Symbolic computation of analytic approximate solutions for nonlinear differential equations with initial conditions

The Adomian decomposition method (ADM) is one of the most effective methods for constructing analytic approximate solutions of nonlinear differential equations. In this paper, based on the new definition of the Adomian polynomials, and the two-step Adomian decomposition method (TSADM) combined with the Padé technique, a new algorithm is proposed to construct accurate analytic approximations of nonlinear differential equations with initial conditions. Furthermore, a MAPLE package is developed, which is user-friendly and efficient. One only needs to input a system, initial conditions and several necessary parameters, then our package will automatically deliver analytic approximate solutions within a few seconds. Several different types of examples are given to illustrate the validity of the package. Our program provides a helpful and easy-to-use tool in science and engineering to deal with initial value problems.Program summaryProgram title: NAPACatalogue identifier: AEJZ_v1_0Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEJZ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen?s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 4060No. of bytes in distributed program, including test data, etc.: 113 498Distribution format: tar.gzProgramming language: MAPLE R13Computer: PCOperating system: Windows XP/7RAM: 2 GbytesClassification: 4.3Nature of problem: Solve nonlinear differential equations with initial conditions.Solution method: Adomian decomposition method and Padé technique.Running time: Seconds at most in routine uses of the program. Special tasks may take up to some minutes.     

CAVE: A package for detection and quantitative analysis of internal cavities in a system of overlapping balls: Application to proteins

We developed a software package (CAVE) in Fortran language to detect internal cavities in proteins which can be applied also to an arbitrary system of balls. The volume, the surface area and other quantitative characteristics of the cavities can be calculated. The code is based on the recently suggested enveloping triangulation algorithm [J. Buša et al., J. Comp. Chem. 30 (2009) 346] for computing volume and surface area of the cavity by analytical equations. Different standard sets of atomic radii can be used. The PDB compatible file containing the atomic coordinates must be stored on the disk in advance. Testing of the code on different proteins and artificial ball systems showed efficiency and accuracy of the algorithm. The program is fast. It can handle a system of several thousands of balls in the order of seconds on contemporary PC's. The code is open source and free.

Program summary

Program title: CAVECatalogue identifier: AEHC_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEHC_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 8670No. of bytes in distributed program, including test data, etc.: 100 131Distribution format: tar.gzProgramming language: FortranComputer: PC Pentium and CoreOperating system: Linux system and Windows XP systemClassification: 16.1Nature of problem: Molecular structure analysis.Solution method: Analytical method for cavities detection, and numerical algorithm for volume and surface area calculation based on the analytical formulas, after using the stereographic transformation.Running time: Depends on the size of the molecule under consideration. The test example included in the distribution takes about 1 minute to run.     

A Maple package for verifying ultradiscrete soliton solutions

We present a computer algebra program for verifying soliton solutions of ultradiscrete equations in which both dependent and independent variables take discrete values. The package is applicable to equations and solutions that include the max function. The program is implemented using Maple software.

Program summary

Program title: UltdeCatalogue identifier: AEDB_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEDB_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 3171No. of bytes in distributed program, including test data, etc.: 13 633Distribution format: tar.gzProgramming language: Maple 10Computer: PC/AT compatible machineOperating system: Windows 2000, Windows XPRAM: Depends on the problem; minimum about 1 GBWord size: 32 bitsClassification: 5Nature of problem: The existence of multi-soliton solutions strongly suggest the integrability of nonlinear evolution equations. However enormous calculation is required to verify multi-soliton solutions of ultradiscrete equations. The use of computer algebra can be helpful in such calculations.Solution method: Simplification by using the properties of max-plus algebra.Restrictions: The program can only handle single ultradiscrete equations.Running time: Depends on the complexity of the equation and solution. For the examples included in the distribution the run times are as follows. (Core 2 Duo 3 GHz, Windows XP)

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Example 1: 2725 sec.

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Example 2: 33 sec.

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Example 3: 1 sec.

     

HidSecSOFTSUSY: Incorporating effects from hidden sectors in the numerical computation of the MSSM spectrum

SOFTSUSY is a software designed to solve the RG equations of the MSSM and compute its low energy spectrum. HidSecSOFTSUSY is an extension of the SOFTSUSY package which modifies the beta functions to include contributions from light dynamic fields in the hidden sector.

Program summary

Program title: HidSecSOFTSUSYCatalogue identifier: AEHP_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEHP_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 4167No. of bytes in distributed program, including test data, etc.: 141 411Distribution format: tar.gzProgramming language: C++, FortranComputer: Personal computerOperating system: Tested on GNU/LinuxWord size: 32 bitsClassification: 11.6External routines: Requires an installed version of SOFTSUSY (http://projects.hepforge.org/softsusy/)Nature of problem: Calculating supersymmetric particle spectrum and mixing parameters while incorporating dynamic modes from the hidden sector into the renormalization group equations. The solution to the equations must be consistent with a high-scale boundary condition on supersymmetry breaking parameters, as well as a weak-scale boundary condition on gauge couplings, Yukawa couplings and the Higgs potential parameters.Solution method: Nested iterative algorithm.Running time: A few seconds per parameter point.     

Scilab software package for the study of dynamical systems

This work presents a new software package for the study of chaotic flows and maps. The codes were written using Scilab, a software package for numerical computations providing a powerful open computing environment for engineering and scientific applications. It was found that Scilab provides various functions for ordinary differential equation solving, Fast Fourier Transform, autocorrelation, and excellent 2D and 3D graphical capabilities. The chaotic behaviors of the nonlinear dynamics systems were analyzed using phase-space maps, autocorrelation functions, power spectra, Lyapunov exponents and Kolmogorov-Sinai entropy. Various well known examples are implemented, with the capability of the users inserting their own ODE.

Program summary

Program title: ChaosCatalogue identifier: AEAP_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAP_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 885No. of bytes in distributed program, including test data, etc.: 5925Distribution format: tar.gzProgramming language: Scilab 3.1.1Computer: PC-compatible running Scilab on MS Windows or LinuxOperating system: Windows XP, LinuxRAM: below 100 MegabytesClassification: 6.2Nature of problem: Any physical model containing linear or nonlinear ordinary differential equations (ODE).Solution method: Numerical solving of ordinary differential equations. The chaotic behavior of the nonlinear dynamical system is analyzed using Poincaré sections, phase-space maps, autocorrelation functions, power spectra, Lyapunov exponents and Kolmogorov-Sinai entropies.Restrictions: The package routines are normally able to handle ODE systems of high orders (up to order twelve and possibly higher), depending on the nature of the problem.Running time: 10 to 20 seconds for problems that do not involve Lyapunov exponents calculation; 60 to 1000 seconds for problems that involve high orders ODE and Lyapunov exponents calculation.     

S@M, a mathematica implementation of the spinor-helicity formalism

In this paper we present the package S@M (Spinors@Mathematica) which implements the spinor-helicity formalism in Mathematica. The package allows the use of complex-spinor algebra along with the multi-purpose features of Mathematica. The package defines the spinor objects with their basic properties along with functions to manipulate them. It also offers the possibility of evaluating the spinorial objects numerically at every computational step. The package is therefore well suited to be used in the context of on-shell technology, in particular for the evaluation of scattering amplitudes at tree- and loop-level.

Program summary

Program title: S@MCatalogue identifier: AEBF_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEBF_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 14 404No. of bytes in distributed program, including test data, etc.: 77 536Distribution format: tar.gzProgramming language: MathematicaComputer: All computers running MathematicaOperating system: Any system running MathematicaClassification: 4.4, 5, 11.1Nature of problem: Implementation of the spinor-helicity formalismSolution method: Mathematica implementationRunning time: The notebooks provided with the package take only a few seconds to run.     

Determining Liouvillian first integrals for dynamical systems in the plane

Here we present/implement a semi-algorithm to find Liouvillian first integrals of dynamical systems in the plane. The algorithm is based on a Darboux-type procedure to find the integrating factor for the system. Since the particular form of such systems allows reducing it to a single rational first order ordinary differential equation (rational first order ODE), the Lsolver package presents a set of software routines in Maple for dealing with rational first order ODEs. The package present commands permitting research investigations of some algebraic properties of the system that is being studied.

Program summary

Program title:LsolverCatalogue number:ADZF_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZF_v1_0.htmlProgram obtainable from:CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions:Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.:1513No. of bytes in distributed program, including test data, etc.:17 367Distribution format:tar.gzProgramming language:MapleComputer:Any computer running MapleOperating system:Windows ME, Windows XPClassification:4.3Nature of problem: Solution of rational first order ordinary differential equations.Solution method:The method of solution is based on a Darboux/PS type approach.Restrictions:If the integrating factor for the ODE under consideration presents Darboux Polynomials of high degree (>3) in the dependent and independent variables, the package may spend an impractical amount of time to obtain the solution.Unusual features:Our implementation not only searches for Liouvillian conserved quantities, but can also be used as a research tool that allows the user to follow all the steps of the procedure (for example, we can calculate the algebraic invariants curves and associated co-factors, the integrating factor etc). In addition, since our package is based on recent theoretical developments [J. Avellar, L.G.S. Duarte, S.E.S. Duarte, L.A.C.P. da Mota, Integrating first-order differential equations with Liouvillian solutions via quadratures: a semi-algorithmic method, J. Comput. Appl. Math. 182 (2005) 327-332], it can successfully solve a class of rational first order ODEs that were not solved by some of the best-known ODE solvers available.Running time:This depends strongly on the ODE, but usually under 4 seconds.     

The Invar tensor package

The Invar package is introduced, a fast manipulator of generic scalar polynomial expressions formed from the Riemann tensor of a four-dimensional metric-compatible connection. The package can maximally simplify any polynomial containing tensor products of up to seven Riemann tensors within seconds. It has been implemented both in Mathematica and Maple algebraic systems.

Program summary

Program title:Invar Tensor PackageCatalogue identifier:ADZK_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZK_v1_0.htmlProgram obtainable from:CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions:Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 136 240No. of bytes in distributed program, including test data, etc.:2 711 923Distribution format:tar.gzProgramming language:Mathematica and MapleComputer:Any computer running Mathematica versions 5.0 to 5.2 or Maple versions 9 and 10Operating system:Linux, Unix, Windows XPRAM:30 MbWord size:64 or 32 bitsClassification:5External routines:The Mathematica version requires the xTensor and xPerm packages. These are freely available at http://metric.iem.csic.es/Martin-Garcia/xActNature of problem:Manipulation and simplification of tensor expressions. Special attention on simplifying scalar polynomial expressions formed from the Riemann tensor on a four-dimensional metric-compatible manifold.Solution method:Algorithms of computational group theory to simplify expressions with tensors that obey permutation symmetries. Tables of syzygies of the scalar invariants of the Riemann tensor.Restrictions:The present versions do not fully address the problem of reducing differential invariants or monomials of the Riemann tensor with free indices.Running time:Less than a second to fully reduce a monomial of the Riemann tensor of degree 7 in terms of independent invariants.     

SMMP v. 3.0—Simulating proteins and protein interactions in Python and Fortran

We describe a revised and updated version of the program package SMMP. SMMP is an open-source FORTRAN package for molecular simulation of proteins within the standard geometry model. It is designed as a simple and inexpensive tool for researchers and students to become familiar with protein simulation techniques. SMMP 3.0 sports a revised API increasing its flexibility, an implementation of the Lund force field, multi-molecule simulations, a parallel implementation of the energy function, Python bindings, and more.

Program summary

Title of program:SMMPCatalogue identifier:ADOJ_v3_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADOJ_v3_0.htmlProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions:Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlProgramming language used:FORTRAN, PythonNo. of lines in distributed program, including test data, etc.:52 105No. of bytes in distributed program, including test data, etc.:599 150Distribution format:tar.gzComputer:Platform independentOperating system:OS independentRAM:2 MbytesClassification:3Does the new version supersede the previous version?:YesNature of problem:Molecular mechanics computations and Monte Carlo simulation of proteins.Solution method:Utilizes ECEPP2/3, FLEX, and Lund potentials. Includes Monte Carlo simulation algorithms for canonical, as well as for generalized ensembles.Reasons for new version:API changes and increased functionality.Summary of revisions:Added Lund potential; parameters used in subroutines are now passed as arguments; multi-molecule simulations; parallelized energy calculation for ECEPP; Python bindings.Restrictions:The consumed CPU time increases with the size of protein molecule.Running time:Depends on the size of the simulated molecule.     

SecDec: A general program for sector decomposition

We present a program for the numerical evaluation of multi-dimensional polynomial parameter integrals. Singularities regulated by dimensional regularisation are extracted using iterated sector decomposition. The program evaluates the coefficients of a Laurent series in the regularisation parameter. It can be applied to multi-loop integrals in Euclidean space as well as other parametric integrals, e.g. phase space integrals.

Program summary

Program title: SecDecCatalogue identifier: AEIR_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEIR_v1_0.htmlProgram obtainable from: CPC Program Library, Queen?s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 57 617No. of bytes in distributed program, including test data, etc.: 895 550Distribution format: tar.gzProgramming language: Wolfram Mathematica, perl, FortranComputer: From a single PC to a cluster, depending on the problemOperating system: Unix, LinuxRAM: Depends on the complexity of the problemClassification: 4.4, 5, 11.1Nature of problem: Extraction of ultraviolet and infrared singularities from parametric integrals appearing in higher order perturbative calculations in gauge theories, e.g. multi-loop Feynman integrals, Wilson loops, phase space integrals.Solution method: Algebraic extraction of singularities in dimensional regularisation using iterated sector decomposition. This leads to a Laurent series in the dimensional regularisation parameter ε, where the coefficients are finite integrals over the unit-hypercube. Those integrals are evaluated numerically by Monte Carlo integration.Restrictions: Depending on the complexity of the problem, limited by memory and CPU time. Multi-scale integrals can only be evaluated at Euclidean points.Running time: Between a few minutes and several days, depending on the complexity of the problem.     

A cascadic monotonic time-discretized algorithm for finite-level quantum control computation

A computer package (CNMS) is presented aimed at the solution of finite-level quantum optimal control problems. This package is based on a recently developed computational strategy known as monotonic schemes.Quantum optimal control problems arise in particular in quantum optics where the optimization of a control representing laser pulses is required. The purpose of the external control field is to channel the system's wavefunction between given states in its most efficient way. Physically motivated constraints, such as limited laser resources, are accommodated through appropriately chosen cost functionals.

Program summary

Program title: CNMSCatalogue identifier: ADEB_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADEB_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 770No. of bytes in distributed program, including test data, etc.: 7098Distribution format: tar.gzProgramming language: MATLAB 6Computer: AMD Athlon 64 × 2 Dual, 2:21 GHz, 1:5 GB RAMOperating system: Microsoft Windows XPWord size: 32Classification: 4.9Nature of problem: Quantum controlSolution method: IterativeRunning time: 60-600 sec     

HFOLD - A program package for calculating two-body MSSM Higgs decays at full one-loop level

HFOLD (Higgs Full One Loop Decays) is a Fortran program package for calculating all MSSM Higgs two-body decay widths and the corresponding branching ratios at full one-loop level. The package is done in the SUSY Parameter Analysis convention and supports the SUSY Les Houches Accord input and output format.

Program summary

Program title: HFOLDCatalogue identifier: AEJG_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEJG_v1_0.htmlProgram obtainable from: CPC Program Library, Queen?s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 340 621No. of bytes in distributed program, including test data, etc.: 1 760 051Distribution format: tar.gzProgramming language: Fortran 77Computer: Workstation, PCOperating system: LinuxRAM: 524 288 000 BytesClassification: 11.1External routines: LoopTools 2.2 (http://www.feynarts.de/looptools/), SLHALib 2.2 (http://www.feynarts.de/slha/). The LoopTools code is included in the distribution package.Nature of problem: A future high-energy e⁺e⁻ linear collider will be the best environment for the precise measurements of masses, cross sections, branching ratios, etc. Experimental accuracies are expected at the per-cent down to the per-mile level. These must be matched from the theoretical side. Therefore higher order calculations are mandatory.Solution method: This program package calculates all MSSM Higgs two-body decay widths and the corresponding branching ratios at full one-loop level. The renormalization is done in the DR scheme following the SUSY Parameter Analysis convention. The program supports the SUSY Les Houches Accord input and output format.Running time: The example provided takes only a few seconds to run.     

xPerm: fast index canonicalization for tensor computer algebra

We present a very fast implementation of the Butler-Portugal algorithm for index canonicalization with respect to permutation symmetries. It is called xPerm, and has been written as a combination of a Mathematica package and a C subroutine. The latter performs the most demanding parts of the computations and can be linked from any other program or computer algebra system. We demonstrate with tests and timings the effectively polynomial performance of the Butler-Portugal algorithm with respect to the number of indices, though we also show a case in which it is exponential. Our implementation handles generic tensorial expressions with several dozen indices in hundredths of a second, or one hundred indices in a few seconds, clearly outperforming all other current canonicalizers. The code has been already under intensive testing for several years and has been essential in recent investigations in large-scale tensor computer algebra.

Program summary

Program title: xPermCatalogue identifier: AEBH_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEBH_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 93 582No. of bytes in distributed program, including test data, etc.: 1 537 832Distribution format: tar.gzProgramming language: C and Mathematica (version 5.0 or higher)Computer: Any computer running C and Mathematica (version 5.0 or higher)Operating system: Linux, Unix, Windows XP, MacOSRAM:: 20 MbyteWord size: 64 or 32 bitsClassification: 1.5, 5Nature of problem: Canonicalization of indexed expressions with respect to permutation symmetries.Solution method: The Butler-Portugal algorithm.Restrictions: Multiterm symmetries are not considered.Running time: A few seconds with generic expressions of up to 100 indices. The xPermDoc.nb notebook supplied with the distribution takes approximately one and a half hours to execute in full.     

HRMC: Hybrid Reverse Monte Carlo method with silicon and carbon potentials

Fortran 77 code is presented for a hybrid method of the Metropolis Monte Carlo (MMC) and Reverse Monte Carlo (RMC) for the simulation of amorphous silicon and carbon structures. In additional to the usual constraints of the pair correlation functions and average coordination, the code also incorporates an optional energy constraint. This energy constraint is in the form of either the Environment Dependent Interatomic Potential (applicable to silicon and carbon) and the original and modified Stillinger-Weber potentials (applicable to silicon). The code also allows porous systems to be modeled via a constraint on porosity and internal surface area using a novel restriction on the available simulation volume.

Program summary

Program title: HRMC version 1.0Catalogue identifier: AEAO_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAO_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 200 894No. of bytes in distributed program, including test data, etc.: 907 557Distribution format: tar.gzProgramming language: FORTRAN 77Computer: Any computer capable of running executables produced by the g77 Fortran compilerOperating system: Unix, WindowsRAM: Depends on the type of empirical potential use, number of atoms and which constraints are employedClassification: 7.7Nature of problem: Atomic modeling using empirical potentials and experimental dataSolution method: Monte CarloAdditional comments: The code is not standard FORTRAN 77 but includes some additional features and therefore generates errors when compiled using the Nag95 compiler. It does compile successfully with the GNU g77 compiler (http://www.gnu.org/software/fortran/fortran.html).Running time: Depends on the type of empirical potential use, number of atoms and which constraints are employed. The test included in the distribution took 37 minutes on a DEC Alpha PC.     

Parallel FFT-based Poisson solver for isolated three-dimensional systems

We describe an implementation to solve Poisson?s equation for an isolated system on a unigrid mesh using FFTs. The method solves the equation globally on mesh blocks distributed across multiple processes on a distributed-memory parallel computer. Test results to demonstrate the convergence and scaling properties of the implementation are presented. The solver is offered to interested users as the library PSPFFT.

Program summary

Program title: PSPFFTCatalogue identifier: AEJK_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEJK_v1_0.htmlProgram obtainable from: CPC Program Library, Queen?s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 110 243No. of bytes in distributed program, including test data, etc.: 16 332 181Distribution format: tar.gzProgramming language: Fortran 95Computer: Any architecture with a Fortran 95 compiler, distributed memory clustersOperating system: Linux, UnixHas the code been vectorized or parallelized?: Yes, using MPI. An arbitrary number of processors may be used (subject to some constraints). The program has been tested on from 1 up to ∼ 13 000 processors. RAM: Depends on the problem size, approximately 170 MBytes for 48³ cells per process.Classification: 4.3, 6.5External routines: MPI (http://www.mcs.anl.gov/mpi/), FFTW (http://www.fftw.org), Silo (https://wci.llnl.gov/codes/silo/) (only necessary for running test problem).Nature of problem: Solving Poisson?s equation globally on unigrid mesh distributed across multiple processes on distributed memory system.Solution method: Numerical solution using multidimensional discrete Fourier Transform in a parallel Fortran 95 code.Unusual features: This code can be compiled as a library to be readily linked and used as a blackbox Poisson solver with other codes.Running time: Depends on the size of the problem, but typically less than 1 second per solve.     

Numeric and symbolic evaluation of the pfaffian of general skew-symmetric matrices

Evaluation of pfaffians arises in a number of physics applications, and for some of them a direct method is preferable to using the determinantal formula. We discuss two methods for the numerical evaluation of pfaffians. The first is tridiagonalization based on Householder transformations. The main advantage of this method is its numerical stability that makes unnecessary the implementation of a pivoting strategy. The second method considered is based on Aitken?s block diagonalization formula. It yields to a kind of LU (similar to Cholesky?s factorization) decomposition (under congruence) of arbitrary skew-symmetric matrices that is well suited both for the numeric and symbolic evaluations of the pfaffian. Fortran subroutines (FORTRAN 77 and 90) implementing both methods are given. We also provide simple implementations in Python and Mathematica for purpose of testing, or for exploratory studies of methods that make use of pfaffians.

Program summary

Program title:PfaffianCatalogue identifier: AEJD_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEJD_v1_0.htmlProgram obtainable from: CPC Program Library, Queen?s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 2281No. of bytes in distributed program, including test data, etc.: 13 226Distribution format: tar.gzProgramming language: Fortran 77 and 90Computer: Any supporting a FORTRAN compilerOperating system: Any supporting a FORTRAN compilerRAM: a few MBClassification: 4.8Nature of problem: Evaluation of the pfaffian of a skew symmetric matrix. Evaluation of pfaffians arises in a number of physics applications involving fermionic mean field wave functions and their overlaps.Solution method: Householder tridiagonalization. Aitken?s block diagonalization formula.Additional comments: Python and Mathematica implementations are provided in the main body of the paper.Running time: Depends on the size of the matrices. For matrices with 100 rows and columns a few milliseconds are required.