Computer Algebra Methods Research Articles

Hamiltonian normalization in the restricted many-body problem by computer algebra methods

A symbolic algorithm for construction of a real canonical transformation that reduces the Hamiltonian determining motion of an autonomous two-degree-of-freedom system in a neighborhood of an equilibrium state to the normal form is discussed. The application of the algorithm to the restricted planar circular three-body problem is demonstrated. The expressions obtained for the coefficients of the Hamiltonian normal form substantiate results derived earlier by A. Deprit. Symbolic computations are performed in the computer algebra system Mathematica.

Symbolic computation and the cyclicity problem for singularities

We show how methods of computational commutative algebra are employed to investigate the local 16th Hilbert Problem, which is to find an upper bound on the number of limit cycles that can bifurcate from singularities in families of polynomial systems of differential equations on R2, and is one step in a program for solving the full 16th Hilbert Problem. We discuss an extension of a well-known theorem, and illustrate the concepts and methods with concrete examples.

Finite quantum models: constructive approach to description of quantum behavior

The universality of quantum mechanics — its applicability to physical systems of quite different nature and scales — indicates that quantum behavior can be a manifestation of general mathematical properties of systems containing indistinguishable elements, i.e., elements lying on the same orbit of some symmetry group. In this paper, we demonstrate that quantum behavior arises naturally in systems with finitely many elements connected by nontrivial symmetry groups. The “finite” approach allows one to see the peculiarities of a quantum description more distinctly without the need for concepts like “wave function collapse,” “Everett’s multiverses,” etc. In particular, under the finiteness assumption, any quantum dynamics is reduced to a simple permutation dynamics. The advantage of finite quantum models is that they can be studied constructively by means of computer algebra and computational group theory methods. Bibliography: 12 titles.

Geometric invariants for initial data sets: analysis, exact solutions, computer algebra, numerics

A personal perspective on the interaction of analytical, numerical and computer algebra methods in classical Relativity is given. This discussion is inspired by the problem of the construction of invariants that characterise key solutions to the Einstein field equations. It is claimed that this kind of ideas will be or importance in the analysis of dynamical black hole spacetimes by either analytical or numerical methods.

Numerical-symbolic exact irreducible decomposition of cyclic-12

AbstractIn 1992, Göran Björck and Ralf Fröberg completely characterized the solution set of cyclic-8. In 2001, Jean-Charles Faugère determined the solution set of cyclic-9, by computer algebra methods and Gröbner basis computation. In this paper, a new theory in matrix analysis of rank-deficient matrices together with algorithms in numerical algebraic geometry enables us to present a symbolic-numerical algorithm to derive exactly the defining polynomials of all prime ideals of positive dimension in primary decomposition of cyclic-12. Empirical evidence together with rigorous proof establishes the fact that the positive-dimensional solution variety of cyclic-12 just consists of 72 quadrics of dimension one.

The [formula omitted] Jones polynomial of the trefoil: A case study of q-holonomic sequences

The sl3 colored Jones polynomial of the trefoil knot is a q-holonomic sequence of two variables with natural origin, namely quantum topology. The paper presents an explicit set of generators for the annihilator ideal of this q-holonomic sequence as a case study. On the one hand, our results are new and useful to quantum topology: this is the first example of a rank 2 Lie algebra computation concerning the colored Jones polynomial of a knot. On the other hand, this work illustrates the applicability and computational power of the employed computer algebra methods.

Experiments with the Site Frequency Spectrum

Evaluating the likelihood function of parameters in highly-structured population genetic models from extant deoxyribonucleic acid (DNA) sequences is computationally prohibitive. In such cases, one may approximately infer the parameters from summary statistics of the data such as the site-frequency-spectrum (SFS) or its linear combinations. Such methods are known as approximate likelihood or Bayesian computations. Using a controlled lumped Markov chain and computational commutative algebraic methods, we compute the exact likelihood of the SFS and many classical linear combinations of it at a non-recombining locus that is neutrally evolving under the infinitely-many-sites mutation model. Using a partially ordered graph of coalescent experiments around the SFS, we provide a decision-theoretic framework for approximate sufficiency. We also extend a family of classical hypothesis tests of standard neutrality at a non-recombining locus based on the SFS to a more powerful version that conditions on the topological information provided by the SFS.

Fortran 90 implementation of the Hartree–Fock approach within the CNDO/2 and INDO models

Despite the tremendous advances made by the ab initio theory of electronic structure of atoms and molecules, its applications are still not possible for very large systems. Therefore, semi-empirical model Hamiltonians based on the zero-differential overlap (ZDO) approach such as the Pariser–Parr–Pople, CNDO, INDO, etc. provide attractive, and computationally tractable, alternatives to the ab initio treatment of large systems. In this paper we describe a Fortran 90 computer program developed by us, that uses CNDO/2 and INDO methods to solve Hartree–Fock (HF) equation for molecular systems. The INDO method can be used for the molecules containing the first-row atoms, while the CNDO/2 method is applicable to those containing both the first-, and the second-row, atoms. We have paid particular attention to computational efficiency while developing the code, and, therefore, it allows us to perform calculations on large molecules such as C 60 on small computers within a matter of seconds. Besides being able to compute the molecular orbitals and total energies, our code is also able to compute properties such as the electric dipole moment, Mulliken population analysis, and linear optical absorption spectrum of the system. We also demonstrate how the program can be used to compute the total energy per unit cell of a polymer. The applications presented in this paper include small organic and inorganic molecules, fullerene C 60, and model polymeric systems, viz., chains containing alternating boron and nitrogen atoms (BN chain), and carbon atoms (C chain). Program summary Program title: cindo.x Catalogue identifier: AECN_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AECN_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 23 304 No. of bytes in distributed program, including test data, etc.: 217 035 Distribution format: tar.gz Programming language: Fortran 90 Computer: PC's/Linux, Program has been tested with Intel Fortran Compiler (noncommercial version 10.1) and gfortran compiler (gcc version 4.3.0) with optimization option -O Operating system: Linux, Code was developed and tested on various recent versions of Fedora including Fedora 9 (kernel version 2.6.25-14) Classification: 16.1 External routines: This program needs to link with LAPACK/BLAS libraries compiled with the same compiler as the program. For the Intel Fortran Compiler we used the ACML library version 3.6.0, while for gfortran compiler we used the libraries supplied with the Fedora distribution. Nature of problem: A good starting description of the electronic structure of extended many-electron systems such as molecules, clusters, and polymers, can be obtained using the Hartree–Fock (HF) method. Solution of HF equations within a fully ab initio formalism for large systems, however, is computationally quite expensive. For such systems, semi-empirical methods such as CNDO and INDO proposed by Pople and collaborators are quite attractive. The present program can solve the HF equations for both open- and closed-shell systems containing first- and second-row atoms using either the INDO model or the CNDO model. Solution method: The single-particle HF orbitals are expressed as linear combinations of the Slater-type orbital (STO) basis set specified by Pople and coworkers. Then using the parameters prescribed for the CNDO/INDO methods, the HF integro-differential equations are transformed into a matrix eigenvalue problem. Thereby, its solutions are obtained in a self-consistent manner, using methods of computational linear algebra. Running time: Each of the examples provided takes only a few seconds to run. For a large molecule or a cluster, however, the run time may be a few minutes.

Hadamard matrices of Williamson type: A challenge for Computer Algebra

We describe new families of challenging polynomial systems of equations arising in the construction of Hadamard matrices of Williamson type. These systems depend on a parameter and exhibit a number of symmetries. We demonstrate that Genetic Algorithms form a suitable tool to apply to these systems, for which the traditional Computer Algebra methods seem to be inappropriate.

A Fortran 90 program to solve the Hartree–Fock equations for interacting spin- [formula omitted] fermions confined in harmonic potentials

A set of weakly interacting spin- 1 2 Fermions, confined by a harmonic oscillator potential, and interacting with each other via a contact potential, is a model system which closely represents the physics of a dilute gas of two-component fermionic atoms confined in a magneto-optic trap. In the present work, our aim is to present a Fortran 90 computer program which, using a basis set expansion technique, solves the Hartree–Fock (HF) equations for spin- 1 2 Fermions confined by a three-dimensional harmonic oscillator potential, and interacting with each other via pair-wise delta-function potentials. Additionally, the program can also account for those anharmonic potentials which can be expressed as a polynomial in the position operators x, y, and z. Both the restricted-HF (RHF), and the unrestricted-HF (UHF) equations can be solved for a given number of Fermions, with either repulsive or attractive interactions among them. The option of UHF solutions for such systems also allows us to study possible magnetic properties of the physics of two-component confined atomic Fermi gases, with imbalanced populations. Using our code we also demonstrate that such a system exhibits shell structure, and follows Hund's rule. Program summary Program title: trap.x Catalogue identifier: AEBB_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEBB_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 17 750 No. of bytes in distributed program, including test data, etc.: 205 138 Distribution format: tar.gz Programming language: mostly Fortran 90 Computer: PCs—SUN, HP Alpha, IBM Operating system: Linux, Solaris, Tru64, AIX Classification: 7.7 Nature of problem: The simplest description of a spin 1 2 ; trapped system at the mean field level is given by the Hartree–Fock method. This program presents an efficient approach to solving these equations. Additionally, this program can solve for time-independent Gross–Pitaevskii and Hartree–Fock equations for bosonic atoms confined in a harmonic trap. Thus the combined program can handle mean-field equations for both the Fermi and the Bose particles. Solution method: The solutions of the Hartree–Fock equation corresponding to the Fermi systems in atomic traps are expanded as linear combinations of simple-harmonic oscillator eigenfunctions. Thus, the Hartree–Fock equations which comprise a set of nonlinear integro-differential equations, are transformed into a matrix eigenvalue problem. Thereby, solutions are obtained in a self-consistent manner, using methods of computational linear algebra. Running time: The run times of example jobs are from a few seconds to a few minutes. For jobs involving very large basis sets, the run time can extend into hours.

An approach to the Sandia workshop static frame challenge problem: A combination of elementary probabilistic, fuzzy set, and worst scenario tools

A model static frame problem with uncertain input data is considered. The Young modulus of the frame material is known only through a limited number of measurements of its local and effective values. In the loaded frame, the displacement w at a given point is to be predicted. To tackle the uncertainty in the displacement magnitude, a set F of conservative Young modulus behavior (YMB) models is constituted and the mean and the variance of w are inferred for each YMB model belonging to F . Moreover, the degree of possibility of each YMB model is assessed by a weight (membership) function ν derived from measured data. The probability of w exceeding a given tolerance is inferred in each YMB model and then weighted by ν. To address the prediction problem in accordance with the worst scenario approach, the maximum is identified in the weighted probabilities. The analysis is based on elementary probabilistic tools as well as on the exploitation of computer algebra and numerical methods. The results are presented in numerous graphs.

Automatic Construction and Verification of Isotopy Invariants

We extend our previous study of the automatic construction of isomorphic classification theorems for algebraic domains by considering the isotopy equivalence relation. Isotopism is an important generalisation of isomorphism, and is studied by mathematicians in domains such as loop theory. This extension was not straightforward, and we had to solve two major technical problems, namely, generating and verifying isotopy invariants. Concentrating on the domain of loop theory, we have developed three novel techniques for generating isotopic invariants, by using the notion of universal identities and by using constructions based on subblocks. In addition, given the complexity of the theorems that verify that a conjunction of the invariants form an isotopy class, we have developed ways of simplifying the problem of proving these theorems. Our techniques employ an interplay of computer algebra, model generation, theorem proving, and satisfiability-solving methods. To demonstrate the power of the approach, we generate isotopic classification theorems for loops of size 6 and 7, which extend the previously known enumeration results. This work was previously beyond the capabilities of automated reasoning techniques.

Symbolic computation techniques for solving large expression problems from mathematics and engineering

This thesis studies the use of computer algebra methods to solve some large-expression problems from mathematics and engineering. We give several strategies for solving problems from symbolic linear algebra and dynamic systems. First, we describe new forms for fraction-free LU factoring and QR factoring. These new forms keep both the computation and the output results in the same domain as the input domain and thereby increase the computational efficiency in applications by delaying the appearance of quotients of the input data. To compute the new forms, we use a fraction-free variant of Gaussian elimination to control the growth of matrix entries. We give a complexity analysis for standard domains. Secondly, we propose a general method, hierarchical representation and signature computing for zero testing, to deal with problems with intermediate or inherent expression swell. For instance, when we use Gaussian elimination to solve large symbolic linear equations, the resulting large expressions can be handled using our general method. We implement a version of LU factoring using hierarchical representation with signature computing for zero testing. The LU factoring is the standard one, rather than the fraction-free one described above. We prove that the improved algorithm is much faster than the classical LU factoring algorithm using Gaussian elimination and give associated time complexity analysis and experimental results. Besides large expression problems from linear algebra, we also explore large expression problems from engineering, especially those arising from analyzing and solving multibody dynamic systems and limit cycle computations. We define a new concept, implicit reduced involutive form, to cope with large expression problems resulting from symbolically pre-processing systems of differential algebraic equations (DAE). We also show how symbolic pre-processing can be combined with numerical from limit cycle computations which could not be directly solved because of large expression swell. The techniques we develop in this thesis are quite general and can be easily applied to other similar areas, such as computing determinants and solving more general DAE models. Keywords. Large Expression Management, Differential Algebraic Equations, RIF-SIMP, LARGE EXPRESSIONS, Multibody Dynamic System, Differentiation and Elimination, Hierarchical Representations, Signature, LII Symbolic Decomposition, Time Complexity, Limit Cycle, Computer Algebra.

Simulation of steady-state NMR of coupled systems using Liouville space and computer algebra methods

A series of repeated pulses and delays applied to a spin system generates a steady state. This is relatively easy to calculate for a single spin, but coupled systems present real challenges. We have used Maple, a computer algebra program to calculate one- and two-spin symbolically, and larger systems numerically. The one-spin calculations illustrate and validate the methods and show how the steady-state free precession method converges to continuous wave NMR. For two-spin systems, we have derived a general formula for the creation of double-quantum signals as a function of irradiation strength, coupling constant, and chemical shift difference. The calculations on three-spin and larger systems reproduce and extend previously published results. In this paper, we have shown that the approach works well for systems in literature. However, the formalism is general and can be extended to more complex spin systems and pulses sequences.

Integrating factors of some classes of third-order ODEs

We present practical computer algebra methods for determining integrating factors of particular classes of third-order non-linear ordinary differential equations (ODEs).

Hypergeometric Summation Techniques for High Order Finite Elements

AbstractThe goal of this paper is to discuss the application of computer algebra methods in the design of a high order finite element solver. The finite element method is nowadays the most popular method for the computer simulation of partial differential equations. The performance of iterative solution methods depends on the condition number of the system matrix, which itself depends on the chosen basis functions. A major goal is to design basis functions minimizing the condition number, and which can be implemented efficiently. A related goal is the application of symbolic summation techniques to derive cheap recurrence relations allowing a simple and efficient implementation of basis functions. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

An algorithm for solving singular perturbation problems with mechanization

In this paper, by using the theories and methods of mathematical analysis and computer algebra, a reliable algorithm of finite difference method for solving singular perturbation problems was established, a new matlab procedure ehssp was implemented, too. Several linear and non-linear problems are presented to illustrate the implementation of the program.

Fast evaluation of the hypergeometric function p F p−1(a; b; z) at the singular point z = 1 by means of the Hurwitz zeta function ζ(α, s)

A method for fast and highly accurate evaluation of the generalized hypergeometric function p F p?1(a 1, ..., a p ; b 1, ..., b p?1; 1) = ? k = 0 ? f k by means of the Hurwitz zeta function ?(?, s) is developed. Based on asymptotic analysis of the coefficients f k , an expansion of p F p?1 is constructed as a combination of the functions ?(?, s) with explicit coefficients expressed in terms of the generalized Bernoulli polynomials. An appropriate selection of the parameter ? and the number of terms of the asymptotic expansion makes it possible to obtain the result with any desired degree of accuracy. The use of computer algebra methods, efficient numerical algorithms, and stochastic optimization methods considerably improve the efficiency of the suggested method.

Methods for justifying arithmetic hypotheses and computer algebra

Computer algebra methods for justifying hypotheses in arithmetic geometry are considered. The specific feature of these methods is that they are designed for parametric problems. The methods developed can be used for solving computational problems and justifying hypotheses on equidistribution, as well as for the theory of algebraic curves over finite fields.

A basis-set based Fortran program to solve the Gross–Pitaevskii equation for dilute Bose gases in harmonic and anharmonic traps

Inhomogeneous boson systems, such as the dilute gases of integral spin atoms in low-temperature magnetic traps, are believed to be well described by the Gross–Pitaevskii equation (GPE). GPE is a nonlinear Schrödinger equation which describes the order parameter of such systems at the mean field level. In the present work, we describe a Fortran 90 computer program developed by us, which solves the GPE using a basis set expansion technique. In this technique, the condensate wave function (order parameter) is expanded in terms of the solutions of the simple-harmonic oscillator (SHO) characterizing the atomic trap. Additionally, the same approach is also used to solve the problems in which the trap is weakly anharmonic, and the anharmonic potential can be expressed as a polynomial in the position operators x, y, and z. The resulting eigenvalue problem is solved iteratively using either the self-consistent-field (SCF) approach, or the imaginary time steepest-descent (SD) approach. Iterations can be initiated using either the simple-harmonic-oscillator ground state solution, or the Thomas–Fermi (TF) solution. It is found that for condensates containing up to a few hundred atoms, both approaches lead to rapid convergence. However, in the strong interaction limit of condensates containing thousands of atoms, it is the SD approach coupled with the TF starting orbitals, which leads to quick convergence. Our results for harmonic traps are also compared with those published by other authors using different numerical approaches, and excellent agreement is obtained. GPE is also solved for a few anharmonic potentials, and the influence of anharmonicity on the condensate is discussed. Additionally, the notion of Shannon entropy for the condensate wave function is defined and studied as a function of the number of particles in the trap. It is demonstrated numerically that the entropy increases with the particle number in a monotonic way. Program summary Title of program:bose.x Catalogue identifier:ADWZ_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADWZ_v1_0 Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Distribution format:tar.gz Computers:PC's/Linux, Sun Ultra 10/Solaris, HP Alpha/Tru64, IBM/AIX Programming language used:mostly Fortran 90 Number of bytes in distributed program, including test data, etc.:27 313 Number of lines in distributed program, including test data, etc.:28 015 Card punching code:ASCII Nature of physical problem:It is widely believed that the static properties of dilute Bose condensates, as obtained in atomic traps, can be described to a fairly good accuracy by the time-independent Gross–Pitaevskii equation. This program presents an efficient approach of solving this equation. Method of solution:The solutions of the Gross–Pitaevskii equation corresponding to the condensates in atomic traps are expanded as linear combinations of simple-harmonic oscillator eigenfunctions. Thus, the Gross–Pitaevskii equation which is a second-order nonlinear differential equation, is transformed into a matrix eigenvalue problem. Thereby, its solutions are obtained in a self-consistent manner, using methods of computational linear algebra. Unusual features of the program:None