Two efficient recursive algorithms epsilon_algorithm and eta_algorithm are introduced to compute the generalized inverse function_valued Padé approximants. The approximants were used to accelerate the convergenc...Two efficient recursive algorithms epsilon_algorithm and eta_algorithm are introduced to compute the generalized inverse function_valued Padé approximants. The approximants were used to accelerate the convergence of the power series with function_valued coefficients and to estimate characteristic value of the integral equations. Famous Wynn identities of the Pad approximants is also established by means of the connection of two algorithms.展开更多
This study proposes a structure-preserving evolutionary framework to find a semi-analytical approximate solution for a nonlinear cervical cancer epidemic(CCE)model.The underlying CCE model lacks a closed-form exact so...This study proposes a structure-preserving evolutionary framework to find a semi-analytical approximate solution for a nonlinear cervical cancer epidemic(CCE)model.The underlying CCE model lacks a closed-form exact solution.Numerical solutions obtained through traditional finite difference schemes do not ensure the preservation of the model’s necessary properties,such as positivity,boundedness,and feasibility.Therefore,the development of structure-preserving semi-analytical approaches is always necessary.This research introduces an intelligently supervised computational paradigm to solve the underlying CCE model’s physical properties by formulating an equivalent unconstrained optimization problem.Singularity-free safe Padérational functions approximate the mathematical shape of state variables,while the model’s physical requirements are treated as problem constraints.The primary model of the governing differential equations is imposed to minimize the error between approximate solutions.An evolutionary algorithm,the Genetic Algorithm with Multi-Parent Crossover(GA-MPC),executes the optimization task.The resulting method is the Evolutionary Safe PadéApproximation(ESPA)scheme.The proof of unconditional convergence of the ESPA scheme on the CCE model is supported by numerical simulations.The performance of the ESPA scheme on the CCE model is thoroughly investigated by considering various orders of non-singular Padéapproximants.展开更多
In this short note, we show the behavior in Orlicz spaces of best approximations by algebraic polynomials pairs on union of neighborhoods, when the measure of them tends to zero.
The generalized inverse function-valued Padé approximant was defined to solve the integral equations. However, it is difficult to compute the approximants by some high-order determinant formulas. In this paper, t...The generalized inverse function-valued Padé approximant was defined to solve the integral equations. However, it is difficult to compute the approximants by some high-order determinant formulas. In this paper, to simplify computation of the function-valued Padé approximants, an efficient Pfaffian formula for the determinants was extended from the matrix form to the function-valued form. As an important application, a Pfaffian formula of [4/4] type Padé approximant was established.展开更多
Given a regular compact set E in , a unit measure μ supported by , a triangular point set , and a function f , holomorphic on E , let πβ,fn,m be the associated multipoint β-Padé approximant of order (n,m) . W...Given a regular compact set E in , a unit measure μ supported by , a triangular point set , and a function f , holomorphic on E , let πβ,fn,m be the associated multipoint β-Padé approximant of order (n,m) . We show that if the sequence πβ,fn,m , n∈Λ , ∧∈n,k are uniformly distributed on with respect to u as n∈Λ . Furthermore, a result about the behavior of the zeros of the exact maximally convergent sequence Λ is provided, under the condition that Λ is “dense enough”.展开更多
The Asymptotic Numerical Method (ANM) is a family of algorithms for path following problems, where each step is based on the computation of truncated vector series [1]. The Vector Padé approximants were introduce...The Asymptotic Numerical Method (ANM) is a family of algorithms for path following problems, where each step is based on the computation of truncated vector series [1]. The Vector Padé approximants were introduced in the ANM to improve the domain of validity of vector series and to reduce the number of steps needed to obtain the entire solution path [1,2]. In this paper and in the framework of the ANM, we define and build a new type of Vector Padé approximant from a truncated vector series by extending the definition of the Padé approximant of a scalar series without any orthonormalization procedure. By this way, we define a new class of Vector Padé approximants which can be used to extend the domain of validity in the ANM algorithms. There is a connection between this type of Vector Padé approximant and Vector Padé type approximant introduced in [3, 4]. We show also that the Vector Padé approximants introduced in the previous works [1,2], are special cases of this class. Applications in 2D nonlinear elasticity are presented.展开更多
An intrinsic extension of Pad′e approximation method, called the generalized Pad′e approximation method, is proposed based on the classic Pad′e approximation theorem. According to the proposed method, the numerator...An intrinsic extension of Pad′e approximation method, called the generalized Pad′e approximation method, is proposed based on the classic Pad′e approximation theorem. According to the proposed method, the numerator and denominator of Pad′e approximant are extended from polynomial functions to a series composed of any kind of function, which means that the generalized Pad′e approximant is not limited to some forms, but can be constructed in different forms in solving different problems. Thus, many existing modifications of Pad′e approximation method can be considered to be the special cases of the proposed method. For solving homoclinic and heteroclinic orbits of strongly nonlinear autonomous oscillators, two novel kinds of generalized Pad′e approximants are constructed. Then, some examples are given to show the validity of the present method. To show the accuracy of the method, all solutions obtained in this paper are compared with those of the Runge–Kutta method.展开更多
To solve Fredholm integral equations of the second kind, a generalized linear functional is introduced and a new function-valued Padé-type approximation is defined. By means of the power series expansion of the s...To solve Fredholm integral equations of the second kind, a generalized linear functional is introduced and a new function-valued Padé-type approximation is defined. By means of the power series expansion of the solution, this method can construct an approximate solution to solve the given integral equation. On the basis of the orthogonal polynomials, two useful determinant expressions of the numerator polynomial and the denominator polynomial for Padé-type approximation are explicitly given.展开更多
A new generalized inverse function-valued Padé approximation (GIFPA) was defined. Existence condition of GIFPA was given and its uniqueness theorem was proved. All possible degeneracy cases of GIFPA were discusse...A new generalized inverse function-valued Padé approximation (GIFPA) was defined. Existence condition of GIFPA was given and its uniqueness theorem was proved. All possible degeneracy cases of GIFPA were discussed and constructed. An example was given to illustrate its application.展开更多
The diagonal Padé approximants for exp ( x ), tan x and tanh x are obtained in a simple manner by using the property of Legendre polynomials that on P r1 (x) is orthogonal to every polynomial o...The diagonal Padé approximants for exp ( x ), tan x and tanh x are obtained in a simple manner by using the property of Legendre polynomials that on P r1 (x) is orthogonal to every polynomial of lower degree. Gauss's quadrature formula is used to find the denominators of some functions.展开更多
Abstract A new function-valued partial Padé-type approximation was introduced in the polynomial space, and an explicit determinant formula was derived by means of some orthogonal polynomials. This method can be a...Abstract A new function-valued partial Padé-type approximation was introduced in the polynomial space, and an explicit determinant formula was derived by means of some orthogonal polynomials. This method can be applied to estimating surplus eigenvalues of the Fredholm integral equation of the second kind when its partial eigenvalues have been known, and at the same time, it can be applied to solving the approximating solution of the given equation.展开更多
Matrix Padé approximation is a widely used method for computing matrix functions. In this paper, we apply matrix Padé-type approximation instead of typical Padé approximation to computing the matrix exp...Matrix Padé approximation is a widely used method for computing matrix functions. In this paper, we apply matrix Padé-type approximation instead of typical Padé approximation to computing the matrix exponential. In our approach the scaling and squaring method is also used to make the approximant more accurate. We present two algorithms for computing and for computing with many espectively. Numerical experiments comparing the proposed method with other existing methods which are MATLAB’s functions expm and funm show that our approach is also very effective and reliable for computing the matrix exponential . Moreover, there are two main advantages of our approach. One is that there is no inverse of a matrix required in this method. The other is that this method is more convenient when computing for a fixed matrix A with many t ≥ 0.展开更多
This paper presents an interpolation-based method(IBM)for approximating some trigonometric functions or their integrals as well.It provides two-sided bounds for each function,which also achieves much better approximat...This paper presents an interpolation-based method(IBM)for approximating some trigonometric functions or their integrals as well.It provides two-sided bounds for each function,which also achieves much better approximation effects than those of prevailing methods.In principle,the IBM can be applied for bounding more bounded smooth functions and their integrals as well,and its applications include approximating the integral of sin(x)/x function and improving the famous square root inequalities.展开更多
In the present paper, the formulae for matrix Padé-type approximation were improved. The mixed model reduction method of matrix Padé-type-Routh for the multivariable linear systems was presented. A well-know...In the present paper, the formulae for matrix Padé-type approximation were improved. The mixed model reduction method of matrix Padé-type-Routh for the multivariable linear systems was presented. A well-known example was given to illustrate that the mixed method is efficient.展开更多
文摘Two efficient recursive algorithms epsilon_algorithm and eta_algorithm are introduced to compute the generalized inverse function_valued Padé approximants. The approximants were used to accelerate the convergence of the power series with function_valued coefficients and to estimate characteristic value of the integral equations. Famous Wynn identities of the Pad approximants is also established by means of the connection of two algorithms.
文摘This study proposes a structure-preserving evolutionary framework to find a semi-analytical approximate solution for a nonlinear cervical cancer epidemic(CCE)model.The underlying CCE model lacks a closed-form exact solution.Numerical solutions obtained through traditional finite difference schemes do not ensure the preservation of the model’s necessary properties,such as positivity,boundedness,and feasibility.Therefore,the development of structure-preserving semi-analytical approaches is always necessary.This research introduces an intelligently supervised computational paradigm to solve the underlying CCE model’s physical properties by formulating an equivalent unconstrained optimization problem.Singularity-free safe Padérational functions approximate the mathematical shape of state variables,while the model’s physical requirements are treated as problem constraints.The primary model of the governing differential equations is imposed to minimize the error between approximate solutions.An evolutionary algorithm,the Genetic Algorithm with Multi-Parent Crossover(GA-MPC),executes the optimization task.The resulting method is the Evolutionary Safe PadéApproximation(ESPA)scheme.The proof of unconditional convergence of the ESPA scheme on the CCE model is supported by numerical simulations.The performance of the ESPA scheme on the CCE model is thoroughly investigated by considering various orders of non-singular Padéapproximants.
文摘In this short note, we show the behavior in Orlicz spaces of best approximations by algebraic polynomials pairs on union of neighborhoods, when the measure of them tends to zero.
文摘The generalized inverse function-valued Padé approximant was defined to solve the integral equations. However, it is difficult to compute the approximants by some high-order determinant formulas. In this paper, to simplify computation of the function-valued Padé approximants, an efficient Pfaffian formula for the determinants was extended from the matrix form to the function-valued form. As an important application, a Pfaffian formula of [4/4] type Padé approximant was established.
文摘Given a regular compact set E in , a unit measure μ supported by , a triangular point set , and a function f , holomorphic on E , let πβ,fn,m be the associated multipoint β-Padé approximant of order (n,m) . We show that if the sequence πβ,fn,m , n∈Λ , ∧∈n,k are uniformly distributed on with respect to u as n∈Λ . Furthermore, a result about the behavior of the zeros of the exact maximally convergent sequence Λ is provided, under the condition that Λ is “dense enough”.
文摘The Asymptotic Numerical Method (ANM) is a family of algorithms for path following problems, where each step is based on the computation of truncated vector series [1]. The Vector Padé approximants were introduced in the ANM to improve the domain of validity of vector series and to reduce the number of steps needed to obtain the entire solution path [1,2]. In this paper and in the framework of the ANM, we define and build a new type of Vector Padé approximant from a truncated vector series by extending the definition of the Padé approximant of a scalar series without any orthonormalization procedure. By this way, we define a new class of Vector Padé approximants which can be used to extend the domain of validity in the ANM algorithms. There is a connection between this type of Vector Padé approximant and Vector Padé type approximant introduced in [3, 4]. We show also that the Vector Padé approximants introduced in the previous works [1,2], are special cases of this class. Applications in 2D nonlinear elasticity are presented.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11172093 and 11372102)the Hunan Provincial Innovation Foundation for Postgraduate,China(Grant No.CX2012B159)
文摘An intrinsic extension of Pad′e approximation method, called the generalized Pad′e approximation method, is proposed based on the classic Pad′e approximation theorem. According to the proposed method, the numerator and denominator of Pad′e approximant are extended from polynomial functions to a series composed of any kind of function, which means that the generalized Pad′e approximant is not limited to some forms, but can be constructed in different forms in solving different problems. Thus, many existing modifications of Pad′e approximation method can be considered to be the special cases of the proposed method. For solving homoclinic and heteroclinic orbits of strongly nonlinear autonomous oscillators, two novel kinds of generalized Pad′e approximants are constructed. Then, some examples are given to show the validity of the present method. To show the accuracy of the method, all solutions obtained in this paper are compared with those of the Runge–Kutta method.
基金Project supported by the National Natural Science Foundation of China (No. 10271074)
文摘To solve Fredholm integral equations of the second kind, a generalized linear functional is introduced and a new function-valued Padé-type approximation is defined. By means of the power series expansion of the solution, this method can construct an approximate solution to solve the given integral equation. On the basis of the orthogonal polynomials, two useful determinant expressions of the numerator polynomial and the denominator polynomial for Padé-type approximation are explicitly given.
文摘A new generalized inverse function-valued Padé approximation (GIFPA) was defined. Existence condition of GIFPA was given and its uniqueness theorem was proved. All possible degeneracy cases of GIFPA were discussed and constructed. An example was given to illustrate its application.
文摘The diagonal Padé approximants for exp ( x ), tan x and tanh x are obtained in a simple manner by using the property of Legendre polynomials that on P r1 (x) is orthogonal to every polynomial of lower degree. Gauss's quadrature formula is used to find the denominators of some functions.
基金Project supported by the National Natural Science Foundation of China(Grant No.10271074)
文摘Abstract A new function-valued partial Padé-type approximation was introduced in the polynomial space, and an explicit determinant formula was derived by means of some orthogonal polynomials. This method can be applied to estimating surplus eigenvalues of the Fredholm integral equation of the second kind when its partial eigenvalues have been known, and at the same time, it can be applied to solving the approximating solution of the given equation.
文摘Matrix Padé approximation is a widely used method for computing matrix functions. In this paper, we apply matrix Padé-type approximation instead of typical Padé approximation to computing the matrix exponential. In our approach the scaling and squaring method is also used to make the approximant more accurate. We present two algorithms for computing and for computing with many espectively. Numerical experiments comparing the proposed method with other existing methods which are MATLAB’s functions expm and funm show that our approach is also very effective and reliable for computing the matrix exponential . Moreover, there are two main advantages of our approach. One is that there is no inverse of a matrix required in this method. The other is that this method is more convenient when computing for a fixed matrix A with many t ≥ 0.
基金Supported by the National Natural Science Foundation of China(61672009,61502130).
文摘This paper presents an interpolation-based method(IBM)for approximating some trigonometric functions or their integrals as well.It provides two-sided bounds for each function,which also achieves much better approximation effects than those of prevailing methods.In principle,the IBM can be applied for bounding more bounded smooth functions and their integrals as well,and its applications include approximating the integral of sin(x)/x function and improving the famous square root inequalities.
基金Project supported by National Natural Science Foundation of China (Grant No .10271074)
文摘In the present paper, the formulae for matrix Padé-type approximation were improved. The mixed model reduction method of matrix Padé-type-Routh for the multivariable linear systems was presented. A well-known example was given to illustrate that the mixed method is efficient.
