermore the new technique insures convergence rates of the preconditioned iteration which are comparable with those obtained with standard implicit preconditioners

new perturbation iteration solutions for bratu type equations perturbation iteration theory is systematically generated for both linear and nonlinear second order differential equations and applied to bratu type equations different perturbation iteration algorithms depending upon the number of taylor expansion terms are proposed using the iteration formulas derived using different perturbation iteration algorithms new solutions of bratu type equations are obtained solutions constructed using different perturbation iteration algorithms are contrasted with each other as well as with numerical solutions it is found that algorithms with more taylor series expansion terms yield more accurate results

approximate inverse techniques for block partitioned matrices this paper proposes some preconditioning options when the system matrix is in block partitioned form this form may arise naturally for example from the incompressible navier stokes equations or may be imposed after a domain decomposition reordering approximate inverse techniques are used to generate sparse approximate solutions whenever these are needed in forming the preconditioner the storage requirements for these preconditioners may be much less than for incomplete lu factorization ilu preconditioners for tough large scale computational fluid dynamics cfd problems the numerical experiments show that these preconditioners can help solve difficult linear systems whose coefficient matrices are highly indefinite

on constraint qualifications in nonlinear programming abstract in this paper we give conditions for deriving the inconsistency of an inequality system of positively homogeneous functions starting from the inconsistency of another one when the impossibility of the starting system represents a necessary optimality condition for an inequality constrained extremum problem and the positively homogeneous functions involved have suitable properties of convexity such conditions collapse into the well known constraint qualifications

polynomial preconditioners for conjugate gradient calculations dubois greenbaum and rodrigue proposed using a truncated neumann series as an approximation to the inverse of a matrix a for the purpose of preconditioning conjugate gradient iterative approximations to ax b if we assume that a has been symmetrically scaled to have unit diagonal and is thus of the form i g then the neumann series is a power series in g with unit coefficients the incomplete inverse was thought of as a replacement of the incomplete cholesky decomposition suggested by meijerink and van der vorst in the family of methods iccg n the motivation for the replacement was the desire to have a preconditioned conjugate gradient method which only involved vector operations and which utilized long vectors we here suggest parameterizing the incomplete inverse to form a preconditioning matrix whose inverse is a polynomial in g we then show how to select the parameters to minimize the condition number of the product of the polynomial and i g theoretically the resulting algorithm

a new approach to bratu s problem in this paper a laplace transform decomposition numerical algorithm is introduced for solving bratu s problem the numerical scheme is based on the application of laplace transform integral operator to the differential equation the nonlinear term is then decomposed and an iterative algorithm is constructed for the determination of the infinite series solution the technique is illustrated with two numerical examples and the results show that the method converges rapidly and approximates the exact solution very accurately using only few iterates of the recursive scheme

a stability analysis of incomplete lu factorizations abstract the combination of iterative methods with preconditionings based on incomplete lu factorizations constitutes an effective class of methods for solving the sparse linear systems arising from the discretization of elliptic partial differential equations in this paper we show that there are some settings in which the incomplete lu preconditioners are not effective and we demonstrate that their poor performance is due to numerical instability our analysis consists of an analytic and numerical study of a sample two dimensional non self adjoint elliptic problem discretized by several finite difference schemes author

solutions of the duffing harmonic oscillator by an iteration procedure a modified iteration procedure is applied to the duffing harmonic oscillator with the procedure the excellent approximate frequencies and the corresponding periodic solutions can be easily obtained

iterative differential quadrature solutions for bratu problem abstract a numerical scheme based on differential quadrature methods is introduced for solving bratu problem the problem is firstly reduced to an iterative one then both of differential quadrature method dqm and moving least squares differential quadrature method mlsdqm are applied to solve iteratively the nonlinear problem the proposed scheme successfully computes multiple solutions to bratu s problem the obtained results agree with the 1d and 2d closed forms further a parametric study is introduced to investigate the computational characteristics of the proposed scheme

left looking version of ainv preconditioner with complete pivoting strategy abstract in this paper we apply a complete pivoting strategy to compute the left looking version of ainv preconditioner for linear systems as the preprocessing the multilevel nested dissection reordering has also been applied we have used this preconditioner as the right preconditioner for several linear systems where the coefficient matrices have been downloaded from the university of florida sparse matrix collection numerical experiments presented in this paper indicate the effectiveness of such a complete pivoting on the quality of left looking version of ainv preconditioner

parallel preconditioning with sparse approximate inverses a parallel preconditioner is presented for the solution of general sparse linear systems of equations a sparse approximate inverse is computed explicitly and then applied as a preconditioner to an iterative method the computation of the preconditioner is inherently parallel and its application only requires a matrix vector product the sparsity pattern of the approximate inverse is not imposed a priori but captured automatically this keeps the amount of work and the number of nonzero entries in the preconditioner to a minimum rigorous bounds on the clustering of the eigenvalues and the singular values are derived for the preconditioned system and the proximity of the approximate to the true inverse is estimated an extensive set of test problems from scientific and industrial applications provides convincing evidence of the effectiveness of this approach