Ground state of the time-independent Gross-Pitaevskii equation

We present a suite of programs to determine the ground state of the time-independent Gross-Pitaevskii equation, used in the simulation of Bose-Einstein condensates. The calculation is based on the Optimal Damping Algorithm, ensuring a fast convergence to the true ground state. Versions are given for the one-, two-, and three-dimensional equation, using either a spectral method, well suited for harmonic trapping potentials, or a spatial grid.

Program summary

Program title: GPODACatalogue identifier: ADZN_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZN_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.: 5339No. of bytes in distributed program, including test data, etc.: 19 426Distribution format: tar.gzProgramming language: Fortran 90Computer: ANY (Compilers under which the program has been tested: Absoft Pro Fortran, The Portland Group Fortran 90/95 compiler, Intel Fortran Compiler)RAM: From <1 MB in 1D to ∼¹⁰² MB for a large 3D gridClassification: 2.7, 4.9External routines: LAPACK, BLAS, DFFTPACKNature of problem: The order parameter (or wave function) of a Bose-Einstein condensate (BEC) is obtained, in a mean field approximation, by the Gross-Pitaevskii equation (GPE) [F. Dalfovo, S. Giorgini, L.P. Pitaevskii, S. Stringari, Rev. Mod. Phys. 71 (1999) 463]. The GPE is a nonlinear Schrödinger-like equation, including here a confining potential. The stationary state of a BEC is obtained by finding the ground state of the time-independent GPE, i.e., the order parameter that minimizes the energy. In addition to the standard three-dimensional GPE, tight traps can lead to effective two- or even one-dimensional BECs, so the 2D and 1D GPEs are also considered.Solution method: The ground state of the time-independent of the GPE is calculated using the Optimal Damping Algorithm [E. Cancès, C. Le Bris, Int. J. Quantum Chem. 79 (2000) 82]. Two sets of programs are given, using either a spectral representation of the order parameter [C.M. Dion, E. Cancès, Phys. Rev. E 67 (2003) 046706], suitable for a (quasi) harmonic trapping potential, or by discretizing the order parameter on a spatial grid.Running time: From seconds in 1D to a few hours for large 3D grids     

Fortran programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap

Here we develop simple numerical algorithms for both stationary and non-stationary solutions of the time-dependent Gross-Pitaevskii (GP) equation describing the properties of Bose-Einstein condensates at ultra low temperatures. In particular, we consider algorithms involving real- and imaginary-time propagation based on a split-step Crank-Nicolson method. In a one-space-variable form of the GP equation we consider the one-dimensional, two-dimensional circularly-symmetric, and the three-dimensional spherically-symmetric harmonic-oscillator traps. In the two-space-variable form we consider the GP equation in two-dimensional anisotropic and three-dimensional axially-symmetric traps. The fully-anisotropic three-dimensional GP equation is also considered. Numerical results for the chemical potential and root-mean-square size of stationary states are reported using imaginary-time propagation programs for all the cases and compared with previously obtained results. Also presented are numerical results of non-stationary oscillation for different trap symmetries using real-time propagation programs. A set of convenient working codes developed in Fortran 77 are also provided for all these cases (twelve programs in all). In the case of two or three space variables, Fortran 90/95 versions provide some simplification over the Fortran 77 programs, and these programs are also included (six programs in all).

Program summary

Program title: (i) imagetime1d, (ii) imagetime2d, (iii) imagetime3d, (iv) imagetimecir, (v) imagetimesph, (vi) imagetimeaxial, (vii) realtime1d, (viii) realtime2d, (ix) realtime3d, (x) realtimecir, (xi) realtimesph, (xii) realtimeaxialCatalogue identifier: AEDU_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEDU_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.: 122 907No. of bytes in distributed program, including test data, etc.: 609 662Distribution format: tar.gzProgramming language: FORTRAN 77 and Fortran 90/95Computer: PCOperating system: Linux, UnixRAM: 1 GByte (i, iv, v), 2 GByte (ii, vi, vii, x, xi), 4 GByte (iii, viii, xii), 8 GByte (ix)Classification: 2.9, 4.3, 4.12Nature of problem: These programs are designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in one-, two- or three-space dimensions with a harmonic, circularly-symmetric, spherically-symmetric, axially-symmetric or anisotropic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Solution method: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation, in either imaginary or real time, over small time steps. The method yields the solution of stationary and/or non-stationary problems.Additional comments: This package consists of 12 programs, see “Program title”, above. FORTRAN77 versions are provided for each of the 12 and, in addition, Fortran 90/95 versions are included for ii, iii, vi, viii, ix, xii. For the particular purpose of each program please see the below.Running time: Minutes on a medium PC (i, iv, v, vii, x, xi), a few hours on a medium PC (ii, vi, viii, xii), days on a medium PC (iii, ix).

Program summary (1)

Title of program: imagtime1d.FTitle of electronic file: imagtime1d.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 1 GByteProgramming language used: Fortran 77Typical running time: Minutes on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in one-space dimension with a harmonic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems.

Program summary (2)

Title of program: imagtimecir.FTitle of electronic file: imagtimecir.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 1 GByteProgramming language used: Fortran 77Typical running time: Minutes on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in two-space dimensions with a circularly-symmetric trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems.

Program summary (3)

Title of program: imagtimesph.FTitle of electronic file: imagtimesph.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 1 GByteProgramming language used: Fortran 77Typical running time: Minutes on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in three-space dimensions with a spherically-symmetric trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems.

Program summary (4)

Title of program: realtime1d.FTitle of electronic file: realtime1d.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 2 GByteProgramming language used: Fortran 77Typical running time: Minutes on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in one-space dimension with a harmonic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and non-stationary problems.

Program summary (5)

Title of program: realtimecir.FTitle of electronic file: realtimecir.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 2 GByteProgramming language used: Fortran 77Typical running time: Minutes on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in two-space dimensions with a circularly-symmetric trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and non-stationary problems.

Program summary (6)

Title of program: realtimesph.FTitle of electronic file: realtimesph.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 2 GByteProgramming language used: Fortran 77Typical running time: Minutes on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in three-space dimensions with a spherically-symmetric trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and non-stationary problems.

Program summary (7)

Title of programs: imagtimeaxial.F and imagtimeaxial.f90Title of electronic file: imagtimeaxial.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 2 GByteProgramming language used: Fortran 77 and Fortran 90Typical running time: Few hours on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in three-space dimensions with an axially-symmetric trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems.

Program summary (8)

Title of program: imagtime2d.F and imagtime2d.f90Title of electronic file: imagtime2d.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 2 GByteProgramming language used: Fortran 77 and Fortran 90Typical running time: Few hours on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in two-space dimensions with an anisotropic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems.

Program summary (9)

Title of program: realtimeaxial.F and realtimeaxial.f90Title of electronic file: realtimeaxial.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 4 GByteProgramming language used: Fortran 77 and Fortran 90Typical running time Hours on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in three-space dimensions with an axially-symmetric trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and non-stationary problems.

Program summary (10)

Title of program: realtime2d.F and realtime2d.f90Title of electronic file: realtime2d.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 4 GByteProgramming language used: Fortran 77 and Fortran 90Typical running time: Hours on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in two-space dimensions with an anisotropic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and non-stationary problems.

Program summary (11)

Title of program: imagtime3d.F and imagtime3d.f90Title of electronic file: imagtime3d.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 4 GByteProgramming language used: Fortran 77 and Fortran 90Typical running time: Few days on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in three-space dimensions with an anisotropic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems.

Program summary (12)

Title of program: realtime3d.F and realtime3d.f90Title of electronic file: realtime3d.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum Ram Memory: 8 GByteProgramming language used: Fortran 77 and Fortran 90Typical running time: Days on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in three-space dimensions with an anisotropic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and non-stationary problems.     

Program package for multicanonical simulations of U(1) lattice gauge theory

We document our Fortran 77 code for multicanonical simulations of 4D U(1) lattice gauge theory in the neighborhood of its phase transition. This includes programs and routines for canonical simulations using biased Metropolis heatbath updating and overrelaxation, determination of multicanonical weights via a Wang-Landau recursion, and multicanonical simulations with fixed weights supplemented by overrelaxation sweeps. Measurements are performed for the action, Polyakov loops and some of their structure factors. Many features of the code transcend the particular application and are expected to be useful for other lattice gauge theory models as well as for systems in statistical physics.

Program summary

Program title: STMC_U1MUCACatalogue identifier: AEET_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEET_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.: 18 376No. of bytes in distributed program, including test data, etc.: 205 183Distribution format: tar.gzProgramming language: Fortran 77Computer: Any capable of compiling and executing Fortran codeOperating system: Any capable of compiling and executing Fortran codeClassification: 11.5Nature of problem: Efficient Markov chain Monte Carlo simulation of U(1) lattice gauge theory close to its phase transition. Measurements and analysis of the action per plaquette, the specific heat, Polyakov loops and their structure factors.Solution method: Multicanonical simulations with an initial Wang-Landau recursion to determine suitable weight factors. Reweighting to physical values using logarithmic coding and calculating jackknife error bars.Running time: The prepared tests runs took up to 74 minutes to execute on a 2 GHz PC.     

Scalable and portable visualization of large atomistic datasets

A scalable and portable code named Atomsviewer has been developed to interactively visualize a large atomistic dataset consisting of up to a billion atoms. The code uses a hierarchical view frustum-culling algorithm based on the octree data structure to efficiently remove atoms outside of the user's field-of-view. Probabilistic and depth-based occlusion-culling algorithms then select atoms, which have a high probability of being visible. Finally a multiresolution algorithm is used to render the selected subset of visible atoms at varying levels of detail. Atomsviewer is written in C++ and OpenGL, and it has been tested on a number of architectures including Windows, Macintosh, and SGI. Atomsviewer has been used to visualize tens of millions of atoms on a standard desktop computer and, in its parallel version, up to a billion atoms.

Program summary

Title of program: AtomsviewerCatalogue identifier: ADUMProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADUMProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandComputer for which the program is designed and others on which it has been tested: 2.4 GHz Pentium 4/Xeon processor, professional graphics card; Apple G4 (867 MHz)/G5, professional graphics cardOperating systems under which the program has been tested: Windows 2000/XP, Mac OS 10.2/10.3, SGI IRIX 6.5Programming languages used: C++, C and OpenGLMemory required to execute with typical data: 1 gigabyte of RAMHigh speed storage required: 60 gigabytesNo. of lines in the distributed program including test data, etc.: 550 241No. of bytes in the distributed program including test data, etc.: 6 258 245Number of bits in a word: ArbitraryNumber of processors used: 1Has the code been vectorized or parallelized: NoDistribution format: tar gzip fileNature of physical problem: Scientific visualization of atomic systemsMethod of solution: Rendering of atoms using computer graphic techniques, culling algorithms for data minimization, and levels-of-detail for minimal renderingRestrictions on the complexity of the problem: NoneTypical running time: The program is interactive in its executionUnusual features of the program: NoneReferences: The conceptual foundation and subsequent implementation of the algorithms are found in [A. Sharma, A. Nakano, R.K. Kalia, P. Vashishta, S. Kodiyalam, P. Miller, W. Zhao, X.L. Liu, T.J. Campbell, A. Haas, Presence—Teleoperators and Virtual Environments 12 (1) (2003)].     

Computing zeros of analytic functions in the complex plane without using derivatives

We present a package in Fortran 90 which solves f(z)=0, where z∈W⊂C without requiring the evaluation of derivatives, f^′(z). W is bounded by a simple closed curve and f(z) must be holomorphic within W.We have developed and tested the package to support our work in the modeling of high frequency and optical wave guiding and resonant structures. The respective eigenvalue problems are particularly challenging because they require the high precision computation of all multiple complex roots of f(z) confined to the specified finite domain. Generally f(z), despite being holomorphic, does not have explicit analytical form thereby inhibiting evaluation of its derivatives.

Program summary

Title of program:EZEROCatalogue identifier:ADXY_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADXY_v1_0Program obtainable from:CPC Program Library, Queen's University of Belfast, N. IrelandComputer:IBM compatible desktop PCOperating system:Fedora Core 2 Linux (with 2.6.5 kernel)Programming languages used:Fortran 90No. of bits in a word:32No. of processors used:oneHas the code been vectorized:noNo. of lines in distributed program, including test data, etc.:21045Number of bytes in distributed program including test data, etc.:223 756Distribution format:tar.gzPeripherals used:noneMethod of solution:Our package uses the principle of the argument to count the number of zeros encompassed by a contour and then computes estimates for the zeros. Refined results for each zero are obtained by application of the derivative-free Halley method with or without Aitken acceleration, as the user wishes.     

An integrated tool for loop calculations: aITALC

aITALC, a new tool for automating loop calculations in high energy physics, is described. The package creates Fortran code for two-fermion scattering processes automatically, starting from the generation and analysis of the Feynman graphs. We describe the modules of the tool, the intercommunication between them and illustrate its use with three examples.

Program summary

Title of the program:aITALC version 1.2.1 (9 August 2005)Catalogue identifier:ADWOProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADWOProgram obtainable from:CPC Program Library, Queen's University of Belfast, N. IrelandComputer:PC i386Operating system:GNU/Linux, tested on different distributions SuSE 8.2 to 9.3, Red Hat 7.2, Debian 3.0, Ubuntu 5.04. Also on SolarisProgramming language used:GNU Make, Diana, Form, Fortran77Additional programs/libraries used:Diana 2.35 (Qgraf 2.0), Form 3.1, LoopTools 2.1 (FF)Memory required to execute with typical data:Up to about 10 MBNo. of processors used:1No. of lines in distributed program, including test data, etc.:40 926No. of bytes in distributed program, including test data, etc.:371 424Distribution format:tar gzip fileHigh-speed storage required:from 1.5 to 30 MB, depending on modules present and unfolding of examplesNature of the physical problem:Calculation of differential cross sections for e⁺e⁻ annihilation in one-loop approximation.Method of solution:Generation and perturbative analysis of Feynman diagrams with later evaluation of matrix elements and form factors.Restriction of the complexity of the problem:The limit of application is, for the moment, the 2→2 particle reactions in the electro-weak standard model.Typical running time:Few minutes, being highly depending on the complexity of the process and the Fortran compiler.     

TiReX: Replica-exchange molecular dynamics using Tinker

We present a driver program for performing replica-exchange molecular dynamics simulations with the Tinker package. Parallelization is based on the Message Passing Interface, with every replica assigned to a separate process. The algorithm is not communication intensive, which makes the program suitable for running even on loosely coupled cluster systems. Particular attention is paid to the practical aspects of analyzing the program output.

Program summary

Program title: TiReXCatalogue identifier: AEEK_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEEK_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.: 43 385No. of bytes in distributed program, including test data, etc.: 502 262Distribution format: tar.gzProgramming language: Fortran 90/95Computer: Most UNIX machinesOperating system: LinuxHas the code been vectorized or parallelized?: parallelized with MPIClassification: 16.13External routines: TINKER version 4.2 or 5.0, built as a libraryNature of problem: Replica-exchange molecular dynamics.Solution method: Each replica is assigned to a separate process; temperatures are swapped between replicas at regular time intervals.Running time: The sample run may take up to a few minutes.     

A Mathematica program for the two-step twelfth-order method with multi-derivative for the numerical solution of a one-dimensional Schrödinger equation

In this paper, we present the detailed Mathematica symbolic derivation and the program which is used to integrate a one-dimensional Schrödinger equation by a new two-step numerical method. We add the fourth- and sixth-order derivatives to raise the precision of the traditional Numerov's method from fourth order to twelfth order, and to expand the interval of periodicity from (0,6) to the one of (0,9.7954) and (9.94792,55.6062). In the program we use an efficient algorithm to calculate the first-order derivative and avoid unnecessarily repeated calculation resulting from the multi-derivatives. We use the well-known Woods-Saxon's potential to test our method. The numerical test shows that the new method is not only superior to the previous lower order ones in accuracy, but also in the efficiency. This program is specially applied to the problem where a high accuracy or a larger step size is required.

Program summary

Title of program: ShdEq.nbCatalogue number: ADTTProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADTTProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions: noneComputer for which the program is designed and others on which it has been tested: The program has been designed for the microcomputer and been tested on the microcomputer.Computers: IBM PCOperating systems under which the program has been tested: Windows XPProgramming language used: Mathematica 4.2Memory required to execute with typical data: 51 712 bytesNo. of bytes in distributed program, including test data, etc.: 45 381No. of lines in distributed program, including test data, etc.: 7311Distribution format: tar gzip fileCPC Program Library subprograms used: noNature of physical problem: Numerical integration of one-dimensional or radial Schrödinger equation to find the eigenvalues for a bound states and phase shift for a continuum state.Method of solution: Using a two-step method twelfth-order method to integrate a Schrödinger equation numerically from both two ends and the connecting conditions at the matching point, an eigenvalue for a bound state or a resonant state with a given phase shift can be found.Restrictions on the complexity of the problem: The analytic form of the potential function and its high-order derivatives must be known.Typical running time: Less than one second.Unusual features of the program: Take advantage of the high-order derivatives of the potential function and efficient algorithm, the program can provide all the numerical solution of a given Schrödinger equation, either a bound or a resonant state, with a very high precision and within a very short CPU time. The program can apply to a very broad range of problems because the method has a very large interval of periodicity.References: [1] T.E. Simos, Proc. Roy. Soc. London A 441 (1993) 283.[2] Z. Wang, Y. Dai, An eighth-order two-step formula for the numerical integration of the one-dimensional Schrödinger equation, Numer. Math. J. Chinese Univ. 12 (2003) 146.[3] Z. Wang, Y. Dai, An twelfth-order four-step formula for the numerical integration of the one-dimensional Schrödinger equation, Internat. J. Modern Phys. C 14 (2003) 1087.     

GenAnneal: Genetically modified Simulated Annealing

A modification of the standard Simulated Annealing (SA) algorithm is presented for finding the global minimum of a continuous multidimensional, multimodal function. We report results of computational experiments with a set of test functions and we compare to methods of similar structure. The accompanying software accepts objective functions coded both in Fortran 77 and C++.

Program summary

Title of program:GenAnnealCatalogue identifier:ADXI_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADXI_v1_0Program available from: CPC Program Library, Queen's University of Belfast, N. IrelandComputer for which the program is designed and others on which it has been tested: The tool is designed to be portable in all systems running the GNU C++ compilerInstallation: University of Ioannina, Greece on Linux based machinesProgramming language used:GNU-C++, GNU-C, GNU Fortran 77Memory required to execute with typical data: 200 KBNo. of bits in a word: 32No. of processors used: 1Has the code been vectorized or parallelized?: NoNo. of bytes in distributed program, including test data, etc.:84 885No. of lines in distributed program, including test data, etc.:14 896Distribution format: tar.gzNature of physical problem: A multitude of problems in science and engineering are often reduced to minimizing a function of many variables. There are instances that a local optimum does not correspond to the desired physical solution and hence the search for a better solution is required. Local optimization techniques are frequently trapped in local minima. Global optimization is hence the appropriate tool. For example, solving a non-linear system of equations via optimization, employing a “least squares” type of objective, one may encounter many local minima that do not correspond to solutions (i.e. they are far from zero).Typical running time: Depending on the objective function.Method of solution: We modified the process of step selection that the traditional Simulated Annealing employs and instead we used a global technique based on grammatical evolution.     

Four-index integral transformation exploiting symmetry

We present a Fortran implementation of four-index integral transformation in the LCAO-MO (linear combination of atomic orbitals-molecular orbitals) framework that exploits symmetry. Electron correlation calculations, such as configuration interaction (CI) calculations, usually require electron repulsion integrals to be transformed to a molecular orbital basis from a basis using atomic orbitals. In large molecular systems it is vital to exploit the sparsity of integrals in making this transformation. By exploiting symmetry, the sparsity of integrals is fully utilized, the size of intermediate file is minimized, and the computational cost is reduced. The present algorithm is simple and can readily be added to existing quantum chemistry program packages.

Program summary

Title of program: SYM4TR (symmetry adapted 4-index integral transformation)Catalogue identifier: ADUWProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADUWProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandComputers: IBM/AIX, HP Alpha server/Tru64, PC's/LinuxProgram language used: Fortran 95Number of lines in distributed program, including test data, etc.: 4519No. of bytes in distributed program, including test data, etc.: 32 095Distributed format: tar gzip fileNature of physical problem: Molecular orbital calculations including electron correlation effects usually require electron repulsion integrals to be transformed from an atomic orbital (AO) basis to a molecular orbital (MO) basis. By exploiting the sparsity of molecular integrals, the computational cost and memory needed for the transformation are minimized.Method of solution: The sparsity of molecular integrals is exploited. The program treats only nonzero integrals. The length of running indices in DO loops is reduced using the block-diagonal form of the MO coefficient matrix. In the present program, the point group is limited to D_2h and its subgroups.     

MinFinder: Locating all the local minima of a function

A new stochastic clustering algorithm is introduced that aims to locate all the local minima of a multidimensional continuous and differentiable function inside a bounded domain. The accompanying software (MinFinder) is written in ANSI C++. However, the user may code his objective function either in C++, C or Fortran 77. We compare the performance of this new method to the performance of Multistart and Topographical Multilevel Single Linkage Clustering on a set of benchmark problems.

Program summary

Title of program:MinFinderCatalogue identifier:ADWUProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADWUProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandComputer for which the program is designed and others on which is has been tested:The tool is designed to be portable in all systems running the GNU C++ compilerInstallation:University of Ioannina, GreeceProgramming language used:GNU-C++, GNU-C, GNU Fortran 77Memory required to execute with typical data:200 KBNo. of bits in a word:32No. of processors used:1Has the code been vectorized or parallelized?:noNo. of lines in distributed program, including test data, etc.:5797No. of bytes in distributed program, including test data, etc.:588 121Distribution format:gzipped tar fileNature of the physical problem:A multitude of problems in science and engineering are often reduced to minimizing a function of many variables. There are instances that a local optimum does not correspond to the desired physical solution and hence the search for a better solution is required. Local optimization techniques can be trapped in any local minimum. Global optimization is then the appropriate tool. For example, solving a non-linear system of equations via optimization, employing a “least squares” type of objective, one may encounter many local minima that do not correspond to solutions, i.e. they are far from zero.Method of solution:Using a uniform pdf, points are sampled from the rectangular search domain. A clustering technique, based on a typical distance and a gradient criterion, is used to decide from which points a local search should be started. The employed local procedure is a BFGS version due to Powell. Further searching is terminated when all the local minima inside the search domain are thought to be found. This is accomplished via the double-box rule.Typical running time:Depending on the objective function     

CADNA: a library for estimating round-off error propagation

The CADNA library enables one to estimate round-off error propagation using a probabilistic approach. With CADNA the numerical quality of any simulation program can be controlled. Furthermore by detecting all the instabilities which may occur at run time, a numerical debugging of the user code can be performed. CADNA provides new numerical types on which round-off errors can be estimated. Slight modifications are required to control a code with CADNA, mainly changes in variable declarations, input and output. This paper describes the features of the CADNA library and shows how to interpret the information it provides concerning round-off error propagation in a code.

Program summary

Program title:CADNACatalogue identifier:AEAT_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAT_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.:53 420No. of bytes in distributed program, including test data, etc.:566 495Distribution format:tar.gzProgramming language:FortranComputer:PC running LINUX with an i686 or an ia64 processor, UNIX workstations including SUN, IBMOperating system:LINUX, UNIXClassification:4.14, 6.5, 20Nature of problem:A simulation program which uses floating-point arithmetic generates round-off errors, due to the rounding performed at each assignment and at each arithmetic operation. Round-off error propagation may invalidate the result of a program. The CADNA library enables one to estimate round-off error propagation in any simulation program and to detect all numerical instabilities that may occur at run time.Solution method:The CADNA library [1] implements Discrete Stochastic Arithmetic [2-4] which is based on a probabilistic model of round-off errors. The program is run several times with a random rounding mode generating different results each time. From this set of results, CADNA estimates the number of exact significant digits in the result that would have been computed with standard floating-point arithmetic.Restrictions:CADNA requires a Fortran 90 (or newer) compiler. In the program to be linked with the CADNA library, round-off errors on complex variables cannot be estimated. Furthermore array functions such as product or sum must not be used. Only the arithmetic operators and the abs, min, max and sqrt functions can be used for arrays.Running time:The version of a code which uses CADNA runs at least three times slower than its floating-point version. This cost depends on the computer architecture and can be higher if the detection of numerical instabilities is enabled. In this case, the cost may be related to the number of instabilities detected.References:

[1]

The CADNA library, URL address: http://www.lip6.fr/cadna.

[2]

J.-M. Chesneaux, L'arithmétique Stochastique et le Logiciel CADNA, Habilitation á diriger des recherches, Université Pierre et Marie Curie, Paris, 1995.

[3]

J. Vignes, A stochastic arithmetic for reliable scientific computation, Math. Comput. Simulation 35 (1993) 233-261.

[4]

J. Vignes, Discrete stochastic arithmetic for validating results of numerical software, Numer. Algorithms 37 (2004) 377-390.

     

GenMin: An enhanced genetic algorithm for global optimization

A new method that employs grammatical evolution and a stopping rule for finding the global minimum of a continuous multidimensional, multimodal function is considered. The genetic algorithm used is a hybrid genetic algorithm in conjunction with a local search procedure. We list results from numerical experiments with a series of test functions and we compare with other established global optimization methods. The accompanying software accepts objective functions coded either in Fortran 77 or in C++.

Program summary

Program title: GenMinCatalogue identifier: AEAR_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAR_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.: 35 810No. of bytes in distributed program, including test data, etc.: 436 613Distribution format: tar.gzProgramming language: GNU-C++, GNU-C, GNU Fortran 77Computer: The tool is designed to be portable in all systems running the GNU C++ compilerOperating system: The tool is designed to be portable in all systems running the GNU C++ compilerRAM: 200 KBWord size: 32 bitsClassification: 4.9Nature of problem: A multitude of problems in science and engineering are often reduced to minimizing a function of many variables. There are instances that a local optimum does not correspond to the desired physical solution and hence the search for a better solution is required. Local optimization techniques are frequently trapped in local minima. Global optimization is hence the appropriate tool. For example, solving a nonlinear system of equations via optimization, employing a least squares type of objective, one may encounter many local minima that do not correspond to solutions (i.e. they are far from zero).Solution method: Grammatical evolution and a stopping rule.Running time: Depending on the objective function. The test example given takes only a few seconds 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)

•

Example 1: 2725 sec.

•

Example 2: 33 sec.

•

Example 3: 1 sec.

     

MontePython: Implementing Quantum Monte Carlo using Python

We present a cross-language C++/Python program for simulations of quantum mechanical systems with the use of Quantum Monte Carlo (QMC) methods. We describe a system for which to apply QMC, the algorithms of variational Monte Carlo and diffusion Monte Carlo and we describe how to implement theses methods in pure C++ and C++/Python. Furthermore we check the efficiency of the implementations in serial and parallel cases to show that the overhead using Python can be negligible.

Program summary

Program title: MontePythonCatalogue identifier: ADZP_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZP_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.: 49 519No. of bytes in distributed program, including test data, etc.: 114 484Distribution format: tar.gzProgramming language: C++, PythonComputer: PC, IBM RS6000/320, HP, ALPHAOperating system: LINUXHas the code been vectorised or parallelized?: Yes, parallelized with MPINumber of processors used: 1-96RAM: Depends on physical system to be simulatedClassification: 7.6; 16.1Nature of problem: Investigating ab initio quantum mechanical systems, specifically Bose-Einstein condensation in dilute gases of ⁸⁷RbSolution method: Quantum Monte CarloRunning time: 225 min with 20 particles (with 4800 walkers moved in 1750 time steps) on 1 AMD Opteron^TM Processor 2218 processor; Production run for, e.g., 200 particles takes around 24 hours on 32 such processors.     

Qprop: A Schrödinger-solver for intense laser-atom interaction

The Qprop package is presented. Qprop has been developed to study laser-atom interaction in the nonperturbative regime where nonlinear phenomena such as above-threshold ionization, high order harmonic generation, and dynamic stabilization are known to occur. In the nonrelativistic regime and within the single active electron approximation, these phenomena can be studied with Qprop in the most rigorous way by solving the time-dependent Schrödinger equation in three spatial dimensions. Because Qprop is optimized for the study of quantum systems that are spherically symmetric in their initial, unperturbed configuration, all wavefunctions are expanded in spherical harmonics. Time-propagation of the wavefunctions is performed using a split-operator approach. Photoelectron spectra are calculated employing a window-operator technique. Besides the solution of the time-dependent Schrödinger equation in single active electron approximation, Qprop allows to study many-electron systems via the solution of the time-dependent Kohn-Sham equations.

Program summary

Program title:QPROPCatalogue number:ADXBProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADXBProgram obtainable from:CPC Program Library, Queen's University of Belfast, N. IrelandComputer on which program has been tested:PC Pentium IV, AthlonOperating system:LinuxProgram language used:C++Memory required to execute with typical data:Memory requirements depend on the number of propagated orbitals and on the size of the orbitals. For instance, time-propagation of a hydrogenic wavefunction in the perturbative regime requires about 64 KB RAM (4 radial orbitals with 1000 grid points). Propagation in the strongly nonperturbative regime providing energy spectra up to high energies may need 60 radial orbitals, each with 30000 grid points, i.e. about 30 MB. Examples are given in the article.No. of bits in a word:Real and complex valued numbers of double precision are usedNo. of lines in distributed program, including test data, etc.:69 995No. of bytes in distributed program, including test data, etc.: 2 927 567Peripheral used:Disk for input-output, terminal for interaction with the userCPU time required to execute test data:Execution time depends on the size of the propagated orbitals and the number of time-stepsDistribution format:tar.gzNature of the physical problem:Atoms put into the strong field of modern lasers display a wealth of novel phenomena that are not accessible to conventional perturbation theory where the external field is considered small as compared to inneratomic forces. Hence, the full ab initio solution of the time-dependent Schrödinger equation is desirable but in full dimensionality only feasible for no more than two (active) electrons. If many-electron effects come into play or effective ground state potentials are needed, (time-dependent) density functional theory may be employed. Qprop aims at providing tools for (i) the time-propagation of the wavefunction according to the time-dependent Schrödinger equation, (ii) the time-propagation of Kohn-Sham orbitals according to the time-dependent Kohn-Sham equations, and (iii) the energy-analysis of the final one-electron wavefunction (or the Kohn-Sham orbitals).Method of solution:An expansion of the wavefunction in spherical harmonics leads to a coupled set of equations for the radial wavefunctions. These radial wavefunctions are propagated using a split-operator technique and the Crank-Nicolson approximation for the short-time propagator. The initial ground state is obtained via imaginary time-propagation for spherically symmetric (but otherwise arbitrary) effective potentials. Excited states can be obtained through the combination of imaginary time-propagation and orthogonalization. For the Kohn-Sham scheme a multipole expansion of the effective potential is employed. Wavefunctions can be analyzed using the window-operator technique, facilitating the calculation of electron spectra, either angular-resolved or integratedRestrictions onto the complexity of the problem:The coupling of the atom to the external field is treated in dipole approximation. The time-dependent Schrödinger solver is restricted to the treatment of a single active electron. As concerns the time-dependent density functional mode of Qprop, the Hartree-potential (accounting for the classical electron-electron repulsion) is expanded up to the quadrupole. Only the monopole term of the Krieger-Li-Iafrate exchange potential is currently implemented. As in any nontrivial optimization problem, convergence to the optimal many-electron state (i.e. the ground state) is not automatically guaranteedExternal routines/libraries used:The program uses the well established libraries blas, lapack, and f2c