A metaheuristic approach to the optimal definition of molecule-fixed axes in rovibrational Hamiltonians

We develop an algorithmic, metaheuristic approach to the definition of molecule-fixed axes orientation in molecules of arbitrary size. The goal is to simplify the treatment of overall rotation and rotation-vibration interaction in rovibrational Hamiltonians. Considering the kinetic elements of the rovibrational Hamiltonian, given by the G matrix, we select the optimal orientation of molecule-fixed axes minimizing specific G matrix elements. To such an end, we develop a global minimization method based in a hybrid Simulated Annealing+Powell's local minimization. The parameters of the method are calibrated using a set of non-rigid molecules: Acetaldehyde, glycolaldehyde, methyl formate and ethyl methyl ether. The results show that the principal axes of inertia do not give the simplest form to the pure rotational contribution. However, minimization of the G matrix rotational element does. Finally, we observe that in the cases considered it is not possible to nullify all the rotation-vibration coupling elements, since the torsional motions are coupled with the overall rotation. However, the treatment yields the optimal solution. The methodology proposed allows also to simplify simultaneously the pure rotational+rotation-vibration coupling elements in rovibrational Hamiltonians.     

A new version of Scilab software package for the study of dynamical systems

This work presents a new version of a software package for the study of chaotic flows, maps and fractals [1]. The codes were written using Scilab, a software package for numerical computations providing a powerful open computing environment for engineering and scientific applications. It was found that Scilab provides various functions for ordinary differential equation solving, Fast Fourier Transform, autocorrelation, and excellent 2D and 3D graphical capabilities. The chaotic behaviors of the nonlinear dynamics systems were analyzed using phase-space maps, autocorrelation functions, power spectra, Lyapunov exponents and Kolmogorov-Sinai entropy. Various well-known examples are implemented, with the capability of the users inserting their own ODE or iterative equations.

New version program summary

Program title: Chaos v2.0Catalogue identifier: AEAP_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAP_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.: 1275No. of bytes in distributed program, including test data, etc.: 7135Distribution format: tar.gzProgramming language: Scilab 5.1.1. Scilab 5.1.1 should be installed before running the program. Information about the installation can be found at http://wiki.scilab.org/howto/install/windows.Computer: PC-compatible running Scilab on MS Windows or LinuxOperating system: Windows XP, LinuxRAM: below 150 MegabytesClassification: 6.2Catalogue identifier of previous version: AEAP_v1_0Journal reference of previous version: Comput. Phys. Comm. 178 (2008) 788Does the new version supersede the previous version?: YesNature of problem: Any physical model containing linear or nonlinear ordinary differential equations (ODE).Solution method:

1.

Numerical solving of ordinary differential equations for the study of chaotic flows. The chaotic behavior of the nonlinear dynamical system is analyzed using Poincare sections, phase-space maps, autocorrelation functions, power spectra, Lyapunov exponents and Kolmogorov-Sinai entropies.

2.

Numerical solving of iterative equations for the study of maps and fractals.

Reasons for new version: The program has been updated to use the new version 5.1.1 of Scilab with new graphical capabilities [2]. Moreover, new use cases have been added which make the handling of the program easier and more efficient.Summary of revisions:

1.

A new use case concerning coupled predator-prey models has been added [3].

2.

Three new use cases concerning fractals (Sierpinsky gasket, Barnsley's Fern and Tree) have been added [3].

3.

The graphical user interface (GUI) of the program has been reconstructed to include the new use cases.

4.

The program has been updated to use Scilab 5.1.1 with the new graphical capabilities.

Additional comments: The program package contains 12 subprograms.

•

interface.sce - the graphical user interface (GUI) that permits the choice of a routine as follows

•

1.sci - Lorenz dynamical system

•

2.sci - Chua dynamical system

•

3.sci - Rosler dynamical system

•

4.sci - Henon map

•

5.sci - Lyapunov exponents for Lorenz dynamical system

•

6.sci - Lyapunov exponent for the logistic map

•

7.sci - Shannon entropy for the logistic map

•

8.sci - Coupled predator-prey model

•

1f.sci - Sierpinsky gasket

•

2f.sci - Barnsley's Fern

•

3f.sci - Barnsley's Tree

Running time: 10 to 20 seconds for problems that do not involve Lyapunov exponents calculation; 60 to 1000 seconds for problems that involve high orders ODE, Lyapunov exponents calculation and fractals.References:

[1]

C.C. Bordeianu, C. Besliu, Al. Jipa, D. Felea, I. V. Grossu, Comput. Phys. Comm. 178 (2008) 788.

[2]

S. Campbell, J.P. Chancelier, R. Nikoukhah, Modeling and Simulation in Scilab/Scicos, Springer, 2006.

[3]

R.H. Landau, M.J. Paez, C.C. Bordeianu, A Survey of Computational Physics, Introductory Computational Science, Princeton University Press, 2008.

     

POTHMF: A program for computing potential curves and matrix elements of the coupled adiabatic radial equations for a hydrogen-like atom in a homogeneous magnetic field

A FORTRAN 77 program is presented which calculates with the relative machine precision potential curves and matrix elements of the coupled adiabatic radial equations for a hydrogen-like atom in a homogeneous magnetic field. The potential curves are eigenvalues corresponding to the angular oblate spheroidal functions that compose adiabatic basis which depends on the radial variable as a parameter. The matrix elements of radial coupling are integrals in angular variables of the following two types: product of angular functions and the first derivative of angular functions in parameter, and product of the first derivatives of angular functions in parameter, respectively. The program calculates also the angular part of the dipole transition matrix elements (in the length form) expressed as integrals in angular variables involving product of a dipole operator and angular functions. Moreover, the program calculates asymptotic regular and irregular matrix solutions of the coupled adiabatic radial equations at the end of interval in radial variable needed for solving a multi-channel scattering problem by the generalized R-matrix method. Potential curves and radial matrix elements computed by the POTHMF program can be used for solving the bound state and multi-channel scattering problems. As a test desk, the program is applied to the calculation of the energy values, a short-range reaction matrix and corresponding wave functions with the help of the KANTBP program. Benchmark calculations for the known photoionization cross-sections are presented.

Program summary

Program title:POTHMFCatalogue identifier:AEAA_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAA_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.:8123No. of bytes in distributed program, including test data, etc.:131 396Distribution format:tar.gzProgramming language:FORTRAN 77Computer:Intel Xeon EM64T, Alpha 21264A, AMD Athlon MP, Pentium IV Xeon, Opteron 248, Intel Pentium IVOperating system:OC Linux, Unix AIX 5.3, SunOS 5.8, Solaris, Windows XPRAM:Depends on

1.

the number of radial differential equations;

2.

the number and order of finite elements;

3.

the number of radial points.

Test run requires 4 MBClassification:2.5External routines:POTHMF uses some Lapack routines, copies of which are included in the distribution (see README file for details).Nature of problem:In the multi-channel adiabatic approach the Schrödinger equation for a hydrogen-like atom in a homogeneous magnetic field of strength γ (γ=B/B₀, B₀≅2.35×¹⁰⁵ T is a dimensionless parameter which determines the field strength B) is reduced by separating the radial coordinate, r, from the angular variables, (θ,φ), and using a basis of the angular oblate spheroidal functions [3] to a system of second-order ordinary differential equations which contain first-derivative coupling terms [4]. The purpose of this program is to calculate potential curves and matrix elements of radial coupling needed for calculating the low-lying bound and scattering states of hydrogen-like atoms in a homogeneous magnetic field of strength 0<γ?1000 within the adiabatic approach [5]. The program evaluates also asymptotic regular and irregular matrix radial solutions of the multi-channel scattering problem needed to extract from the R-matrix a required symmetric shortrange open-channel reaction matrix K [6] independent from matching point [7]. In addition, the program computes the dipole transition matrix elements in the length form between the basis functions that are needed for calculating the dipole transitions between the low-lying bound and scattering states and photoionization cross sections [8].Solution method:The angular oblate spheroidal eigenvalue problem depending on the radial variable is solved using a series expansion in the Legendre polynomials [3]. The resulting tridiagonal symmetric algebraic eigenvalue problem for the evaluation of selected eigenvalues, i.e. the potential curves, is solved by the LDL^T factorization using the DSTEVR program [2]. Derivatives of the eigenfunctions with respect to the radial variable which are contained in matrix elements of the coupled radial equations are obtained by solving the inhomogeneous algebraic equations. The corresponding algebraic problem is solved by using the LDL^T factorization with the help of the DPTTRS program [2]. Asymptotics of the matrix elements at large values of radial variable are computed using a series expansion in the associated Laguerre polynomials [9]. The corresponding matching points between the numeric and asymptotic solutions are found automatically. These asymptotics are used for the evaluation of the asymptotic regular and irregular matrix radial solutions of the multi-channel scattering problem [7]. As a test desk, the program is applied to the calculation of the energy values of the ground and excited bound states and reaction matrix of multi-channel scattering problem for a hydrogen atom in a homogeneous magnetic field using the KANTBP program [10].Restrictions:The computer memory requirements depend on:

1.

the number of radial differential equations;

2.

the number and order of finite elements;

3.

the total number of radial points.

Restrictions due to dimension sizes can be changed by resetting a small number of PARAMETER statements before recompiling (see Introduction and listing for details).Running time:The running time depends critically upon:

1.

the number of radial differential equations;

2.

the number and order of finite elements;

3.

the total number of radial points on interval [rmin,rmax].

The test run which accompanies this paper took 7 s required for calculating of potential curves, radial matrix elements, and dipole transition matrix elements on a finite-element grid on interval [rmin=0, rmax=100] used for solving discrete and continuous spectrum problems and obtaining asymptotic regular and irregular matrix radial solutions at rmax=100 for continuous spectrum problem on the Intel Pentium IV 2.4 GHz. The number of radial differential equations was equal to 6. The accompanying test run using the KANTBP program took 2 s for solving discrete and continuous spectrum problems using the above calculated potential curves, matrix elements and asymptotic regular and irregular matrix radial solutions. Note, that in the accompanied benchmark calculations of the photoionization cross-sections from the bound states of a hydrogen atom in a homogeneous magnetic field to continuum we have used interval [rmin=0, rmax=1000] for continuous spectrum problem. The total number of radial differential equations was varied from 10 to 18.References:

[1]

W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, 1986.

[2]

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

[3]

M. Abramovits, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965.

[4]

U. Fano, Colloq. Int. C.N.R.S. 273 (1977) 127; A.F. Starace, G.L. Webster, Phys. Rev. A 19 (1979) 1629-1640; C.V. Clark, K.T. Lu, A.F. Starace, in: H.G. Beyer, H. Kleinpoppen (Eds.), Progress in Atomic Spectroscopy, Part C, Plenum, New York, 1984, pp. 247-320; U. Fano, A.R.P. Rau, Atomic Collisions and Spectra, Academic Press, Florida, 1986.

[5]

M.G. Dimova, M.S. Kaschiev, S.I. Vinitsky, J. Phys. B 38 (2005) 2337-2352; O. Chuluunbaatar, A.A. Gusev, V.L. Derbov, M.S. Kaschiev, V.V. Serov, T.V. Tupikova, S.I. Vinitsky, Proc. SPIE 6537 (2007) 653706-1-18.

[6]

M.J. Seaton, Rep. Prog. Phys. 46 (1983) 167-257.

[7]

M. Gailitis, J. Phys. B 9 (1976) 843-854; J. Macek, Phys. Rev. A 30 (1984) 1277-1278; S.I. Vinitsky, V.P. Gerdt, A.A. Gusev, M.S. Kaschiev, V.A. Rostovtsev, V.N. Samoylov, T.V. Tupikova, O. Chuluunbaatar, Programming and Computer Software 33 (2007) 105-116.

[8]

H. Friedrich, Theoretical Atomic Physics, Springer, New York, 1991.

[9]

R.J. Damburg, R.Kh. Propin, J. Phys. B 1 (1968) 681-691; J.D. Power, Phil. Trans. Roy. Soc. London A 274 (1973) 663-702.

[10]

O. Chuluunbaatar, A.A. Gusev, A.G. Abrashkevich, A. Amaya-Tapia, M.S. Kaschiev, S.Y. Larsen, S.I. Vinitsky, Comput. Phys. Comm. 177 (2007) 649-675.

     

A new version of Visual tool for estimating the fractal dimension of images

This work presents a new version of a Visual Basic 6.0 application for estimating the fractal dimension of images (Grossu et al., 2009 [1]). The earlier version was limited to bi-dimensional sets of points, stored in bitmap files. The application was extended for working also with comma separated values files and three-dimensional images.

New version program summary

Program title: Fractal Analysis v02Catalogue identifier: AEEG_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEEG_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.: 9999No. of bytes in distributed program, including test data, etc.: 4 366 783Distribution format: tar.gzProgramming language: MS Visual Basic 6.0Computer: PCOperating system: MS Windows 98 or laterRAM: 30 MClassification: 14Catalogue identifier of previous version: AEEG_v1_0Journal reference of previous version: Comput. Phys. Comm. 180 (2009) 1999Does the new version supersede the previous version?: YesNature of problem: Estimating the fractal dimension of 2D and 3D images.Solution method: Optimized implementation of the box-counting algorithm.Reasons for new version:

1.

The previous version was limited to bitmap image files. The new application was extended in order to work with objects stored in comma separated values (csv) files. The main advantages are:

a)

Easier integration with other applications (csv is a widely used, simple text file format);

b)

Less resources consumed and improved performance (only the information of interest, the “black points”, are stored);

c)

Higher resolution (the points coordinates are loaded into Visual Basic double variables [2]);

d)

Possibility of storing three-dimensional objects (e.g. the 3D Sierpinski gasket).

2.

In this version the optimized box-counting algorithm [1] was extended to the three-dimensional case.

Summary of revisions:

1.

The application interface was changed from SDI (single document interface) to MDI (multi-document interface).

2.

One form was added in order to provide a graphical user interface for the new functionalities (fractal analysis of 2D and 3D images stored in csv files).

Additional comments: User friendly graphical interface; Easy deployment mechanism.Running time: In the first approximation, the algorithm is linear.References:

[1] I.V. Grossu, C. Besliu, M.V. Rusu, Al. Jipa, C.C. Bordeianu, D. Felea, Comput. Phys. Comm. 180 (2009)  1999-2001.

[2] F. Balena, Programming Microsoft Visual Basic 6.0, Microsoft Press, US, 1999.

     

Code OK3 - An upgraded version of OK2 with beam wobbling function

For computer simulations on heavy ion beam (HIB) irradiation onto a target with an arbitrary shape and structure in heavy ion fusion (HIF), the code OK2 was developed and presented in Computer Physics Communications 161 (2004). Code OK3 is an upgrade of OK2 including an important capability of wobbling beam illumination. The wobbling beam introduces a unique possibility for a smooth mechanism of inertial fusion target implosion, so that sufficient fusion energy is released to construct a fusion reactor in future.

New version program summary

Program title: OK3Catalogue identifier: ADST_v3_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADST_v3_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.: 221 517No. of bytes in distributed program, including test data, etc.: 2 471 015Distribution format: tar.gzProgramming language: C++Computer: PC (Pentium 4, 1 GHz or more recommended)Operating system: Windows or UNIXRAM: 2048 MBytesClassification: 19.7Catalogue identifier of previous version: ADST_v2_0Journal reference of previous version: Comput. Phys. Comm. 161 (2004) 143Does the new version supersede the previous version?: YesNature of problem: In heavy ion fusion (HIF), ion cancer therapy, material processing, etc., a precise beam energy deposition is essentially important [1]. Codes OK1 and OK2 have been developed to simulate the heavy ion beam energy deposition in three-dimensional arbitrary shaped targets [2, 3]. Wobbling beam illumination is important to smooth the beam energy deposition nonuniformity in HIF, so that a uniform target implosion is realized and a sufficient fusion output energy is released.Solution method: OK3 code works on the base of OK1 and OK2 [2, 3]. The code simulates a multi-beam illumination on a target with arbitrary shape and structure, including beam wobbling function.Reasons for new version: The code OK3 is based on OK2 [3] and uses the same algorithm with some improvements, the most important one is the beam wobbling function.Summary of revisions:

1.

In the code OK3, beams are subdivided on many bunches. The displacement of each bunch center from the initial beam direction is calculated.

2.

Code OK3 allows the beamlet number to vary from bunch to bunch. That reduces the calculation error especially in case of very complicated mesh structure with big internal holes.

3.

The target temperature rises during the time of energy deposition.

4.

Some procedures are improved to perform faster.

5.

The energy conservation is checked up on each step of calculation process and corrected if necessary.

New procedures included in OK3

1.

Procedure BeamCenterRot( ) rotates the beam axis around the impinging direction of each beam.

2.

Procedure BeamletRot( ) rotates the beamlet axes that belong to each beam.

3.

Procedure Rotation( ) sets the coordinates of rotated beams and beamlets in chamber and pellet systems.

4.

Procedure BeamletOut( ) calculates the lost energy of ions that have not impinged on the target.

5.

Procedure TargetT( ) sets the temperature of the target layer of energy deposition during the irradiation process.

6.

Procedure ECL( ) checks up the energy conservation law at each step of the energy deposition process.

7.

Procedure ECLt( ) performs the final check up of the energy conservation law at the end of deposition process.

Modified procedures in OK3

1.

Procedure InitBeam( ): This procedure initializes the beam radius and coefficients A1, A2, A3, A4 and A5 for Gauss distributed beams [2]. It is enlarged in OK3 and can set beams with radii from 1 to 20 mm.

2.

Procedure kBunch( ) is modified to allow beamlet number variation from bunch to bunch during the deposition.

3.

Procedure ijkSp( ) and procedure Hole( ) are modified to perform faster.

4.

Procedure Espl( ) and procedure ChechE( ) are modified to increase the calculation accuracy.

5.

Procedure SD( ) calculates the total relative root-mean-square (RMS) deviation and the total relative peak-to-valley (PTV) deviation in energy deposition non-uniformity. This procedure is not included in code OK2 because of its limited applications (for spherical targets only). It is taken from code OK1 and modified to perform with code OK3.

Running time: The execution time depends on the pellet mesh number and the number of beams in the simulated illumination as well as on the beam characteristics (beam radius on the pellet surface, beam subdivision, projectile particle energy and so on). In almost all of the practical running tests performed, the typical running time for one beam deposition is about 30 s on a PC with a CPU of Pentium 4, 2.4 GHz.References:

[1]

A.I. Ogoyski, et al., Heavy ion beam irradiation non-uniformity in inertial fusion, Phys. Lett. A 315 (2003) 372-377.

[2]

A.I. Ogoyski, et al., Code OK1 - Simulation of multi-beam irradiation on a spherical target in heavy ion fusion, Comput. Phys. Comm. 157 (2004) 160-172.

[3]

A.I. Ogoyski, et al., Code OK2 - A simulation code of ion-beam illumination on an arbitrary shape and structure target, Comput. Phys. Comm. 161 (2004) 143-150.

     

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.

     

DIRAC: A new version of computer algebra tools for studying the properties and behavior of hydrogen-like ions

During recent years, the Dirac package has proved to be an efficient tool for studying the structural properties and dynamic behavior of hydrogen-like ions. Originally designed as a set of Maple procedures, this package provides interactive access to the wave and Green's functions in the non-relativistic and relativistic frameworks and supports analytical evaluation of a large number of radial integrals that are required for the construction of transition amplitudes and interaction cross sections. We provide here a new version of the Dirac program which is developed within the framework of Mathematica (version 6.0). This new version aims to cater to a wider community of researchers that use the Mathematica platform and to take advantage of the generally faster processing times therein. Moreover, the addition of new procedures, a more convenient and detailed help system, as well as source code revisions to overcome identified shortcomings should ensure expanded use of the new Dirac program over its predecessor.

New version program summary

Program title: DIRACCatalogue identifier: ADUQ_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADUQ_v2_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC license, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 45 073No. of bytes in distributed program, including test data, etc.: 285 828Distribution format: tar.gzProgramming language: Mathematica 6.0 or higherComputer: All computers with a license for the computer algebra package Mathematica (version 6.0 or higher)Operating system: Mathematica is O/S independentClassification: 2.1Catalogue identifier of previous version: ADUQ_v1_0Journal reference of previous version: Comput. Phys. Comm. 165 (2005) 139Does the new version supersede the previous version?: YesNature of problem: Since the early days of quantum mechanics, the “hydrogen atom” has served as one of the key models for studying the structure and dynamics of various quantum systems. Its analytic solutions are frequently used in case studies in atomic and molecular physics, quantum optics, plasma physics, or even in the field of quantum information and computation. Fast and reliable access to functions and properties of the hydrogenic systems are frequently required, in both the non-relativistic and relativistic frameworks. Despite all the knowledge about one-electron ions, providing such an access is not a simple task, owing to the rather complicated mathematical structure of the Schrödinger and especially Dirac equations. Moreover, for analyzing experimental results as well as for performing advanced theoretical studies one often needs (apart from the detailed information on atomic wave- and Green's functions) to be able to calculate a number of integrals involving these functions. Although for many types of transition operators these integrals can be evaluated analytically in terms of special mathematical functions, such an evaluation is usually rather involved and prone to mistakes.Solution method: A set of Mathematica procedures is developed which provides both the non-relativistic and relativistic solutions of the “Hydrogen atom model”. It facilitates, moreover, the symbolic evaluation of integrals involved in the calculations of cross sections and transition amplitudes. These procedures are based on a large number of relations among special mathematical functions, information about their integral representations, recurrence formulae and series expansions. Based on this knowledge, the DIRAC tools provide a fast and reliable algebraic (and if necessary, numeric) manipulation of functions and properties of one-electron systems, thus helping to obtain further insight into the behavior of quantum physical systems.Reasons for new version: The original version of the DIRAC program was developed as a toolbox of Maple procedures and was submitted to the CPC library in 2004 (cf. Ref. [1]). Since then DIRAC has found its niche in advanced theoretical studies carried out in realm of heavy ion physics. With the help of this program detailed analysis has been performed, in particular, for the various excitation and ionization processes occurring in relativistic ion-atom collisions [2], the polarization of the characteristic X-ray radiation following radiative electron capture [3], the correlation properties of the two-photon emission from few-electron heavy ions [4], the spin entanglement phenomena in atomic photoionization [5] and even for exploring the vibrational excitations of the heavy nuclei [6]. Although these studies have conclusively proven the potential of the program, they have also illuminated routes for its further enhancement. Apart from certain source code revisions, demand has grown for a new version of DIRAC compatible with the Mathematica platform. The version presented here includes a wider ranging and more user friendly interactive help system, a number of new procedures and reprogramming for greater computational efficiency.Summary of revisions: The most important new capabilities of the DIRAC program since the previous version are:

1.

The utilization of the Mathematica (version 6.0) platform.

2.

The addition of a number of new procedures. Since the complete list of the new (and updated) procedures can be found in the interactive help library of the program, we mention here only the most important ones:

○

DiracGlobal[] - Displays a list of the current global settings which specify the framework, nuclear charge and the units which are to be used by the DIRAC program.

○

DiracRadialOrbitalMomentum[] - Returns a non-relativistic radial orbital in momentum space for both, the bound and free electron states.

○

DiracSlaterRadial[] - Evaluates the radial Slater integral both, with the non-relativistic and relativistic wavefunctions. In the previous version of the program this procedure was restricted to the non-relativistic framework only.

○

DiracGreensIntegralRadial[] - Evaluates the two-dimensional radial integrals with the wave- and Green's functions both in non-relativistic and relativistic frameworks.

○

DiracAngularMatrixElement[] - Calculates the angular matrix elements for various irreducible tensor operators.

3.

The elimination of some redundant procedures. In particular, the previous version supported evaluation of the spherical Bessel functions, Wigner 3j symbols, Clebsch-Gordan coefficients and spherical harmonics functions. These tools are now superseded by in-built procedures of Mathematica.

4.

The development of a full featured interactive help system which follows the style of the Mathematica Help Pages.

5.

Extensive revision of the source code in order to correct a number of bugs and inconsistencies that have been identified during use of the previous version of Dirac.

The DIRAC package is distributed as a compressed tar file from which the DIRAC root directory can be (re-)generated. The root directory contains the source code and help libraries, a “Readme” file, Dirac_Installation_Instructions, as well as the notebook DemonstrationNotebook.nb that includes a number of test cases to illustrate the use of the program. These test cases, which concern the theoretical analysis of wavefunctions and the fine-structure of hydrogen-like ions, has already been discussed in detail in Ref. [1] and are provided here in order to underline the continuity between the previous (Maple) and new (Mathematica) versions of the DIRAC program.Unusual features: Even though all basic features of the previous Maple version have been retained in as close to the original form as possible, some small syntax changes became necessary in the new version of DIRAC in order to follow Mathematica standards. First of all, these changes concern naming conventions for DIRAC's procedures. As was discussed in Ref. [1], previously rather long names were employed in which each word was separated by an underscore. For example, when running the Maple version of the program one had to call the procedure Dirac_Slater_radial() in order to evaluate the Slater integral. Such a naming convention however, cannot be used in the Mathematica framework where the underscore character is reserved to represent Blank, a built-in symbol. In the new version of DIRAC we therefore follow the Mathematica convention of delimiting each word in a procedure's name by capitalization. Evaluation of the Slater determinant can be accomplished now simply by entering DiracSlaterRadial[]. Besides procedure names, a new convention is introduced to represent fundamental physical constants. In this version of DIRAC the group of (preset) global variables has changed to resemble their conventional symbols, specifically α, a₀, ec, me, c and ?, being the fine structure constant, Bohr radius, electron charge, electron mass, speed of light and the Planck constant respectively. If the numerical evaluator N is wrapped around any of these constants, their numerical values are returned.Running time: Although the program replies promptly upon most requests, the running time also depends on the particular task. For example, computation of (radial) matrix elements involving components of relativistic wavefunctions might require a few seconds of a runtime. A number of test calculations performed regarding this and other tasks clearly indicate that the new version of Dirac requires up to 90% less evaluation time compared to its predecessor.References:

[1]

A. Surzhykov, P. Koval, S. Fritzsche, Comput. Phys. Comm. 165 (2005) 139.

[2]

H. Ogawa, et al., Phys. Rev. A 75 (2007) 1.

[3]

A.V. Maiorova, et al., J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 125003.

[4]

L. Borowska, A. Surzhykov, Th. Stöhlker, S. Fritzsche, Phys. Rev. A 74 (2006) 062516.

[5]

T. Radtke, S. Fritzsche, A. Surzhykov, Phys. Rev. A 74 (2006) 032709.

[6]

A. Pálffy, Z. Harman, A. Surzhykov, U.D. Jentschura, Phys. Rev. A 75 (2007) 012712.

     

ODPEVP: A program for computing eigenvalues and eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm-Liouville problem

A FORTRAN 77 program is presented for calculating with the given accuracy eigenvalues, eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm-Liouville problem with the parametric third type boundary conditions on the finite interval. The program calculates also potential matrix elements - integrals of the eigenfunctions multiplied by their first derivatives with respect to the parameter. Eigenvalues and matrix elements computed by the ODPEVP program can be used for solving the bound state and multi-channel scattering problems for a system of the coupled second-order ordinary differential equations with the help of the KANTBP programs [O. Chuluunbaatar, A.A. Gusev, A.G. Abrashkevich, A. Amaya-Tapia, M.S. Kaschiev, S.Y. Larsen, S.I. Vinitsky, Comput. Phys. Commun. 177 (2007) 649-675; O. Chuluunbaatar, A.A. Gusev, S.I. Vinitsky, A.G. Abrashkevich, Comput. Phys. Commun. 179 (2008) 685-693]. As a test desk, the program is applied to the calculation of the potential matrix elements for an integrable 2D-model of three identical particles on a line with pair zero-range potentials, a 3D-model of a hydrogen atom in a homogeneous magnetic field and a hydrogen atom on a three-dimensional sphere.

Program summary

Program title: ODPEVPCatalogue identifier: AEDV_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEDV_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC license, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 3001No. of bytes in distributed program, including test data, etc.: 24 195Distribution format: tar.gzProgramming language: FORTRAN 77Computer: Intel Xeon EM64T, Alpha 21264A, AMD Athlon MP, Pentium IV Xeon, Opteron 248, Intel Pentium IVOperating system: OC Linux, Unix AIX 5.3, SunOS 5.8, Solaris, Windows XPRAM: depends on

1.

the number and order of finite elements;

2.

the number of points; and

3.

the number of eigenfunctions required.

Test run requires 4 MBClassification: 2.1, 2.4External routines: GAULEG [3]Nature of problem: The three-dimensional boundary problem for the elliptic partial differential equation with an axial symmetry similar to the Schrödinger equation with the Coulomb and transverse oscillator potentials is reduced to the two-dimensional one. The latter finds wide applications in modeling of photoionization and recombination of oppositively charged particles (positrons, antiprotons) in the magnet-optical trap [4], optical absorption in quantum wells [5], and channeling of likely charged particles in thin doped films [6,7] or neutral atoms and molecules in artificial waveguides or surfaces [8,9]. In the adiabatic approach [10] known in mathematics as Kantorovich method [11] the solution of the two-dimensional elliptic partial differential equation is expanded over basis functions with respect to the fast variable (for example, angular variable) and depended on the slow variable (for example, radial coordinate ) as a parameter. An averaging of the problem by such a basis leads to a system of the second-order ordinary differential equations which contain potential matrix elements and the first-derivative coupling terms (see, e.g., [12,13,14]). The purpose of this paper is to present the finite element method procedure based on the use of high-order accuracy approximations for calculating eigenvalues, eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm-Liouville problem with the parametric third type boundary conditions on the finite interval. The program developed calculates potential matrix elements - integrals of the eigenfunctions multiplied by their derivatives with respect to the parameter. These matrix elements can be used for solving the bound state and multi-channel scattering problems for a system of the coupled second-order ordinary differential equations with the help of the KANTBP programs [1,2].Solution method: The parametric self-adjoined Sturm-Liouville problem with the parametric third type boundary conditions is solved by the finite element method using high-order accuracy approximations [15]. The generalized algebraic eigenvalue problem AF=EBF with respect to a pair of unknown (E,F) arising after the replacement of the differential problem by the finite-element approximation is solved by the subspace iteration method using the SSPACE program [16]. First derivatives of the eigenfunctions with respect to the parameter which contained in potential matrix elements of the coupled system equations are obtained by solving the inhomogeneous algebraic equations. As a test desk, the program is applied to the calculation of the potential matrix elements for an integrable 2D-model of three identical particles on a line with pair zero-range potentials described in [1,17,18], a 3D-model of a hydrogen atom in a homogeneous magnetic field described in [14,19] and a hydrogen atom on a three-dimensional sphere [20].Restrictions: The computer memory requirements depend on:

1.

the number and order of finite elements;

2.

the number of points; and

3.

the number of eigenfunctions required.

Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see sections below and listing for details). The user must also supply DOUBLE PRECISION functions POTCCL and POTCC1 for evaluating potential function U(ρ,z) of Eq. (1) and its first derivative with respect to parameter ρ. The user should supply DOUBLE PRECISION functions F1FUNC and F2FUNC that evaluate functions f₁(z) and f₂(z) of Eq. (1). The user must also supply subroutine BOUNCF for evaluating the parametric third type boundary conditions.Running time: The running time depends critically upon:

1.

the number and order of finite elements;

2.

the number of points on interval [zmin,zmax]; and

3.

the number of eigenfunctions required.

The test run which accompanies this paper took 2 s with calculation of matrix potentials on the Intel Pentium IV 2.4 GHz.References:

[1]

O. Chuluunbaatar, A.A. Gusev, A.G. Abrashkevich, A. Amaya-Tapia, M.S. Kaschiev, S.Y. Larsen, S.I. Vinitsky, Comput. Phys. Comm. 177 (2007) 649-675

[2]

O. Chuluunbaatar, A.A. Gusev, S.I. Vinitsky, A.G. Abrashkevich, Comput. Phys. Comm. 179 (2008) 685-693.

[3]

W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, 1986.

[4]

O. Chuluunbaatar, A.A. Gusev, S.I. Vinitsky, V.L. Derbov, L.A. Melnikov, V.V. Serov, Phys. Rev. A 77 (2008) 034702-1-4.

[5]

E.M. Kazaryan, A.A. Kostanyan, H.A. Sarkisyan, Physica E 28 (2005) 423-430.

[6]

Yu.N. Demkov, J.D. Meyer, Eur. Phys. J. B 42 (2004) 361-365.

[7]

P.M. Krassovitskiy, N.Zh. Takibaev, Bull. Russian Acad. Sci. Phys. 70 (2006) 815-818.

[8]

V.S. Melezhik, J.I. Kim, P. Schmelcher, Phys. Rev. A 76 (2007) 053611-1-15.

[9]

F.M. Pen'kov, Phys. Rev. A 62 (2000) 044701-1-4.

[10]

M. Born, X. Huang, Dynamical Theory of Crystal Lattices, The Clarendon Press, Oxford, England, 1954.

[11]

L.V. Kantorovich, V.I. Krylov, Approximate Methods of Higher Analysis, Wiley, New York, 1964.

[12]

U. Fano, Colloq. Int. C.N.R.S. 273 (1977) 127;

A.F. Starace, G.L. Webster, Phys. Rev. A 19 (1979) 1629-1640.

[13]

C.V. Clark, K.T. Lu, A.F. Starace, in: H.G. Beyer, H. Kleinpoppen (eds.), Progress in Atomic Spectroscopy, Part C, Plenum, New York, 1984, pp. 247-320.

[14]

O. Chuluunbaatar, A.A. Gusev, V.L. Derbov, M.S. Kaschiev, L.A. Melnikov, V.V. Serov, S.I. Vinitsky, J. Phys. A 40 (2007) 11485-11524.

[15]

A.G. Abrashkevich, D.G. Abrashkevich, M.S. Kaschiev, I.V. Puzynin, Comput. Phys. Comm. 85 (1995) 40-64.

[16]

K.J. Bathe, Finite Element Procedures in Engineering Analysis, Englewood Cliffs, Prentice-Hall, New York, 1982.

[17]

O. Chuluunbaatar, A.A. Gusev, M.S. Kaschiev, V.A. Kaschieva, A. Amaya-Tapia, S.Y. Larsen, S.I. Vinitsky, J. Phys. B 39 (2006) 243-269.

[18]

Yu.A. Kuperin, P.B. Kurasov, Yu.B. Melnikov, S.P. Merkuriev, Ann. Phys. 205 (1991) 330-361.

[19]

O. Chuluunbaatar, A.A. Gusev, V.P. Gerdt, V.A. Rostovtsev, S.I. Vinitsky, A.G. Abrashkevich, M.S. Kaschiev, V.V. Serov, Comput. Phys. Comm. 178 (2008) 301-330.

[20]

A.G. Abrashkevich, M.S. Kaschiev, S.I. Vinitsky, J. Comp. Phys. 163 (2000) 328-348.

     

A spectral lower bound for the treewidth of a graph and its consequences

We give a lower bound for the treewidth of a graph in terms of the second smallest eigenvalue of its Laplacian matrix. We use this lower bound to show that the treewidth of a d-dimensional hypercube is at least ⌊3·2^d/(2(d+4))⌋−1. The currently known upper bound is . We generalize this result to Hamming graphs. We also observe that every graph G on n vertices, with maximum degree Δ

(1)

contains an induced cycle (chordless cycle) of length at least 1+log_Δ(μn/8) (provided G is not acyclic),

(2)

has a clique minor K_h for some ,

where μ is the second smallest eigenvalue of the Laplacian matrix of G.     

micrOMEGAs 2.0.7: a program to calculate the relic density of dark matter in a generic model

micrOMEGAs2.0.7 is a code which calculates the relic density of a stable massive particle in an arbitrary model. The underlying assumption is that there is a conservation law like R-parity in supersymmetry which guarantees the stability of the lightest odd particle. The new physics model must be incorporated in the notation of CalcHEP, a package for the automatic generation of squared matrix elements. Once this is done, all annihilation and coannihilation channels are included automatically in any model. Cross-sections at v=0, relevant for indirect detection of dark matter, are also computed automatically. The package includes three sample models: the minimal supersymmetric standard model (MSSM), the MSSM with complex phases and the NMSSM. Extension to other models, including non supersymmetric models, is described.

Program summary

Title of program:micrOMEGAs2.0.7Catalogue identifier:ADQR_v2_1Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADQR_v2_1.htmlProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions:Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.:216 529No. of bytes in distributed program, including test data, etc.:1 848 816Distribution format:tar.gzProgramming language used:C and FortranComputer:PC, Alpha, Mac, SunOperating system:UNIX (Linux, OSF1, SunOS, Darwin, Cygwin)RAM:17 MB depending on the number of processes requiredClassification:1.9, 11.6Catalogue identifier of previous version:ADQR_v2_0Journal version of previous version:Comput. Phys. Comm. 176 (2007) 367Does the new version supersede the previous version?:YesNature of problem:Calculation of the relic density 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 formulae 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. Higher-order QCD corrections to Higgs couplings to quark pairs are included.Reasons for new version:The main changes in this new version consist, on the one hand, in improvements of the user interface and treatment of error codes when using spectrum calculators in the MSSM and, on the other hand, on a completely revised code for the calculation of the relic density in the NMSSM based on the code NMSSMTools1.0.2 for the computation of the spectrum.Summary of revisions:

•

The version of CalcHEP was updated to CalcHEP 2.4.

•

The procedure for shared library generation has been improved. Now the libraries are recalculated each time the model is modified.

•

The default value for the top quark mass has been set to 171.4 GeV.

Changes specific to the MSSM model.

•

The deltaMb correction is now included in the B,t,H-vertex and is always included for other Higgs vertices.

•

In case of a fatal error in an RGE program, micrOMEGAs now continues operation while issuing a warning that the given point is not valid. This is important when running scans over parameter space. However this means that the standard ˆC command that could be used to cancel a job now only cancels the RGE program. To cancel a job, use “kill -9 -N” where N is the micrOMEGAs process id, all child processes launched by micrOMEGAs will be killed at once.

•

Following the last SLHA2 release, we use key=26 item of EXTPAR block for the pole mass of the CP-odd Higgs so that micrOMEGAs can now use SoftSUSY for spectrum calculation with EWSB input. The Isajet interface was corrected too, so the user has to recompile the isajet_slha executable. For SuSpect we still support an old “wrong” interface where key=24 is used for the mass of the CP-odd Higgs.

•

In the non-universal SUGRA model, we set the value of M₀ (M1/2,A₀) to the value of the largest subset of equal parameters among scalar masses (gaugino masses, trilinear couplings). In the previous version these parameters were set arbitrarily to be equal to MH2, MG2 and At respectively. The spectrum calculators need an input value for M₀,M1/2 and A₀ for initialisation purposes.

•

We have removed bugs in micrOMEGAs-Isajet interface in case of non-universal SUGRA.

•

$(FFLAGS) is added to compilation instruction of suspect.exe. It was omitted in version 2.0.

•

The treatment of errors in reading of the LesHouches accord file is improved. Now, if the SPINFO block is absent in the SLHA output it is considered as a fatal error.

•

Instructions for calculation of Δρ, μ(g−2), Br(b→sγ) and Br(Bs→μ⁺μ⁻) constraints are included in EWSB sample main programs omg.c/omg.cpp/omg.F.

•

We have corrected the name of the library for neutralino-neutralino annihilation in our sample files MSSM/cs br.*.

Changes specific to the NMSSM model.

•

The NMSSM has been completely revised. Now it is based on NMSSMTools_1.0.2.

•

The deltaMb corrections in the NMSSM are included in the Higgs potential.

CP violation model.

•

We have included in our package the MSSM with CP violation. Our implementation was described in Phys. Rev. D 73 (2006) 115007. It is based on the CPSUPERH package published in Comput. Phys. Comm. 156 (2004) 283.

Unusual features:Depending on the parameters of the model, the program generates additional new code, compiles it and loads it dynamically.Running time:0.2 seconds     

LCG MCDB—a knowledgebase of Monte-Carlo simulated events

In this paper we report on LCG Monte-Carlo Data Base (MCDB) and software which has been developed to operate MCDB. The main purpose of the LCG MCDB project is to provide a storage and documentation system for sophisticated event samples simulated for the LHC Collaborations by experts. In many cases, the modern Monte-Carlo simulation of physical processes requires expert knowledge in Monte-Carlo generators or significant amount of CPU time to produce the events. MCDB is a knowledgebase mainly dedicated to accumulate simulated events of this type. The main motivation behind LCG MCDB is to make the sophisticated MC event samples available for various physical groups. All the data from MCDB is accessible in several convenient ways. LCG MCDB is being developed within the CERN LCG Application Area Simulation project.

Program summary

Program title: LCG Monte-Carlo Data BaseCatalogue identifier: ADZX_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZX_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: GNU General Public LicenceNo. of lines in distributed program, including test data, etc.: 30 129No. of bytes in distributed program, including test data, etc.: 216 943Distribution format: tar.gzProgramming language: PerlComputer: CPU: Intel Pentium 4, RAM: 1 Gb, HDD: 100 GbOperating system: Scientific Linux CERN 3/4RAM: 1 073 741 824 bytes (1 Gb)Classification: 9External routines:

•

perl >= 5.8.5;

•

Perl modules

○

DBD-mysql >= 2.9004,

○

File::Basename,

○

GD::SecurityImage,

○

GD::SecurityImage::AC,

○

Linux::Statistics,

○

XML::LibXML > 1.6,

○

XML::SAX,

○

XML::NamespaceSupport;

•

Apache HTTP Server >= 2.0.59;

•

mod auth external >= 2.2.9;

•

edg-utils-system RPM package;

•

gd >= 2.0.28;

•

rpm package CASTOR-client >= 2.1.2-4;

•

arc-server (optional)

Nature of problem: Often, different groups of experimentalists prepare similar samples of particle collision events or turn to the same group of authors of Monte-Carlo (MC) generators to prepare the events. For example, the same MC samples of Standard Model (SM) processes can be employed for the investigations either in the SM analyses (as a signal) or in searches for new phenomena in Beyond Standard Model analyses (as a background). If the samples are made available publicly and equipped with corresponding and comprehensive documentation, it can speed up cross checks of the samples themselves and physical models applied. Some event samples require a lot of computing resources for preparation. So, a central storage of the samples prevents possible waste of researcher time and computing resources, which can be used to prepare the same events many times.Solution method: Creation of a special knowledgebase (MCDB) designed to keep event samples for the LHC experimental and phenomenological community. The knowledgebase is realized as a separate web-server (http://mcdb.cern.ch). All event samples are kept on types at CERN. Documentation describing the events is the main contents of MCDB. Users can browse the knowledgebase, read and comment articles (documentation), and download event samples. Authors can upload new event samples, create new articles, and edit own articles.Restrictions: The software is adopted to solve the problems, described in the article and there are no any additional restrictions.Unusual features: The software provides a framework to store and document large files with flexible authentication and authorization system. Different external storages with large capacity can be used to keep the files. The WEB Content Management System provides all of the necessary interfaces for the authors of the files, end-users and administrators.Running time: Real time operations.References:[1] The main LCG MCDB server, http://mcdb.cern.ch/.[2] P. Bartalini, L. Dudko, A. Kryukov, I.V. Selyuzhenkov, A. Sherstnev, A. Vologdin, LCG Monte-Carlo data base, hep-ph/0404241.[3] J.P. Baud, B. Couturier, C. Curran, J.D. Durand, E. Knezo, S. Occhetti, O. Barring, CASTOR: status and evolution, cs.oh/0305047.     

A new version of the CADNA library for estimating round-off error propagation in Fortran programs

The CADNA library enables one to estimate, using a probabilistic approach, round-off error propagation in any simulation program. CADNA provides new numerical types, the so-called stochastic types, on which round-off errors can be estimated. Furthermore CADNA contains the definition of arithmetic and relational operators which are overloaded for stochastic variables and the definition of mathematical functions which can be used with stochastic arguments. On 64-bit processors, depending on the rounding mode chosen, the mathematical library associated with the GNU Fortran compiler may provide incorrect results or generate severe bugs. Therefore the CADNA library has been improved to enable the numerical validation of programs on 64-bit processors.

New version program summary

Program title: CADNACatalogue identifier: AEAT_v1_1Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAT_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.: 28 488No. of bytes in distributed program, including test data, etc.: 463 778Distribution format: tar.gzProgramming language: FortranNOTE: A C++ version of this program is available in the Library as AEGQ_v1_0Computer: PC running LINUX with an i686 or an ia64 processor, UNIX workstations including SUN, IBMOperating system: LINUX, UNIXClassification: 6.5Catalogue identifier of previous version: AEAT_v1_0Journal reference of previous version: Comput. Phys. Commun. 178 (2008) 933Does the new version supersede the previous version?: YesNature 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 [1-3] implements Discrete Stochastic Arithmetic [4,5] 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.Reasons for new version: On 64-bit processors, the mathematical library associated with the GNU Fortran compiler may provide incorrect results or generate severe bugs with rounding towards −∞ and +∞, which the random rounding mode is based on. Therefore a particular definition of mathematical functions for stochastic arguments has been included in the CADNA library to enable its use with the GNU Fortran compiler on 64-bit processors.Summary of revisions: If CADNA is used on a 64-bit processor with the GNU Fortran compiler, mathematical functions are computed with rounding to the nearest, otherwise they are computed with the random rounding mode. It must be pointed out that the knowledge of the accuracy of the stochastic argument of a mathematical function is never lost.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.Additional comments: In the library archive, users are advised to read the INSTALL file first. The doc directory contains a user guide named ug.cadna.pdf which shows how to control the numerical accuracy of a program using CADNA, provides installation instructions and describes test runs. The source code, which is located in the src directory, consists of one assembly language file (cadna_rounding.s) and eighteen Fortran language files. cadna_rounding.s is a symbolic link to the assembly file corresponding to the processor and the Fortran compiler used. This assembly file contains routines which are frequently called in the CADNA Fortran files to change the rounding mode. The Fortran language files contain the definition of the stochastic types on which the control of accuracy can be performed, CADNA specific functions (for instance to enable or disable the detection of numerical instabilities), the definition of arithmetic and relational operators which are overloaded for stochastic variables and the definition of mathematical functions which can be used with stochastic arguments. The examples directory contains seven test runs which illustrate the use of the CADNA library and the benefits of Discrete Stochastic Arithmetic.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]

F. Jézéquel, J.-M. Chesneaux, CADNA: a library for estimating round-off error propagation, Comput. Phys. Commun. 178 (12) (2008) 933-955.

[3]

N.S. Scott, F. Jézéquel, C. Denis, J.-M. Chesneaux, Numerical ‘health check’ for scientific codes: the CADNA approach, Comput. Phys. Commun. 176 (8) (2007) 507-521.

[4]

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

[5]

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

     

Ten commandments for good default expression simplification

This article provides goals for the design and improvement of default computer algebra expression simplification. These goals can also help users recognize and partially circumvent some limitations of their current computer algebra systems. Although motivated by computer algebra, many of the goals are also applicable to manual simplification, indicating what transformations are necessary and sufficient for good simplification when no particular canonical result form is required.After motivating the ten goals, the article then explains how the Altran partially factored form for rational expressions was extended for Derive and for the computer algebra in Texas Instruments products to help fulfill these goals. In contrast to the distributed Altran representation, this recursive partially factored semi-fraction form:

•

does not unnecessarily force common denominators,

•

discovers and preserves significantly more factors,

•

can represent general expressions, and

•

can produce an entire spectrum from fully factored over a common denominator through complete multivariate partial fractions, including a dense subset of all intermediate forms.

     

AutoDipole - Automated generation of dipole subtraction terms -

We present an automated generation of the subtraction terms for next-to-leading order QCD calculations in the Catani-Seymour dipole formalism. For a given scattering process with n external particles our Mathematica package generates all dipole terms, allowing for both massless and massive dipoles. The numerical evaluation of the subtraction terms proceeds with MadGraph, which provides Fortran code for the necessary scattering amplitudes. Checks of the numerical stability are discussed.

Program summary

Program title: AutoDipoleCatalogue identifier: AEGO_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEGO_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.: 138 042No. of bytes in distributed program, including test data, etc.: 1 117 665Distribution format: tar.gzProgramming language: Mathematica and FortranComputer: Computers running Mathematica (version 7.0)Operating system: The package should work on every Linux system supported by Mathematica. Detailed tests have been performed on Scientific Linux as supported by DESY and CERN and on openSUSE and Debian.RAM: Depending on the complexity of the problem, recommended at least 128 MB RAMClassification: 11.5External routines: MadGraph (including HELAS library) available under http://madgraph.hep.uiuc.edu/ or http://madgraph.phys.ucl.ac.be/ or http://madgraph.roma2.infn.it/. A copy of the tar file, MG_ME_SA_V4.4.30, is included in the AutoDipole distribution package.Nature of problem: Computation of next-to-leading order QCD corrections to scattering cross sections, regularization of real emission contributions.Solution method: Catani-Seymour subtraction method for massless and massive partons [1,2]; Numerical evaluation of subtracted matrix elements interfaced to MadGraph [3-5] (stand-alone version) using helicity amplitudes and the HELAS library [6,7] (contained in MadGraph).Restrictions: Limitations of MadGraph are inherited.Running time: Dependent on the complexity of the problem with typical run times of the order of minutes.References:

[1]

S. Catani, M.H. Seymour, Nuclear Phys. B 485 (1997) 291, hep-ph/9605323.

[2]

S. Catani, et al., Nuclear Phys. B 627 (2002) 189, hep-ph/0201036.

[3]

T. Stelzer, W.F. Long, Comput. Phys. Comm. 81 (1994) 357, hep-ph/9401258.

[4]

F. Maltoni, T. Stelzer, JHEP 0302 (2003) 027, hep-ph/0208156.

[5]

J. Alwall, et al., JHEP 0709 (2007) 028, arXiv:0706.2334 [hep-ph].

[6]

K. Hagiwara, H. Murayama, I. Watanabe, Nuclear Phys. B 367 (1991) 257.

[7]

H. Murayama, I. Watanabe, K. Hagiwara, KEK-91-11.

     

An updated version of the Motion4D-library

We present an updated version of the Motion4D-library that can be used for the newly developed GeodesicViewer application.

New version program summary

Program title: Motion4D-libraryCatalogue identifier: AEEX_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEEX_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.: 153 757No. of bytes in distributed program, including test data, etc.: 5 178 439Distribution format: tar.gzProgramming language: C++Computer: All platforms with a C++ compilerOperating system: Linux, Unix, WindowsRAM: 31 MBytesCatalogue identifier of previous version: AEEX_v1_0Journal reference of previous version: Comput. Phys. Comm. 180 (2009) 2355Classification: 1.5External routines: Gnu Scientific Library (GSL) (http://www.gnu.org/software/gsl/)Does the new version supersede the previous version?: YesNature of problem: Solve geodesic equation, parallel and Fermi-Walker transport in four-dimensional Lorentzian spacetimes.Solution method: Integration of ordinary differential equations.Reasons for new version: To be applicable for the GeodesicViewer (accepted for publication in Comput. Phys. Comm. (COMPHY) 3941, doi:10.1016/j.cpc.2009.10.010 [program AEFP_v1_0]), there were several minor adjustments to be done.Summary of revisions:

1.

Calculation of embedding diagrams are improved.

2.

Physical units can be used for some metrics.

3.

Tests for the constraint equation within the metric classes are slightly modified.

4.

New metrics: AlcubierreWarp, GoedelScaled, GoedelScaledCart, Kasner.

Running time: The test runs provided with the distribution require only a few seconds to run.