tmLQCD: A program suite to simulate Wilson twisted mass lattice QCD

We discuss a program suite for simulating Quantum Chromodynamics on a 4-dimensional space–time lattice. The basic Hybrid Monte Carlo algorithm is introduced and a number of algorithmic improvements are explained. We then discuss the implementations of these concepts as well as our parallelisation strategy in the actual simulation code. Finally, we provide a user guide to compile and run the program.

Program summary

Program title: tmLQCDCatalogue identifier: AEEH_v1_0Program summary URL::http://cpc.cs.qub.ac.uk/summaries/AEEH_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: GNU General Public Licence (GPL)No. of lines in distributed program, including test data, etc.: 122 768No. of bytes in distributed program, including test data, etc.: 931 042Distribution format: tar.gzProgramming language: C and MPIComputer: anyOperating system: any with a standard C compilerHas the code been vectorised or parallelised?: Yes. One or optionally any even number of processors may be used. Tested with up to 32 768 processorsRAM: no typical values availableClassification: 11.5External routines: LAPACK [1] and LIME [2] libraryNature of problem: Quantum ChromodynamicsSolution method: Markov Chain Monte Carlo using the Hybrid Monte Carlo algorithm with mass preconditioning and multiple time scales [3]. Iterative solver for large systems of linear equations.Restrictions: Restricted to an even number of (not necessarily mass degenerate) quark flavours in the Wilson or Wilson twisted mass formulation of lattice QCD.Running time: Depending on the problem size, the architecture and the input parameters from a few minutes to weeks.References:

• [1] 

  http://www.netlib.org/lapack/.

• [2] 

  USQCD, http://usqcd.jlab.org/usqcd-docs/c-lime/.

• [3] 

  C. Urbach, K. Jansen, A. Shindler, U. Wenger, Comput. Phys. Commun. 174 (2006) 87, hep-lat/0506011.

     

2DRMP: A suite of two-dimensional R-matrix propagation codes

The R-matrix method has proved to be a remarkably stable, robust and efficient technique for solving the close-coupling equations that arise in electron and photon collisions with atoms, ions and molecules. During the last thirty-four years a series of related R-matrix program packages have been published periodically in CPC. These packages are primarily concerned with low-energy scattering where the incident energy is insufficient to ionise the target. In this paper we describe 2DRMP, a suite of two-dimensional R-matrix propagation programs aimed at creating virtual experiments on high performance and grid architectures to enable the study of electron scattering from H-like atoms and ions at intermediate energies.

Program summary

Program title: 2DRMPCatalogue identifier: AEEA_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEEA_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.: 196 717No. of bytes in distributed program, including test data, etc.: 3 819 727Distribution format: tar.gzProgramming language: Fortran 95, MPIComputer: Tested on CRAY XT4 [1]; IBM eServer 575 [2]; Itanium II cluster [3]Operating system: Tested on UNICOS/lc [1]; IBM AIX [2]; Red Hat Linux Enterprise AS [3]Has the code been vectorised or parallelised?: Yes. 16 cores were used for small test runClassification: 2.4External routines: BLAS, LAPACK, PBLAS, ScaLAPACKSubprograms used: ADAZ_v1_1Nature of problem: 2DRMP is a suite of programs aimed at creating virtual experiments on high performance architectures to enable the study of electron scattering from H-like atoms and ions at intermediate energies.Solution method: Two-dimensional R-matrix propagation theory. The (r₁,r₂) space of the internal region is subdivided into a number of subregions. Local R-matrices are constructed within each subregion and used to propagate a global R-matrix, ℜ, across the internal region. On the boundary of the internal region ℜ is transformed onto the IERM target state basis. Thus, the two-dimensional R-matrix propagation technique transforms an intractable problem into a series of tractable problems enabling the internal region to be extended far beyond that which is possible with the standard one-sector codes. A distinctive feature of the method is that both electrons are treated identically and the R-matrix basis states are constructed to allow for both electrons to be in the continuum. The subregion size is flexible and can be adjusted to accommodate the number of cores available.Restrictions: The implementation is currently restricted to electron scattering from H-like atoms and ions.Additional comments: The programs have been designed to operate on serial computers and to exploit the distributed memory parallelism found on tightly coupled high performance clusters and supercomputers. 2DRMP has been systematically and comprehensively documented using ROBODoc [4] which is an API documentation tool that works by extracting specially formatted headers from the program source code and writing them to documentation files.Running time: The wall clock running time for the small test run using 16 cores and performed on [3] is as follows: bp (7 s); rint2 (34 s); newrd (32 s); diag (21 s); amps (11 s); prop (24 s).References:

• [1] 

  HECToR, CRAY XT4 running UNICOS/lc, http://www.hector.ac.uk/, accessed 22 July, 2009.

• [2] 

  HPCx, IBM eServer 575 running IBM AIX, http://www.hpcx.ac.uk/, accessed 22 July, 2009.

• [3] 

  HP Cluster, Itanium II cluster running Red Hat Linux Enterprise AS, Queen s University Belfast, http://www.qub.ac.uk/directorates/InformationServices/Research/HighPerformanceComputing/Services/Hardware/HPResearch/, accessed 22 July, 2009.

• [4] 

  Automating Software Documentation with ROBODoc, http://www.xs4all.nl/~rfsber/Robo/, accessed 22 July, 2009.

     

FARM_2DRMP: A version of FARM for use with 2DRMP

To complete the 2DRMP package an asymptotic program, such as FARM, is needed. The original version of FARM is designed to construct the physical R-matrix, R, from surface amplitudes contained in the H-file. However, in 2DRMP, R has already been constructed for each scattering energy during propagation. Therefore, this modified version of FARM, known as FARM_2DRMP, has been developed solely for use with 2DRMP.

New version program summary

Program title: FARM_2DRMPCatalogue identifier: ADAZ_v1_1Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADAZ_v1_1.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.: 13 806No. of bytes in distributed program, including test data, etc.: 134 462Distribution format: tar.gzProgramming language: Fortran 95 and MPIComputer: Tested on CRAY XT4 [1]; IBM eServer 575 [2]; Itanium II cluster [3]Operating system: Tested on UNICOS/lc [1]; IBM AIX [2]; Red Hat Linux Enterprise AS [3]Has the code been vectorized or parallelized?: Yes. 16 cores were used for the small test runClassification: 2.4External routines: BLAS, LAPACKDoes the new version supersede the previous version?: NoNature of problem: The program solves the scattering problem in the asymptotic region of R-matrix theory where exchange is negligible.Solution method: A radius is determined at which the wave function, calculated as a Gailitis expansion [4] with accelerated summing [5] over terms, converges. The R-matrix is propagated from the boundary of the internal region to this radius and the K-matrix calculated. Collision strengths or cross sections may be calculated.Reasons for new version: To complete the 2DRMP package [6] an asymptotic program, such as FARM [7], is needed. The original version of FARM is designed to construct the physical R-matrix, R, from surface amplitudes contained in the H-file. However, in 2DRMP, R, has already been constructed for each scattering energy during propagation and each R is stored in one of the RmatT files described in Fig. 8 of [6]. Therefore, this modified version of FARM, known as FARM_2DRMP, has been developed solely for use with 2DRMP. Instructions on its use and corresponding test data is provided with 2DRMP [6].Summary of revisions: FARM_2DRMP contains two codes, farm.f and farm_par.f90. The former is a serial code while the latter is a parallel F95 code that employs an MPI harness to enable the nenergy energies to be computed simultaneously across ncore cores, with each core processing either ⌊nenergy/ncore⌋ or ⌈nenergy/ncore⌉ energies. The input files, input.d and H, and the output file farm.out are as described in [7]. Both codes read R directly from RmatT.Restrictions: FARM_2DRMP is for use solely with 2DRMP and for a specified L,S and Π combination. The energy range specified in input.d must match that specified in energies.data.Running time: The wall clock running time for the small test run using 16 cores and performed on [3] is 9 secs.References:

• [1] 

  HECToR, CRAY XT4 running UNICOS/lc, http://www.hector.ac.uk/, visited 22 July, 2009.

• [2] 

  HPCx, IBM eServer 575 running IBM AIX, http://www.hpcx.ac.uk/, visited 22 July, 2009.

• [3] 

  HP Cluster, Itanium II cluster running Red Hat Linux Enterprise AS, Queen's University Belfast, http://www.qub.ac.uk/directorates/InformationServices/Research/HighPerformanceComputing/Services/Hardware/HPResearch/, visited 22 July, 2009.

• [4] 

  M. Gailitis, J. Phys. B 9 (1976) 843.

• [5] 

  C.J. Noble, R.K. Nesbet, Comput. Phys. Comm. 33 (1984) 399.

• [6] 

  N.S. Scott, M.P. Scott, P.G. Burke, T. Stitt, V. Faro-Maza, C. Denis, A. Maniopoulou, Comput. Phys. Comm. 180 (12) (2009) 2424–2449, this issue.

• [7] 

  V.M. Burke, C.J. Noble, Comput. Phys. Comm. 85 (1995) 471.

     

Automated generation of lattice QCD Feynman rules

The derivation of the Feynman rules for lattice perturbation theory from actions and operators is complicated, especially for highly improved actions such as HISQ. This task is, however, both important and particularly suitable for automation. We describe a suite of software to generate and evaluate Feynman rules for a wide range of lattice field theories with gluons and (relativistic and/or heavy) quarks. Our programs are capable of dealing with actions as complicated as (m)NRQCD and HISQ. Automated differentiation methods are used to calculate also the derivatives of Feynman diagrams.

Program summary

Program title: HiPPY, HPsrcCatalogue identifier: AEDX_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEDX_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: GPLv2 (see Additional comments below)No. of lines in distributed program, including test data, etc.: 513 426No. of bytes in distributed program, including test data, etc.: 4 893 707Distribution format: tar.gzProgramming language: Python, Fortran95Computer: HiPPy: Single-processor workstations. HPsrc: Single-processor workstations and MPI-enabled multi-processor systemsOperating system: HiPPy: Any for which Python v2.5.x is available. HPsrc: Any for which a standards-compliant Fortran95 compiler is availableHas the code been vectorised or parallelised?: YesRAM: Problem specific, typically less than 1 GB for either codeClassification: 4.4, 11.5Nature of problem: Derivation and use of perturbative Feynman rules for complicated lattice QCD actions.Solution method: An automated expansion method implemented in Python (HiPPy) and code to use expansions to generate Feynman rules in Fortran95 (HPsrc).Restrictions: No general restrictions. Specific restrictions are discussed in the text.Additional comments: The HiPPy and HPsrc codes are released under the second version of the GNU General Public Licence (GPL v2). Therefore anyone is free to use or modify the code for their own calculations. As part of the licensing, we ask that any publications including results from the use of this code or of modifications of it cite Refs. [1,2] as well as this paper. Finally, we also ask that details of these publications, as well as of any bugs or required or useful improvements of this core code, would be communicated to us.Running time: Very problem specific, depending on the complexity of the Feynman rules and the number of integration points. Typically between a few minutes and several weeks. The installation tests provided with the program code take only a few seconds to run.References:

• [1] 

  A. Hart, G.M. von Hippel, R.R. Horgan, L.C. Storoni, Automatically generating Feynman rules for improved lattice eld theories, J. Comput. Phys. 209 (2005) 340–353, doi:10.1016/j.jcp.2005.03.010, arXiv:hep-lat/0411026.

• [2] 

  M. Lüscher, P. Weisz, Efficient Numerical Techniques for Perturbative Lattice Gauge Theory Computations, Nucl. Phys. B 266 (1986) 309, doi:10.1016/0550-3213(86)90094-5.

     

HONEI: A collection of libraries for numerical computations targeting multiple processor architectures

We present HONEI, an open-source collection of libraries offering a hardware oriented approach to numerical calculations. HONEI abstracts the hardware, and applications written on top of HONEI can be executed on a wide range of computer architectures such as CPUs, GPUs and the Cell processor. We demonstrate the flexibility and performance of our approach with two test applications, a Finite Element multigrid solver for the Poisson problem and a robust and fast simulation of shallow water waves. By linking against HONEI's libraries, we achieve a two-fold speedup over straight forward C++ code using HONEI's SSE backend, and additional 3–4 and 4–16 times faster execution on the Cell and a GPU. A second important aspect of our approach is that the full performance capabilities of the hardware under consideration can be exploited by adding optimised application-specific operations to the HONEI libraries. HONEI provides all necessary infrastructure for development and evaluation of such kernels, significantly simplifying their development.

Program summary

Program title: HONEICatalogue identifier: AEDW_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEDW_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: GPLv2No. of lines in distributed program, including test data, etc.: 216 180No. of bytes in distributed program, including test data, etc.: 1 270 140Distribution format: tar.gzProgramming language: C++Computer: x86, x86_64, NVIDIA CUDA GPUs, Cell blades and PlayStation 3Operating system: LinuxRAM: at least 500 MB freeClassification: 4.8, 4.3, 6.1External routines: SSE: none; [1] for GPU, [2] for Cell backendNature of problem: Computational science in general and numerical simulation in particular have reached a turning point. The revolution developers are facing is not primarily driven by a change in (problem-specific) methodology, but rather by the fundamental paradigm shift of the underlying hardware towards heterogeneity and parallelism. This is particularly relevant for data-intensive problems stemming from discretisations with local support, such as finite differences, volumes and elements.Solution method: To address these issues, we present a hardware aware collection of libraries combining the advantages of modern software techniques and hardware oriented programming. Applications built on top of these libraries can be configured trivially to execute on CPUs, GPUs or the Cell processor. In order to evaluate the performance and accuracy of our approach, we provide two domain specific applications; a multigrid solver for the Poisson problem and a fully explicit solver for 2D shallow water equations.Restrictions: HONEI is actively being developed, and its feature list is continuously expanded. Not all combinations of operations and architectures might be supported in earlier versions of the code. Obtaining snapshots from http://www.honei.org is recommended.Unusual features: The considered applications as well as all library operations can be run on NVIDIA GPUs and the Cell BE.Running time: Depending on the application, and the input sizes. The Poisson solver executes in few seconds, while the SWE solver requires up to 5 minutes for large spatial discretisations or small timesteps.References:

• [1] 

  http://www.nvidia.com/cuda.

• [2] 

  http://www.ibm.com/developerworks/power/cell.

     

XtalOpt version r7: An open-source evolutionary algorithm for crystal structure prediction

A new version of XtalOpt, a user-friendly GPL-licensed evolutionary algorithm for crystal structure prediction, is available for download from the CPC library or the XtalOpt website, http://xtalopt.openmolecules.net. The new version now supports four external geometry optimization codes (VASP, GULP, PWSCF, and CASTEP), as well as three queuing systems: PBS, SGE, SLURM, and “Local”. The local queuing system allows the geometry optimizations to be performed on the user?s workstation if an external computational cluster is unavailable. Support for the Windows operating system has been added, and a Windows installer is provided. Numerous bugfixes and feature enhancements have been made in the new release as well.

New version program summary

Program title:XtalOptCatalogue identifier: AEGX_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEGX_v2_0.htmlProgram obtainable from: CPC Program Library, Queen?s University, Belfast, N. IrelandLicensing provisions: GPL v2.1 or later [1]No. of lines in distributed program, including test data, etc.: 125 383No. of bytes in distributed program, including test data, etc.: 11 607 415Distribution format: tar.gzProgramming language: C++Computer: PCs, workstations, or clustersOperating system: Linux, MS WindowsClassification: 7.7External routines: Qt [2], Open Babel [3], Avogadro [4], and one of: VASP [5], PWSCF [6], GULP [7], CASTEP [8]Catalogue identifier of previous version: AEGX_v1_0Journal reference of previous version: Comput. Phys. Comm. 182 (2011) 372Does the new version supersede the previous version?: YesNature of problem: Predicting the crystal structure of a system from its stoichiometry alone remains a grand challenge in computational materials science, chemistry, and physics.Solution method: Evolutionary algorithms are stochastic search techniques which use concepts from biological evolution in order to locate the global minimum of a crystalline structure on its potential energy surface. Our evolutionary algorithm, XtalOpt, is freely available for use and collaboration under the GNU Public License. See the original publication on XtalOpt?s implementation [11] for more information on the method.Reasons for new version: Since XtalOpt?s initial release in June 2010, support for additional optimizers, queuing systems, and an operating system has been added. XtalOpt can now use VASP, GULP, PWSCF, or CASTEP to perform local geometry optimizations. The queue submission code has been rewritten, and now supports running any of the above codes on ssh-accessible computer clusters that use the Portable Batch System (PBS), Sun Grid Engine (SGE), or SLURM queuing systems for managing the optimization jobs. Alternatively, geometry optimizations may be performed on the user?s workstation using the new internal “Local” queuing system if high performance computing resources are unavailable. XtalOpt has been built and tested on the Microsoft Windows operating system (XP or later) in addition to Linux, and a Windows installer is provided. The installer includes a development version of Avogadro that contains expanded crystallography support [12] that is not available in the mainline Avogadro releases. Other notable new developments include:

• • 

  LIBSSH [10] is distributed with the XtalOpt sources and used for communication with the remote clusters, eliminating the previous requirement to set up public-key authentication;

• • 

  Plotting enthalpy (or energy) vs. structure number in the plot tab will trace out the history of the most stable structure as the search progresses A read-only mode has been added to allow inspection of previous searches through the user interface without connecting to a cluster or submitting new jobs;

• • 

  The tutorial [13] has been rewritten to reflect the changes to the interface and the newly supported codes. Expanded sections on optimizations schemes and save/resume have been added;

• • 

  The included version of SPGLIB has been updated. An option has been added to set the Cartesian tolerance of the space group detection. A new option has been added to the Progress table?s right-click menu that copies the selected structure?s POSCAR formatted representation to the clipboard;

• • 

  Numerous other small bugfixes/enhancements.

Summary of revisions: See “Reasons for new version” above.Running time: User dependent. The program runs until stopped by the user.References:

•  [1] 

  http://www.gnu.org/licenses/gpl.html.

•  [2] 

  http://www.trolltech.com/.

•  [3] 

  http://openbabel.org/.

•  [4] 

  http://avogadro.openmolecules.net.

•  [5] 

  http://cms.mpi.univie.ac.at/vasp.

•  [6] 

  http://www.quantum-espresso.org.

•  [7] 

  https://www.ivec.org/gulp.

•  [8] 

  http://www.castep.org.

•  [9] 

  http://spglib.sourceforge.net.

• [10] 

  http://www.libssh.org.

• [11] 

  D. Lonie, E. Zurek, Comp. Phys. Comm. 182 (2011) 372–387, doi:10.1016/j.cpc.2010.07.048.

• [12] 

  http://davidlonie.blogspot.com/2011/03/new-avogadro-crystallography-extension.html.

• [13] 

  http://xtalopt.openmolecules.net/globalsearch/docs/tut-xo.html.

     

XtalOpt: An open-source evolutionary algorithm for crystal structure prediction

The implementation and testing of XtalOpt, an evolutionary algorithm for crystal structure prediction, is outlined. We present our new periodic displacement (ripple) operator which is ideally suited to extended systems. It is demonstrated that hybrid operators, which combine two pure operators, reduce the number of duplicate structures in the search. This allows for better exploration of the potential energy surface of the system in question, while simultaneously zooming in on the most promising regions. A continuous workflow, which makes better use of computational resources as compared to traditional generation based algorithms, is employed. Various parameters in XtalOpt are optimized using a novel benchmarking scheme. XtalOpt is available under the GNU Public License, has been interfaced with various codes commonly used to study extended systems, and has an easy to use, intuitive graphical interface.

Program summary

Program title:XtalOptCatalogue identifier: AEGX_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEGX_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: GPL v2.1 or later [1]No. of lines in distributed program, including test data, etc.: 36 849No. of bytes in distributed program, including test data, etc.: 1 149 399Distribution format: tar.gzProgramming language: C++Computer: PCs, workstations, or clustersOperating system: LinuxClassification: 7.7External routines: QT [2], OpenBabel [3], AVOGADRO [4], SPGLIB [8] and one of: VASP [5], PWSCF [6], GULP [7].Nature of problem: Predicting the crystal structure of a system from its stoichiometry alone remains a grand challenge in computational materials science, chemistry, and physics.Solution method: Evolutionary algorithms are stochastic search techniques which use concepts from biological evolution in order to locate the global minimum on their potential energy surface. Our evolutionary algorithm, XtalOpt, is freely available to the scientific community for use and collaboration under the GNU Public License.Running time: User dependent. The program runs until stopped by the user.References:

• [1] 

  http://www.gnu.org/licenses/gpl.html.

• [2] 

  http://www.trolltech.com/.

• [3] 

  http://openbabel.org/.

• [4] 

  http://avogadro.openmolecules.net.

• [5] 

  http://cms.mpi.univie.ac.at/vasp.

• [6] 

  http://www.quantum-espresso.org.

• [7] 

  https://www.ivec.org/gulp.

• [8] 

  http://spglib.sourceforge.net.

     

GeodesicViewer – A tool for exploring geodesics in the theory of relativity

The GeodesicViewer realizes exocentric two- and three-dimensional illustrations of lightlike and timelike geodesics in the general theory of relativity. By means of an intuitive graphical user interface, all parameters of a spacetime as well as the initial conditions of the geodesics can be modified interactively.

New version program summary

Program title: GeodesicViewerCatalogue identifier: AEFP_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEFP_v2_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.: 76 202No. of bytes in distributed program, including test data, etc.: 1 722 290Distribution format: tar.gzProgramming language: C++, OpenGLComputer: All platforms with a C++ compiler, Qt, OpenGLOperating system: Linux, Mac OS X, WindowsRAM: 24 MBytesClassification: 1.5External routines:

• • 

  Motion4D (included in the package)

• • 

  Gnu Scientific Library (GSL) (http://www.gnu.org/software/gsl/)

• • 

  Qt (http://qt.nokia.com/downloads)

• • 

  OpenGL (http://www.opengl.org/)

Catalogue identifier of previous version: AEFP_v1_0Journal reference of previous version: Comput. Phys. Comm. 181 (2010) 413Does the new version supersede the previous version?: YesNature of problem: Illustrate geodesics in four-dimensional Lorentzian spacetimes.Solution method: Integration of ordinary differential equations. 3D-Rendering via OpenGL.Reasons for new version: The main reason for the new version was to visualize the parallel transport of the Sachs legs and to show the influence of curved spacetime on a bundle of light rays as is realized in the new version of the Motion4D library (http://cpc.cs.qub.ac.uk/summaries/AEEX_v3_0.html).Summary of revisions:

• • 

  By choosing the new geodesic type “lightlike_sachs”, the parallel transport of the Sachs basis and the integration of the Jacobi equation can be visualized.

• • 

  The 2D representation via Qwt was replaced by an OpenGL 2D implementation to speed up the visualization.

• • 

  Viewing parameters can now be stored in a configuration file (.cfg).

• • 

  Several new objects can be used in 3D and 2D representation.

• • 

  Several predefined local tetrads can be choosen.

• • 

  There are some minor modifications: new mouse control (rotate on sphere); line smoothing; current last point in coordinates is shown; mutual-coordinate representation extended; current cursor position in 2D; colors for 2D view.

Running time: Interactive. The examples given take milliseconds.     

Continuous-time quantum Monte Carlo impurity solvers

Continuous-time quantum Monte Carlo impurity solvers are algorithms that sample the partition function of an impurity model using diagrammatic Monte Carlo techniques. The present paper describes codes that implement the interaction expansion algorithm originally developed by Rubtsov, Savkin, and Lichtenstein, as well as the hybridization expansion method developed by Werner, Millis, Troyer, et al. These impurity solvers are part of the ALPS-DMFT application package and are accompanied by an implementation of dynamical mean-field self-consistency equations for (single orbital single site) dynamical mean-field problems with arbitrary densities of states.

Program summary

Program title: dmftCatalogue identifier: AEIL_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEIL_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: ALPS LIBRARY LICENSE version 1.1No. of lines in distributed program, including test data, etc.: 899 806No. of bytes in distributed program, including test data, etc.: 32 153 916Distribution format: tar.gzProgramming language: C++Operating system: The ALPS libraries have been tested on the following platforms and compilers:

• • 

  Linux with GNU Compiler Collection (g++ version 3.1 and higher), and Intel C++ Compiler (icc version 7.0 and higher)

• • 

  MacOS X with GNU Compiler (g++ Apple-version 3.1, 3.3 and 4.0)

• • 

  IBM AIX with Visual Age C++ (xlC version 6.0) and GNU (g++ version 3.1 and higher) compilers

• • 

  Compaq Tru64 UNIX with Compq C++ Compiler (cxx)

• • 

  SGI IRIX with MIPSpro C++ Compiler (CC)

• • 

  HP-UX with HP C++ Compiler (aCC)

• • 

  Windows with Cygwin or coLinux platforms and GNU Compiler Collection (g++ version 3.1 and higher)

RAM: 10 MB–1 GBClassification: 7.3External routines: ALPS [1], BLAS/LAPACK, HDF5Nature of problem: (See [2].) Quantum impurity models describe an atom or molecule embedded in a host material with which it can exchange electrons. They are basic to nanoscience as representations of quantum dots and molecular conductors and play an increasingly important role in the theory of “correlated electron” materials as auxiliary problems whose solution gives the “dynamical mean field” approximation to the self-energy and local correlation functions.Solution method: Quantum impurity models require a method of solution which provides access to both high and low energy scales and is effective for wide classes of physically realistic models. The continuous-time quantum Monte Carlo algorithms for which we present implementations here meet this challenge. Continuous-time quantum impurity methods are based on partition function expansions of quantum impurity models that are stochastically sampled to all orders using diagrammatic quantum Monte Carlo techniques. For a review of quantum impurity models and their applications and of continuous-time quantum Monte Carlo methods for impurity models we refer the reader to [2].Additional comments: Use of dmft requires citation of this paper. Use of any ALPS program requires citation of the ALPS [1] paper.Running time: 60 s–8 h per iteration.References:

• [1] 

  A. Albuquerque, F. Alet, P. Corboz, et al., J. Magn. Magn. Mater. 310 (2007) 1187.

• [2] 

  http://arxiv.org/abs/1012.4474, Rev. Mod. Phys., in press.

     

C-library raft: Reconstruction algorithms for tomography. Applications to X-ray fluorescence tomography

There are many reconstruction algorithms for tomography, raft for short, and some of them are considered “classic” by researchers. The so-called raft library, provide a set of useful and basic tools, usually needed in many inverse problems that are related to medical imaging. The subroutines in raft are free software and written in C language; portable to any system with a working C compiler. This paper presents source codes written according to raft routines, applied to a new imaging modality called X-ray fluorescence tomography.

Program summary

Program title: raftCatalogue identifier: AEJY_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEJY_v1_0.htmlProgram obtainable from: CPC Program Library, Queen?s University, Belfast, N. IrelandLicensing provisions: GNU General Public Licence, version 2No. of lines in distributed program, including test data, etc.: 218 844No. of bytes in distributed program, including test data, etc.: 3 562 902Distribution format: tar.gzProgramming language: Standard C.Computer: Any with a standard C compilerOperating system: Linux and WindowsClassification: 2.4, 2.9, 3, 4.3, 4.7External routines:

•  

  raft:   autoconf 2.60 or later – http://www.gnu.org/software/autoconf/   GSL scientific library – http://www.gnu.org/software/gsl/   Confuse parser library – http://www.nongnu.org/confuse/

raft-fun: gengetopt – http://www.gnu.org/software/gengetopt/gengetopt.htmlNature of problem: Reconstruction algorithms for tomography, specially in X-ray fluorescence tomography.Solution method: As a library, raft covers the standard reconstruction algorithms like filtered backprojection, Novikov?s inversion, Hogan?s formula, among others. The input data set is represented by a complete sinogram covering a determined angular range. Users are allowed to set solid angle range for fluorescence emission at each algorithm.Running time: 1 second to 15 minutes, depending on the data size.     

FIESTA 2: Parallelizeable multiloop numerical calculations

The program FIESTA has been completely rewritten. Now it can be used not only as a tool to evaluate Feynman integrals numerically, but also to expand Feynman integrals automatically in limits of momenta and masses with the use of sector decompositions and Mellin–Barnes representations. Other important improvements to the code are complete parallelization (even to multiple computers), high-precision arithmetics (allowing to calculate integrals which were undoable before), new integrators, Speer sectors as a strategy, the possibility to evaluate more general parametric integrals.

Program summary

Program title:FIESTA 2Catalogue identifier: AECP_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AECP_v2_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: GNU GPL version 2No. of lines in distributed program, including test data, etc.: 39 783No. of bytes in distributed program, including test data, etc.: 6 154 515Distribution format: tar.gzProgramming language: Wolfram Mathematica 6.0 (or higher) and CComputer: From a desktop PC to a supercomputerOperating system: Unix, Linux, Windows, Mac OS XHas the code been vectorised or parallelized?: Yes, the code has been parallelized for use on multi-kernel computers as well as clusters via Mathlink over the TCP/IP protocol. The program can work successfully with a single processor, however, it is ready to work in a parallel environment and the use of multi-kernel processor and multi-processor computers significantly speeds up the calculation; on clusters the calculation speed can be improved even further.RAM: Depends on the complexity of the problemClassification: 4.4, 4.12, 5, 6.5Catalogue identifier of previous version: AECP_v1_0Journal reference of previous version: Comput. Phys. Comm. 180 (2009) 735External routines: QLink [1], Cuba library [2], MPFR [3]Does the new version supersede the previous version?: YesNature of problem: The sector decomposition approach to evaluating Feynman integrals falls apart into the sector decomposition itself, where one has to minimize the number of sectors; the pole resolution and epsilon expansion; and the numerical integration of the resulting expression.Solution method: The sector decomposition is based on a new strategy as well as on classical strategies such as Speer sectors. The sector decomposition, pole resolution and epsilon-expansion are performed in Wolfram Mathematica 6.0 or, preferably, 7.0 (enabling parallelization) [4]. The data is stored on hard disk via a special program, QLink [1]. The expression for integration is passed to the C-part of the code, that parses the string and performs the integration by one of the algorithms in the Cuba library package [2]. This part of the evaluation is perfectly parallelized on multi-kernel computers.Reasons for new version:

• 1. 

  The first version of FIESTA had problems related to numerical instability, so for some classes of integrals it could not produce a result.

• 2. 

  The sector decomposition method can be applied not only for integral calculation.

Summary of revisions:

• 1. 

  New integrator library is used.

• 2. 

  New methods to deal with numerical instability (MPFR library).

• 3. 

  Parallelization in Mathematica.

• 4. 

  Parallelization on multiple computers via TCP-IP.

• 5. 

  New sector decomposition strategy (Speer sectors).

• 6. 

  Possibility of using FIESTA to for integral expansion.

• 7. 

  Possibility of using FIESTA to discover poles in d.

• 8. 

  New negative terms resolution strategies.

Restrictions: The complexity of the problem is mostly restricted by CPU time required to perform the evaluation of the integralRunning time: Depends on the complexity of the problemReferences:

• [1] 

  http://qlink08.sourceforge.net, open source.

• [2] 

  http://www.feynarts.de/cuba/, open source.

• [3] 

  http://www.mpfr.org/, open source.

• [4] 

  http://www.wolfram.com/products/mathematica/index.html.

     

Indirect search for dark matter with micrOMEGAs_2.4

We present a new module of micrOMEGAs devoted to the computation of indirect signals from dark matter annihilation in any new model with a stable weakly interacting particle. The code provides the mass spectrum, cross-sections, relic density and exotic fluxes of gamma rays, positrons and antiprotons. The propagation of charged particles in the Galactic halo is handled with a new module that allows to easily modify the propagation parameters.

Program summary

Program title: micrOMEGAs2.4Catalogue identifier: ADQR_v2_3Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADQR_v2_3.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.: 401 126No. of bytes in distributed program, including test data, etc.: 6 583 596Distribution format: tar.gzProgramming language: C and FortranComputer: PC, Alpha, Mac, SunOperating system: UNIX (Linux, OSF1, SunOS, Darwin, Cygwin)RAM: 50 MB depending on the number of processes requiredClassification: 1.9, 11.6Catalogue identifier of previous version: ADQR_v2_3Journal reference of previous version: Comput. Phys. Comm. 180 (2009) 747Does the new version supersede the previous version?: YesNature of problem: Calculation of the relic density and detection rates of the lightest stable particle in a generic new model of particle physics.Solution method: In numerically solving the evolution equation for the density of dark matter, relativistic formulas for the thermal average are used. All tree-level processes for annihilation and coannihilation of new particles in the model are included. The cross-sections for all processes are calculated exactly with CalcHEP after definition of a model file. The propagation of the charged cosmic rays is solved within a semi-analytical two-zone model.Reasons for new version: There are many experiments that are currently searching for the remnants of dark matter annihilation. In this version we perform the computation of indirect signals from dark matter annihilation in any new model with a stable weakly interacting particle. We include the propagation of charged particles in the Galactic halo.Summary of revisions:

• • 

  Annihilation cross-sections for all 2-body tree-level processes and for radiative emission of a photon for all models.

• • 

  Annihilation cross-sections into polarised gauge bosons.

• • 

  Annihilation cross-sections for the loop induced processes γγ and γZ0 in the MSSM.

• • 

  Modelling of the DM halo with a general parameterization and with the possibility of including DM clumps.

• • 

  Computation of the propagation of charged particles through the Galaxy, including the possibility of modifying the propagation parameters.

• • 

  Effect of solar modulation on the charged particle spectrum.

• • 

  Model independent predictions of the indirect detection signals.

Unusual features: Depending on the parameters of the model, the program generates additional new code, compiles it and loads it dynamically.Running time: 3 sec     

The nonlinear gyro-kinetic flux tube code GKW

A new nonlinear gyro-kinetic flux tube code (GKW) for the simulation of micro instabilities and turbulence in magnetic confinement plasmas is presented in this paper. The code incorporates all physics effects that can be expected from a state of the art gyro-kinetic simulation code in the local limit: kinetic electrons, electromagnetic effects, collisions, full general geometry with a coupling to a MHD equilibrium code, and E×B shearing. In addition the physics of plasma rotation has been implemented through a formulation of the gyro-kinetic equation in the co-moving system. The gyro-kinetic model is five-dimensional and requires a massive parallel approach. GKW has been parallelised using MPI and scales well up to 8192+ cores. The paper presents the set of equations solved, the numerical methods, the code structure, and the essential benchmarks.

Program summary

Program title: GKWCatalogue identifier: AEES_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEES_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: GNU GPL v3No. of lines in distributed program, including test data, etc.: 29 998No. of bytes in distributed program, including test data, etc.: 206 943Distribution format: tar.gzProgramming language: Fortran 95Computer: Not computer specificOperating system: Any for which a Fortran 95 compiler is availableHas the code been vectorised or parallelised?: Yes. The program can efficiently utilise 8192+ processors, depending on problem and available computer. 128 processors is reasonable for a typical nonlinear kinetic run on the latest x86-64 machines.RAM:∼128 MB–1 GB for a linear run; 25 GB for typical nonlinear kinetic run (30 million grid points)Classification: 19.8, 19.9, 19.11External routines: None required, although the functionality of the program is somewhat limited without a MPI implementation (preferably MPI-2) and the FFTW3 library.Nature of problem: Five-dimensional gyro-kinetic Vlasov equation in general flux tube tokamak geometry with kinetic electrons, electro-magnetic effects and collisionsSolution method: Pseudo-spectral and finite difference with explicit time integrationAdditional comments: The MHD equilibrium code CHEASE [1] is used for the general geometry calculations. This code has been developed in CRPP Lausanne and is not distributed together with GKW, but can be downloaded separately. The geometry module of GKW is based on the version 7.1 of CHEASE, which includes the output for Hamada coordinates.Running time: (On recent x86-64 hardware) ∼10 minutes for a short linear problem; 48 hours for typical nonlinear kinetic run.References:

•  

  [1] H. Lütjens, A. Bondeson, O. Sauter, Comput. Phys. Comm. 97 (1996) 219, http://cpc.cs.qub.ac.uk/summaries/ADDH_v1_0.html.

     

Fast computation of close-coupling exchange integrals using polynomials in a tree representation

The semi-classical atomic-orbital close-coupling method is a well-known approach for the calculation of cross sections in ion–atom collisions. It strongly relies on the fast and stable computation of exchange integrals. We present an upgrade to earlier implementations of the Fourier-transform method.For this purpose, we implement an extensive library for symbolic storage of polynomials, relying on sophisticated tree structures to allow fast manipulation and numerically stable evaluation. Using this library, we considerably speed up creation and computation of exchange integrals. This enables us to compute cross sections for more complex collision systems.

Program summary

Program title: TXINTCatalogue identifier: AEHS_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEHS_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.: 12 332No. of bytes in distributed program, including test data, etc.: 157 086Distribution format: tar.gzProgramming language: Fortran 95Computer: All with a Fortran 95 compilerOperating system: All with a Fortran 95 compilerRAM: Depends heavily on input, usually less than 100 MiBClassification: 16.10Nature of problem: Analytical calculation of one- and two-center exchange matrix elements for the close-coupling method in the impact parameter model.Solution method: Similar to the code of Hansen and Dubois [1], we use the Fourier-transform method suggested by Shakeshaft [2] to compute the integrals. However, we heavily speed up the calculation using a library for symbolic manipulation of polynomials.Restrictions: We restrict ourselves to a defined collision system in the impact parameter model.Unusual features: A library for symbolic manipulation of polynomials, where polynomials are stored in a space-saving left-child right-sibling binary tree. This provides stable numerical evaluation and fast mutation while maintaining full compatibility with the original code.Additional comments: This program makes heavy use of the new features provided by the Fortran 90 standard, most prominently pointers, derived types and allocatable structures and a small portion of Fortran 95. Only newer compilers support these features. Following compilers support all features needed by the program.

• • 

  GNU Fortran Compiler “gfortran” from version 4.3.0

• • 

  GNU Fortran 95 Compiler “g95” from version 4.2.0

• • 

  Intel Fortran Compiler “ifort” from version 11.0

Running time: Heavily dependent on input, usually less than one CPU second.References:

• [1] 

  J.-P. Hansen, A. Dubois, Comput. Phys. Commun. 67 (1992) 456.

• [2] 

  R. Shakeshaft, J. Phys. B: At. Mol. Opt. Phys. 8 (1975) L134.

     

QCDMAPT_F: Fortran version of QCDMAPT package

The QCDMAPT program package facilitates computations in the framework of dispersive approach to Quantum Chromodynamics. The QCDMAPT_F version of this package enables one to perform such computations with Fortran, whereas the previous version was developed for use with Maple system. The QCDMAPT_F package possesses the same basic features as its previous version. Namely, it embodies the calculated explicit expressions for relevant spectral functions up to the four–loop level and the subroutines for necessary integrals.

New version program summary

Program title: QCDMAPT_FCatalogue identifier: AEGP_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEGP_v2_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.: 10 786No. of bytes in distributed program, including test data, etc.: 332 329Distribution format: tar.gzProgramming language: Fortran 77 and higherComputer: Any which supports Fortran 77Operating system: Any which supports Fortran 77Classification: 11.1, 11.5, 11.6External routines: MATHLIB routine RADAPT (D102) from CERNLIB Program Library [1]Catalogue identifier of previous version: AEGP_v1_0Journal reference of previous version: Comput. Phys. Comm. 181 (2010) 1769Does the new version supersede the previous version?: No. This version provides an alternative to the previous, Maple, version.Nature of problem: A central object of the dispersive (or “analytic”) approach to Quantum Chromodynamics [2,3] is the so-called spectral function, which can be calculated by making use of the strong running coupling. At the one-loop level the latter has a quite simple form and the relevant spectral function can easily be calculated. However, at the higher loop levels the strong running coupling has a rather cumbersome structure. Here, the explicit calculation of corresponding spectral functions represents a somewhat complicated task (see Section 3 and Appendix B of Ref. [4]), whereas their numerical evaluation requires a lot of computational resources and essentially slows down the overall computation process.Solution method: The developed package includes the calculated explicit expressions for relevant spectral functions up to the four-loop level and the subroutines for necessary integrals.Reasons for new version: The previous version of the package (Ref. [4]) was developed for use with Maple system. The new version is developed for Fortran programming language.Summary of revisions: The QCDMAPT_F package consists of the main program (QCDMAPT_F.f) and two samples of the file containing the values of input parameters (QCDMAPT_F.i1 and QCDMAPT_F.i2). The main program includes the definitions of relevant spectral functions and subroutines for necessary integrals. The main program also provides an example of computation of the values of (M)APT spacelike/timelike expansion functions for the specified set of input parameters and (as an option) generates the output data files with values of these functions over the given kinematic intervals.Additional comments: For the proper functioning of QCDMAPT_F package, the “MATHLIB” CERNLIB library [1] has to be installed.Running time: The running time of the main program with sample set of input parameters specified in the file QCDMAPT_F.i2 is about a minute (depends on CPU).References:

• [1] 

  Subroutine D102 of the “MATHLIB” CERNLIB library, URL addresses: http://cernlib.web.cern.ch/cernlib/mathlib.html, http://wwwasdoc.web.cern.ch/wwwasdoc/shortwrupsdir/d102/top.html.

• [2] 

  D.V. Shirkov, I.L. Solovtsov, Phys. Rev. Lett. 79 (1997) 1209;   K.A. Milton, I.L. Solovtsov, Phys. Rev. D 55 (1997) 5295;   K.A. Milton, I.L. Solovtsov, Phys. Rev. D 59 (1999) 107701;   I.L. Solovtsov, D.V. Shirkov, Theor. Math. Phys. 120 (1999) 1220;   D.V. Shirkov, I.L. Solovtsov, Theor. Math. Phys. 150 (2007) 132.

• [3] 

  A.V. Nesterenko, Phys. Rev. D 62 (2000) 094028;   A.V. Nesterenko, Phys. Rev. D 64 (2001) 116009;   A.V. Nesterenko, Int. J. Mod. Phys. A 18 (2003) 5475;   A.V. Nesterenko, J. Papavassiliou, J. Phys. G 32 (2006) 1025;   A.V. Nesterenko, Nucl. Phys. B (Proc. Suppl.) 186 (2009) 207.

• [4] 

  A.V. Nesterenko, C. Simolo, Comput. Phys. Comm. 181 (2010) 1769.

     

ROOT — A C++ framework for petabyte data storage, statistical analysis and visualization

A new stable version (“production version”) v5.28.00 of ROOT [1] has been published [2]. It features several major improvements in many areas, most noteworthy data storage performance as well as statistics and graphics features. Some of these improvements have already been predicted in the original publication Antcheva et al. (2009) [3]. This version will be maintained for at least 6 months; new minor revisions (“patch releases”) will be published [4] to solve problems reported with this version.

New version program summary

Program title: ROOTCatalogue identifier: AEFA_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEFA_v2_0.htmlProgram obtainable from: CPC Program Library, Queen?s University, Belfast, N. IrelandLicensing provisions: GNU Lesser Public License v.2.1No. of lines in distributed program, including test data, etc.: 2 934 693No. of bytes in distributed program, including test data, etc.: 1009Distribution format: tar.gzProgramming language: C++Computer: Intel i386, Intel x86-64, Motorola PPC, Sun Sparc, HP PA-RISCOperating system: GNU/Linux, Windows XP/Vista/7, Mac OS X, FreeBSD, OpenBSD, Solaris, HP-UX, AIXHas the code been vectorized or parallelized?: YesRAM: > 55 MbytesClassification: 4, 9, 11.9, 14Catalogue identifier of previous version: AEFA_v1_0Journal reference of previous version: Comput. Phys. Commun. 180 (2009) 2499Does the new version supersede the previous version?: YesNature of problem: Storage, analysis and visualization of scientific dataSolution method: Object store, wide range of analysis algorithms and visualization methodsReasons for new version: Added features and corrections of deficienciesSummary of revisions: The release notes at http://root.cern.ch/root/v528/Version528.news.html give a module-oriented overview of the changes in v5.28.00. Highlights include

• • 

  File format Reading of TTrees has been improved dramatically with respect to CPU time (30%) and notably with respect to disk space.

• • 

  Histograms A new TEfficiency class has been provided to handle the calculation of efficiencies and their uncertainties, TH2Poly for polygon-shaped bins (e.g. maps), TKDE for kernel density estimation, and TSVDUnfold for singular value decomposition.

• • 

  Graphics Kerning is now supported in TLatex, PostScript and PDF; a table of contents can be added to PDF files. A new font provides italic symbols. A TPad containing GL can be stored in a binary (i.e. non-vector) image file; add support for full-scene anti-aliasing. Usability enhancements to EVE.

• • 

  Math New interfaces for generating random number according to a given distribution, goodness of fit tests of unbinned data, binning multidimensional data, and several advanced statistical functions were added.

• • 

  RooFit Introduction of HistFactory; major additions to RooStats.

• • 

  TMVA Updated to version 4.1.0, adding e.g. the support for simultaneous classification of multiple output classes for several multivariate methods.

• • 

  PROOF Many new features, adding to PROOF?s usability, plus improvements and fixes.

• • 

  PyROOT Support of Python 3 has been added.

• • 

  Tutorials Several new tutorials were provided for above new features (notably RooStats).

A detailed list of all the changes is available at http://root.cern.ch/root/htmldoc/examples/V5.Additional comments: For an up-to-date author list see: http://root.cern.ch/drupal/content/root-development-team and http://root.cern.ch/drupal/content/former-root-developers.The distribution file for this program is over 30 Mbytes and therefore is not delivered directly when download or E-mail is requested. Instead a html file giving details of how the program can be obtained is sent.Running time: Depending on the data size and complexity of analysis algorithms.References:

• [1] 

  http://root.cern.ch.

• [2] 

  http://root.cern.ch/drupal/content/production-version-528.

• [3] 

  I. Antcheva, M. Ballintijn, B. Bellenot, M. Biskup, R. Brun, N. Buncic, Ph. Canal, D. Casadei, O. Couet, V. Fine, L. Franco, G. Ganis, A. Gheata, D. Gonzalez Maline, M. Goto, J. Iwaszkiewicz, A. Kreshuk, D. Marcos Segura, R. Maunder, L. Moneta, A. Naumann, E. Offermann, V. Onuchin, S. Panacek, F. Rademakers, P. Russo, M. Tadel, ROOT — A C++ framework for petabyte data storage, statistical analysis and visualization, Comput. Phys. Commun. 180 (2009) 2499.

• [4] 

  http://root.cern.ch/drupal/content/root-version-v5-28-00-patch-release-notes.