CP methods of higher order for Sturm-Liouville and Schrödinger equations

The algorithm upon which the code SLCPM12, described in Computer Physics Communications 118 (1999) 259-277, is based, is extended to higher order. The implementation of the original algorithm, which was of order {12,10} (meaning order 12 at low energies and order 10 at high energies), was more efficient than the well-established codes SL02F, SLEDGE and SLEIGN. In the new algorithm the orders {14,12}, {16,14} and {18,16} are introduced. Besides regular Sturm-Liouville and one-dimensional Schrödinger problems also radial Schrödinger equations are considered with potentials of the form V(r)=S(r)/r+R(r), where S(r) and R(r) are well behaved functions which tend to some (not necessarily equal) constants when r→0 and r→∞. Numerical illustrations are given showing the accuracy, the robustness and the CPU-time gain of the proposed algorithms.     

Homotopy perturbation technique for solving two-point boundary value problems - comparison with other methods

The two-point boundary value problems occur in a wide variety of problems in engineering and science. In this paper, we implement the homotopy perturbation method for solving the linear and nonlinear two-point boundary value problems. The main aim of this paper is to compare the performance of the homotopy perturbation method with extended Adomian decomposition method and shooting method. As a result, for the same number of terms, the homotopy perturbation method yields relatively more accurate results with rapid convergence than other methods. The computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations.     

Solving problems with unilateral constraints by DAE methods

We study trajectories of mechanical systems with unilateral constraints under the additional assumption that always a given number of constraints is active. A reformulation as a problem with bilateral conditions yields a drastic reduction in the number of constraints, but in general, we are faced with regularity problems. We illustrate our approach in the special case of a dynamical rigid body contact problem. In particular, we present a regularization technique which leads to the definition of generalized solutions and a quite effective numerical method on the basis of algorithms for differential–algebraic systems. The results are applied to a wheel–rail contact problem of actual interest to railway engineers.     

Multiparameter singular perturbation problems: Iterative expansions and asymptotic stability

Recursive formulae are derived which yield asymptotic expansions for the eigenvalues of multiparameter singular perturbation problems. These formulae follow readily from an exact expression for the eigenvalues which involves an implicit matrix function. The implicit function satisfies an algebraic matrix Riccati equation reminiscent of a similar equation of the single parameter theory. The results also explicate the ‘block D-stability’ criterion for asymptotic stability previously introduced by Khalil and Kokotovic.     

Solving discretized optimization problems by partially reduced SQP methods

The solution of discretized optimization problems is a major task in many application areas from engineering and science. These optimization problems present various challenges which result from the high number of variables involved as well as from the properties of the underlying process to be optimized. They also provide several strucures which have to be exploited by efficient numerical solution approaches. In this paper we focus on partially reduced SQP methods which are shown to be particularly well suited for this problem class. In practical applications the efficiency of this approach is demonstrated for optimization problems resulting from discretized DAE as well as from discretized PDE. The practically important issues of inexact solution of linearized subproblems and of working range validation are tackled as well. Received: 17 April 1997 / Accepted: 2 September 1997     

Numerical solution of initial-value problems by collocation methods using generalized piecewise functions

A general class of piecewise functions is described which leads to the same order of convergence of collocation methods as piecewise polynomials. This order only depends on the collocation points used.     

Solution of the Schrödinger equation over an infinite integration interval by perturbation methods, revisited

We consider the solution of the one-dimensional Schrödinger problem over an infinite integration interval. The infinite problem is regularized by truncating the integration interval and imposing the appropriate boundary conditions at the truncation points. The Schrödinger problem is then solved on the truncated integration interval using one of the piecewise perturbation methods developed for the regular Schrödinger problem.We select the truncation points using a procedure based on the WKB approximation. However for problems which behave as a Coulomb problem both around the origin and in the asymptotic range, a more accurate treatment of the numerical boundaries is possible. Taking into account the asymptotic form of the Coulomb equation, adapted boundary conditions can be constructed and as a consequence smaller truncation points can be chosen. To deal with the singularity of the Coulomb-like problem around the origin, a special perturbative algorithm is applied in a small interval around the origin.     

用割线法迭代求解对称三对角矩阵特征值问题

为对称三对角矩阵特征值问题，提出一种新的分而治之的算法。新算法以二分法，割线法迭代为基础，不同于Ｃｕｐｐｅｎ的方法和Ｌａｎｇｕｅｒｒｅ迭代法。理论分析和数据实验的结果表明：新算法的收敛速度明显比文［１］中的Ｌａｇｕｅｒｒｅ迭代法快。     

Solving singular nonlinear boundary value problems by combining the homotopy perturbation method and reproducing kernel Hilbert space method

This paper investigates singular nonlinear boundary value problems (BVPs). The numerical solutions are developed by combining He's homotopy perturbation method (HPM) and reproducing kernel Hilbert space method (RKHSM). He's HPM is based on the use of traditional perturbation method and homotopy technique. The HPM can reduce a nonlinear problem to a sequence of linear problems and generate a rapid convergent series solution in most cases. RKHSM is also an analytical technique, which can solve powerfully singular linear BVPs. Therefore, we solve singular nonlinear BVPs using advantages of these two methods. Three numerical examples are presented to illustrate the strength of the method.     

Solving certain queueing problems modelled by toeplitz matrices

By using the concept of generating function associated with a Toeplitz matrix, we analyze existence conditions for the probability invariant vector π of certain stochastic semi-infinite Toeplitz-like matrices. An application to the shortest queue problem is shown. By exploiting the functional formulation given in terms of generating functions, we devise a weakly numerically stable algorithm for computing the probability invariant vector π. The algorithm is divided into three stages. At the first stage the zeros of a complex function are numerically computed by means of an extension of the Aberth method. At the second stage the first k components of π are computed by solving an interpolation problem, where k is a suitable constant associated with the matrix. Finally, at the third stage a triangular Toeplitz system is solved and its solution is refined by applying the power method or any other refinement method based on regular splittings. In the solution of the triangular Toeplitz system and at each step of the refinement method, special FFT-based techniques are applied in order to keep the arithmetic cost within the O(n log n) bound, where n is an upper bound to the number of the computed components. Numerical comparisons with the available algorithms show the effectiveness of our algorithm in a wide set of cases.     

光滑神经网络解决非李普西茨优化问题的研究

为寻求满足约束条件的优化问题的最优解,针对目标函数是非李普西茨函数,可行域由线性不等式或非线性不等式约束函数组成的区域的优化问题,构造了一种光滑神经网络模型。此模型通过引进光滑逼近技术将目标函数由非光滑函数转换成相应的光滑函数以及结合惩罚函数方法所构造而成。通过详细的理论分析证明了不论初始点在可行域内还是在可行域外,光滑神经网络的解都具有一致有界性和全局性,以及光滑神经网络的任意聚点都是原始优化问题的稳定点等结论。最后通过几个简单的仿真实验证明了理论的正确性。     

Solving fuzzy shortest path problems by neural networks

In this paper, we introduce the neural networks for solving fuzzy shortest path problems. The penalization of the neural networks is realized after transforming into crisp shortest path model. The procedure and efficiency of this approach are shown with numerical simulations.     

用口诀法学会解网络图计算试题

网络图计算一直以来是计算机软件资格与水平考试中信息系统集成项目管理工程师和项目管理师考试的重点,也是考生的难点。对此,给出了一种口诀法,逐步引导考生理解术语、学会做图、学会找出关键路径、学会计算,最后给出了一个运用口诀法解题的案例。     

Solving inverse eigenvalue problems by a projected Newton method

The inverse eigenvalue problem for symmetric matrices (IEP) can be formulated as a system of two matrix equations. For solving the system a variation of Newton's method is used which has been proposed by Fusco and Zecca [Calcolo XXIII (1986), pp. 285–303] for the simultaneous computation of eigenvalues and eigenvectors of a given symmetric matrix. An iteration step of this method consists of a Newton step followed by an orthonormalization with the consequence that each iterate satisfies one of the given equations. The method is proved to convergence locally quadratically to regular solutions. The algorithm and some numerical examples are presented. In addition, it is shown that the so-called Method III proposed by Friedland, Nocedal, and Overton [SIAM J. Numer. Anal., 24 (1987), pp. 634–667] for solving IEP may be constructed similarly to the method presented here.     

Solving inverse initial-value, boundary-value problems via genetic algorithm

There is a growing interest in inverse initial-value, boundary-value (inverse IVBV) problems, and in the development of robust, computationally efficient methods suitable for their solution. Inverse problems are prominent in science and engineering where often an effect is measured and the cause is not known; scientists and engineers observe the response of a system and desire to know the particulars of the system that elicited such a response. IVBV problems result when the equations that govern the behavior of a system are partial differential equations (wave phenomena, diffusion, potential of all kinds, etc.). Thus, inverse IVBV problems stem from systems governed by partial differential equations in which a response has been measured and a characteristic of the system must be computed. In this paper, an approach to solving inverse IVBV problems is presented in which the stated problem is transformed into a nonlinear optimization problem which is then solved using a genetic algorithm. Results are presented demonstrating the effectiveness of this approach for solving inverse problems that result from systems governed by three specific partial differential (1) the heat equation, (2) the wave equation, and (3) Poisson’s equation.     

Solving 2D-wave problems by the iterative differential quadrature method

In this paper, the numerical stability of an iterative method based on differential quadrature (DQ) rules when applied to solve a two-dimensional (2D) wave problem is discussed. The physical model of a vibrating membrane, with different initial conditions, is considered. The stability analysis is performed by the matrix method generalized for a 2D space-time domain. This method was presented few years ago by the same author as an analytical support to check the stability of the iterative differential quadrature method in 1D space-time domains. The stability analysis confirms here the conditionally stable nature of the method. The accuracy of the solution is discussed too.