Algorithm For Symbolic Computation Research Articles

A symbolic computational algorithm for designing feedback stabilizers of polynomial non-linear systems

A symbolic computational algorithm for designing feedback stabilizers of polynomial non-linear systems

HBFTrans2: A Maple Package to Construct Hirota Bilinear Form for Nonlinear Equations

An improved algorithm for symbolic computation of Hirota bilinear form of nonlinear equations by a logarithm transformation is presented. The improved algorithm is more efficient by using the property of Hirota-D operator. The software package HBFTrans2 is written in Maple and its running efficiency is tested by a variety of soliton equations.

Symbolic algorithm for inverting cyclic pentadiagonal matrices recursively — Derivation and implementation

In this paper, by using parallel computing along with recursion, we describe a reliable symbolic computational algorithm for inverting cyclic pentadiagonal matrices. The algorithm is implemented in MAPLE. Two other symbolic algorithms are developed and the computational costs for all algorithms are given. An example is presented for the sake of illustration.

New Exact Solutions of (2+1)-Dimensional Generalization of Shallow Water Wave Equation by (G′/G)-Expansion Method

Extending a symbolic computation algorithm, namely, (G′/G)-expansion method, a new series of exact solutions are constructed for (2+1)-dimensional generalization of shallow water wave equation. These solutions included hyperbolic function solution, trigonometric function solution and rational function solution. The procedure can illustrate that the new algorithm is concise, powerful and straightforward, and it can also be applied to find exact solutions for other high dimensional nonlinear evolution equations.

A Maple Package on Symbolic Computation of Hirota Bilinear Form for Nonlinear Equations

An improved algorithm for symbolic computation of Hirota bilinear forms of KdV-type equations with logarithmic transformations is presented. In the algorithm, the general assumption of Hirota bilinear form is successfully reduced based on the property of uniformity in rank. Furthermore, we discard the integral operation in the traditional algorithm. The software package HBFTrans is written in Maple and its running effectiveness is tested by a variety soliton equations.

On the Leverrier-Faddeev algorithm for computing the Moore-Penrose inverse

We present an improvement in the implementation of the Leverrier-Faddeev algorithm for symbolic computation of the Moore-Penrose inverse of one-variable polynomial matrices, introduced in Linear Algebra Appl. 252, 35–60 (1997). Complexity analysis of the original algorithm and its improvement is presented. Algorithm and its improvement are implemented and compared in the symbolic computational package MATHEMATICA. We compare CPU time required for computation of some test matrices by means of the original algorithm and its improvement.

Observer Design for Nonlinear Systems With Time-Periodic Coefficients via Normal Form Theory

For most complex dynamic systems, it is not always possible to measure all system states by a direct measurement technique. Thus for dynamic characterization and controller design purposes, it is often necessary to design an observer in order to get an estimate of those states, which cannot be measured directly. In this work, the problem of designing state observers for free systems (linear as well as nonlinear) with time-periodic coefficients is addressed. It is shown that, for linear periodic systems, the observer design problem is the duality of the controller design problem. The state observer is constructed using a symbolic controller design method developed earlier using a Chebyshev expansion technique where the Floquet multipliers can be placed in the desired locations within the unit circle. For nonlinear time-periodic systems, an observer design methodology is developed using the Lyapunov–Floquet transformation and the Poincaré normal form technique. First, a set of time-periodic near identity coordinate transformations are applied to convert the nonlinear problem to a linear observer design problem. The conditions for existence of such invertible maps and their computations are discussed. Then the local identity observers are designed and implemented using a symbolic computational algorithm. Several illustrative examples are included to show the effectiveness of the proposed methods.

Two-stage complex Rosenbrock schemes for stiff systems

New two-stage Rosenbrock schemes with complex coefficients are proposed for stiff systems of differential equations. The schemes are fourth-order accurate and satisfy enhanced stability requirements. A one-parameter family of L1-stable schemes with coefficients explicitly calculated by formulas involving only fractions and radicals is constructed. A single L2-stable scheme is found in this family. The coefficients of the fourth-order accurate L4-stable scheme previously obtained by P.D Shirkov are refined. Several fourth-order schemes are constructed that are high-order accurate for linear problems and possess the limiting order of L-decay. The schemes proposed are proved to converge. A symbolic computation algorithm is developed that constructs order conditions for multistage Rosenbrock schemes with complex coefficients. This algorithm is used to design the schemes proposed and to obtain fifth-order accurate conditions.

A new recursive algorithm for inverting general periodic pentadiagonal and anti-pentadiagonal matrices

In the current article, the authors present a new recursive symbolic computational algorithm, that will never break down, for inverting general periodic pentadiagonal and anti-pentadiagonal matrices. It is a natural generalization of the work presented in [M.E.A. El-Mikkawy, E.D. Rahmo, A new recursive algorithm for inverting general tridiagonal and anti-tridiagonal matrices, Appl. Math. Comput. 204 (2008) 368–372]. The algorithm is suited for implementation using computer algebra systems (CAS) such as Mathematica, Macsyma and Maple. An illustrative example is given.

A new recursive algorithm for inverting general tridiagonal and anti-tridiagonal matrices

In the current article, the authors present a new recursive symbolic computational algorithm for inverting general tridiagonal and anti-tridiagonal matrices. An illustrative example is given.

Calculation of a formal moment generating function by using a differential operator

A differential form in a formal moment generating function is given by the decomposition of powers in terms of the Hermite polynomials. This paper shows that this differential form for calculating the expectation of normal and χ 2 distributions has the benefit of avoiding divergence for Edgeworth type approximations from the viewpoint of a formal power series ring. A symbolic computational algorithm is also discussed, within the distribution theory of statistics.

A Maple Package on Symbolic Computation of ConservedDensities for (1+1)-Dimensional Nonlinear EvolutionSystems

A new algorithm for symbolic computation of polynomial-typeconserved densities for nonlinear evolution systems is presented.The algorithm is implemented in Maple. The improved algorithm ismore efficient not only in removing the redundant terms of thegeneral form of the conserved densities but also in solving theconserved densities with the associated flux synchronously withoutusing Euler operator. Furthermore, the program conslaw.mpl can beused to determine the preferences for a given parameterizednonlinear evolution systems. The code is tested on severalwell-known nonlinear evolution equations from the soliton theory.

Symbolic computation in studying stability of solutions of linear differential equations with periodic coefficients

An algorithm for symbolic computation of characteristic exponents of a linear system of differential equations with a periodic matrix represented by a power series in terms of a small parameter is discussed. The algorithm is based on the infinite determinant method. The corresponding procedures implemented in the Mathematica system and computation results related to the elliptic restricted four-body problem are presented.

Some symbolic computation algorithms in cosmic dynamics problems

An algorithm for computing fundamental solutions to a linear system of differential equations with a periodic matrix represented by a power series in terms of a small parameter is discussed. An algorithm based on the infinite determinant method for determining boundaries between regions of stability and instability for such a system in the parameter space is also discussed. The corresponding procedures implemented in the Mathematica system and computation results related to the elliptic restricted five-body problem are presented.

A finite algorithm for generalized inverses of polynomial and rational matrices

There are proposed two types of finite algorithms for symbolic computation of outer inverses of a given polynomial or rational matrix, based on the extension of the Leverrier–Faddeev algorithm. The implementation of the algorithm corresponding to rational matrices is developed in the symbolic package MATHEMATICA.

Reliable Padé analytical continuation method based on a high-accuracy symbolic computation algorithm

We critique a Pade analytic continuation method whereby a rational polynomial function is fit to a set of input points by means of a single matrix inversion. This procedure is accomplished to an extremely high accuracy using a novel symbolic computation algorithm. As an example of this method in action we apply it to the problem of determining the spectral function of a one-particle thermal Green's function known only at a finite number of Matsubara frequencies with two example self energies drawn from the T-matrix theory of the Hubbard model. We present a systematic analysis of the effects of error in the input points on the analytic continuation, and this leads us to propose a procedure to test quantitatively the reliability of the resulting continuation, thus eliminating the black magic label frequently attached to this procedure.

Symbolic Computation of Local Stability and Bifurcation Surfaces for Nonlinear Time-Periodic Systems

A new technique is presented for symbolic computation of local stability boundaries and bifurcation surfaces for nonlinear multidimensional time-periodic dynamical systems as an explicit function of the system parameters. This is made possible by the recent development of a symbolic computational algorithm for approximating the parameter-dependent fundamental solution matrix of linear time-periodic systems. By evaluating this matrix at the end of the principal period, the parameter-dependent Floquet Transition Matrix (FTM), or the linear part of the Poincare map, is obtained. The subsequent use of well-known criteria for the local stability and bifurcation conditions of equilibria and periodic solutions enables one to obtain the equations for the bifurcation surfaces in the parameter space as polynomials of the system parameters. Further, the method may be used in conjunction with a series expansion to obtain perturbation-like expressions for the bifurcation boundaries. Because this method is not based on expansion in terms of a small parameter, it can be successfully applied to periodic systems whose internal excitation is strong. Also, the proposed method appears to be more efficient in terms of cpu time than the truncated point mapping method. Two illustrative example problems, viz., a parametrically excited simple pendulum and a double inverted pendulum subjected to a periodic follower force, are included.

RELIABILITY EVALUATION IN OPERATIONAL PLANNING OF POWER SYSTEMS

This paper presents a computer approach to evaluate the power system reliability considering generation/transmission capacities. The proposed technique is a 2-step approach. Step 1 is to modify the system minimal cutsets to a capacity related cutsets CRC. CRC is defined as a minimum set of system elements, which if failed, the rest of the elements are not enough for the system to survive, and this set is a subset of any minimal cutset of the system. Step 2 deals with the evaluation of the power system reliability based on CRC of the system, enumerated in step 1, and a symbolic reliability computation algorithm based on journal article(Fong, et al.,1987). The above method is illustrated by numerical examples and compared with the results tabulated in journal article(Rai, et at.,1991). A reinforcement strategy to improve the system reliability is suggested. The strategy is to select the important link and reinforce it.

The design of linear controllers with symbolic algebra

Modern transfer function methods use quite a lot of abstract algebra in the design of control systems. In computing the final results, symbolic algebra turns out to be an important tool. In this paper, symbolic computation algorithms will be developed to enable computations directly in the theoretically useful transfer function rings. A symbolic software package, called Hinf, will be developed and used both for educational and theoretical purposes.

A Graph Oriented Knowledge-Based System to Assist Restoration of Distribution Systems

This paper describes a knowledge-based system designed to assist electrical distribution system operators in restoring the network after a fault. A hybrid approach that links symbolic and numerical computation algorithms is proposed. Features that depend on system operators’ experience and norms of the utiIity arc included in a knowledge base. The reasoning process is guided by an inference engine with backward chaining strategy. Electrical overload checks are carried out by a simulator based on the method of momentums. The adoption of concepts and data structure borrowed from network flow algorithms leads to the development of a very effcient procedure.