Computer Algebra System REDUCE Research Articles

ReLie: A Reduce Program for Lie Group Analysis of Differential Equations

Lie symmetry analysis provides a general theoretical framework for investigating ordinary and partial differential equations. The theory is completely algorithmic even if it usually involves lengthy computations. For this reason, along the years many computer algebra packages have been developed to automate the computation. In this paper, we describe the program ReLie, written in the Computer Algebra System Reduce, since 2008 an open source program for all platforms. ReLie is able to perform almost automatically the needed computations for Lie symmetry analysis of differential equations. Its source code is freely available too. The use of the program is illustrated by means of some examples; nevertheless, it is to be underlined that it proves effective also for more complex computations where one has to deal with very large expressions.

On the Euler-Imshenetskii-Darboux transformation of linear second-order equations

It is shown how the linear Euler-Imshenetskii-Darboux (EID) differential transformation can be used for generating infinite sequences of linear second-order ordinary differential equations starting from certain standard equations. In so doing, the method of factorization of differential operators and operator identities obtained by means of this method are used. Generalizations of some well-known integrable cases of the Schrodinger equation are found. An example of an integrable equation with the Liouville coefficients, which apparently cannot be solved by the well-known Kovacic and Singer algorithms and their modifications, is constructed. An algorithm for solving the constructed class of equations has been created and implemented in the computer algebra system REDUCE. The corresponding procedure GENERATE is a supplement to the ODESOLVE procedure available in REDUCE. Solutions of some equations by means of the GENERATE procedure in REDUCE 3.8, as well as those obtained by means of DSOLVE in Maple 10, are presented. Although the algorithm based on the Euler-Imshenetskii-Darboux transformation is not an alternative to the existing algorithms for solving linear second-order ordinary differential equations, it is rather efficient within the limits of its applicability.

A REDUCE package for finding conserved densities of systems of implicit difference?difference equations*1

A computer algebra program for finding polynomial conserved densities of implicit difference–difference equations is presented. The algorithm is based on scaling properties and implemented in computer algebra system REDUCE. The package is applicable to systems of any number of nonlinear difference–difference equations of polynomial type. Title of program: TXCD Catalogue identifier: ADTS Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADTS Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Computers: PC/AT compatible machine Operating systems: Windows 2000 Programming language used: REDUCE 3.6, RLISP Memory required to execute with typical data: Depends on the problem, minimum about 2 M bytes. No. of bits in a word: 32 No. of bytes in distributed program, including test data, etc.: 10 005 No. of lines in distributed program, including test data, etc.: 1739 Distribution format: tar gzip file Nature of physical problem: The existence of conserved densities for difference–difference equations is of interest for their classification and for understanding the stability of their solutions. Restriction on the complexity of the problem: The program can handle difference–difference equations which can be transformed to polynomial ones, and determine the homogeneous conservation laws. Typical running time: It depends on the equation and the rank of the conserved density. It increases exponentially with the rank of the conserved density. The running times on the PC Pentium with operating systems Windows 2000 (Xeon, 1.7 GHz) are shown in the table below. Timings are given in milliseconds. Performance on Windows Example Rank 0 1 2 3 4 5 6 7 8 9 1(i) 15 15 15 31 150 718 5483 28176 127914 493092 1(ii) 15 15 16 63 170 2264 11171 103938 299001 1386468 1(iii) 15 15 15 46 250 5686 29210 203190 924372 1(iv) 15 15 15 31 47 156 1031 8905 40485 194595 2 15 15 45 187 2358 36673 433794 3(i) 15 63 1780 66518 1390030 ∗∗ 3(ii) 15 47 829 37640 786594 ∗∗ The cases ∗∗ were rejected by memory error.

A REDUCE package for finding conserved densities of systems of implicit difference–difference equations

A computer algebra program for finding polynomial conserved densities of implicit difference–difference equations is presented. The algorithm is based on scaling properties and implemented in computer algebra system REDUCE. The package is applicable to systems of any number of nonlinear difference–difference equations of polynomial type. Program summary Title of program: TXCD Catalogue identifier: ADTS Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADTS Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Computers: PC/AT compatible machine Operating systems: Windows 2000 Programming language used: REDUCE 3.6, RLISP Memory required to execute with typical data: Depends on the problem, minimum about 2 M bytes. No. of bits in a word: 32 No. of bytes in distributed program, including test data, etc.: 10 005 No. of lines in distributed program, including test data, etc.: 1739 Distribution format: tar gzip file Nature of physical problem: The existence of conserved densities for difference–difference equations is of interest for their classification and for understanding the stability of their solutions. Restriction on the complexity of the problem: The program can handle difference–difference equations which can be transformed to polynomial ones, and determine the homogeneous conservation laws. Typical running time: It depends on the equation and the rank of the conserved density. It increases exponentially with the rank of the conserved density. The running times on the PC Pentium with operating systems Windows 2000 (Xeon, 1.7 GHz) are shown in the table below. Timings are given in milliseconds. Performance on Windows Example Rank 0 1 2 3 4 5 6 7 8 9 1(i) 15 15 15 31 150 718 5483 28176 127914 493092 1(ii) 15 15 16 63 170 2264 11171 103938 299001 1386468 1(iii) 15 15 15 46 250 5686 29210 203190 924372 1(iv) 15 15 15 31 47 156 1031 8905 40485 194595 2 15 15 45 187 2358 36673 433794 3(i) 15 63 1780 66518 1390030 ∗∗ 3(ii) 15 47 829 37640 786594 ∗∗ The cases ∗∗ were rejected by memory error.

A Comparison of Algorithms for the Normalization and Quantization of Polynomial Hamiltonians

Algorithms and programs for the normalization of polynomial Hamiltonians of classical mechanics by the Birkhoff–Gustavson and Deprit–Hori, as well as quasi-classical quantization procedures for normal forms, are compared. The algorithms and programs are represented in a universal pseudocode and implemented in the computer algebra systems REDUCE, MAPLE, and MATHEMATICA. Examples that illustrate the operation of these algorithms and programs for polynomial Hamiltonians of atomic systems in external electromagnetic fields are considered.

A REDUCE package for finding conserved densities of systems of nonlinear difference–difference equations

Abstract A computer algebra program for finding polynomial conserved densities of nonlinear difference–difference equations is presented. The algorithm is based on scaling properties and implemented in computer algebra system REDUCE. The package is applicable to systems of any number of nonlinear difference–difference equations.

OpenMath support under CSL-hosted REDUCE

OpenMath is a standard for representing mathematical objects. This paper describes an attempt to include OpenMath support into the computer algebra system R EDUCE . This support is limited to the version based upon Cambridge Standard Lisp and is based upon the OpenMath C Library developed under Esprit project 24.969. The rationale for this effort was to avoid having to rewrite substantial amounts of code due to changes in the OpenMath standard but instead relying on the underlying C Library to provide the necessary modifications.

Group and renormgroup symmetry of a simple model for nonlinear phenomena in optics, gas dynamics, and plasma theory

Lie-Bäcklund symmetries of the quasi-Chaplygin system of equations are found as linear forms in hodograph variables. These symmetries are used to obtain invariant solutions and to construct a symmetry group of a boundary value problem (renormgroup). The implementation of an algorithm for solving the determining equations using the special package DIMSYM and the computer algebra system REDUCE is discussed.

GITA: A REDUCE program for the normalization of polynomial Hamiltonians

For a given polynomial Hamiltonian near an equilibrium point the program GITA calculates analytically the normal Birkhoff-Gustavson form in Cartesian as well as angle-action coordinates and the formal integral of the motion. These quantities are presented in the form of a truncated power series, the highest degree of which determines the degree of the approximation. The program performs the calculation of the modified normal Birkhoff-Gustavson form for the Hamiltonian restricted by the resonance condition. The program package is written in the computer algebra system REDUCE.

Application of algebraic methods to the calculation of steady states in continuous culture

For the calculation of steady states in continuous culture and the application to optimization, numerical methods for solving the related nonlinear system of equations are usually employed. Such a system may have more than one solution butnumerical algorithms find — at best — one solution per computational run depending on the starting point. There is no proof that all the solutions are actually found. To overcome these problems algebraic methods can be employed. Many biotechnological models can be transformed into a polynomial form. In contrast to nonlinear systems in general, polynomial systems are well investigated and all solutions can be calculated with given accuracy for example by using the Gröbner Bases representation. This allows even for sensitivity studies which are difficult to handle with numerical methods. The process of transforming and solving the system of equations is automated with the computer algebra system REDUCE. In this contribution the approach is presented and examples are shown.

Displacement analysis of the 6-6 stewart platform mechanisms

A method for the closed-form forward displacement analysis (FDA) of the Stewart platform mechanism (SPM) is presented in this paper. Meanwhile, the desired FDA in closed form of the 6-6/6-5/6-4 SPM with coplanar ball-joints has been performed. The analysis leads to a 20/2016th order polynomial equation in one unknown, from which with other equations 40/40/32 different locations of the platform can be derived. This theory has been programmed in personal computer, all solutions can be gained in 4 s with seven digits of precision. The symbolic computation is carried out by computer algebra system REDUCE.

HEAT-KERNEL CALCULATION OF QUARK DETERMINANT AND COMPUTER ALGEBRA

In this paper we describe the calculational background of deriving a strong meson Lagrangian from the Nambu–Jona-Lasinio quark model using the computer algebra systems FORM and REDUCE in recursive algorithms, based on the heat-kernel method for the calculation of the quark determinant.

Application of Algebraic Methods to the Calculation of Steady States in Continuous Culture

For the calculation of steady states in continuous culture and the application to optimization usually numerical methods for solving the related nonlinear system of equations are employed. Such a system may have more than one solution but numerical algorithms find - at best cases - one solution per computation run depending on the starting point. There is no proof that really all the solutions are found. To overcome these problems algebraic methods can be employed. Many biotechnological models can be transformed into a polynomial form. In contrast to nonlinear systems in general polynomial systems are well investigated and all solutions can be calculated with given accuracy for example by using the Grobner Bases representation. This allows even for sensitivity studies which are difficult to handle with numerical methods. The process of transforming and solving the system of equations is automized with the computer algebra system REDUCE. In this contribution the approach is presented and examples are shown.

Support-operators method for PDE discretization: symbolic algorithms and realization

The paper deals with the development and implementation of symbolic algorithms for the support-operators method for construction of finite-difference schemes. i.e., discretization of partial differential equations (PDEs), on a nonorthogonal logically rectangular grid. The general description of the support-operators method in the framework of the symbolic manipulation approach is presented. On the base of fundamental algorithms, developed earlier, algorithms for the construction of an adjoint difference operator are designed. These algorithms play a key role in the realization of the support-operators method. The algorithms have been implemented in the computer algebra system REDUCE. A 2D example demonstrates the power of the developed symbolic computation tool.

The Study of Motion for a Gyroscope with Gimbal Suspension: Obtaining the Highest Approximations for a Drift of Magnus

Here we give a solution to the classic problem of the applied mechanics of the gyroscope. The problem is reduced to using the computer algebra system REDUCE. We show that symbolic computation is very convenient for applied mechanics.

Automatic differentiation and implementation of thermodynamic models using a computer algebra system

A computational package (DIFTHER) for the analytical differentiation of thermodynamic models is presented. The package is based on the Computer Algebra System REDUCE, which is extended by means of new mathematical rules and procedures. In its current version, given the expression of a model of the excess Gibbs free energy, DIFTHER obtains the activity coefficients through partial differentiation outputting them in a complete FORTRAN subroutine, valid for any number of components. Several procedures have been developed in Standard LISP to promote the automatic generation of efficient codes. The performance of the system is illustrated with Wilson's gE model. It is concluded that, by using the DIFTHER package, considerable time savings can be achieved when implementing new thermodynamic models.

Computing centre conditions for certain cubic systems

We present necessary and sufficient conditions for a critical point of certain two-dimensional cubic differential systems to be a centre. Extensive use of the computer algebra system REDUCE is involved. The search for necessary and sufficient conditions for a centre has long been of considerable interest in the theory of nonlinear differential equations. It has proved to be a difficult problem, and full conditions are known for very few classes of systems. Such conditions are also required in the investigation of Hilbert's sixteenth problem concerning the number of limit cycles of polynomial systems.

Modelling Waves with Computer Algebra

A sophisticated model for linear waves in an inhomogeneous plasma is tackled completely using the computer algebra system REDUCE. The algebra code mirrors the mathematics, and is structured in a simple and straightforward manner. In so doing, the solution technique is made obvious, and the overall philosophy of the approach is intuitive to the non-specialist computer algebra user.

Two practical packages for computational physics — GCPM, RLFI

Two handy computer-program packages for technical support of the work in two different branches of the computational physics are reported: 1. (1) A genral Package for the symbolic and numerical transformation of expressions from one system of units to another. 2. (2) A package allowing high-quality two-dimensional of mathematical formulas from the computer-algebra system REDUCE.

Computer classification of integrable coupled KdV-like systems

The foundations of the symmetry approach to the classification problem of integrable non-linear evolution systems are briefly described. Within the framework of the symmetry approach the ten-parametric family of the third order non-linear evolution coupled KdV-like systems is investigated. The necessary integrability conditions lead to an over-determined non-linear algebraic system. To solve that system an effective method based on its structure has been used. This allows us to obtain the complete list of integrable systems of a given type. All computation has been completed on the basis of computer algebra systems FORMAC and REDUCE.