Gauss-Legendre and Chebyshev quadratures for singular integrals

Exact expressions are presented for efficient computation of the weights in Gauss-Legendre and Chebyshev quadratures for selected singular integrands. The singularities may be of Cauchy type, logarithmic type or algebraic-logarithmic end-point branching points. We provide Fortran 90 routines for computing the weights for both the Gauss-Legendre and the Chebyshev (Fejér-1) meshes whose size can be set by the user.

New program summary

Program title: SINGQUADCatalogue identifier: AEBR_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEBR_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.: 4128No. of bytes in distributed program, including test data, etc.: 25 815Distribution format: tar.gzProgramming language: Fortran 90Computer: Any with a Fortran 90 compilerOperating system: Linux, Windows, MacRAM: Depending on the complexity of the problemClassification: 4.11Nature of problem: Program provides Gauss-Legendre and Chebyshev (Fejér-1) weights for various singular integrands.Solution method: The weights are obtained from the condition that the quadrature of order N must be exact for a polynomial of degree?(N−1). The weights are expressed as moments of the singular kernels associated with Legendre or Chebyshev polynomials. These moments are obtained in analytic form amenable for computation.Additional comments: If the NAGWare f95 compiler is used, the option, “-kind = byte”, must be included in the compile command lines of the Makefile.Running time: The test run supplied with the distribution takes a couple of seconds to execute.     

BSR: B-spline atomic R-matrix codes

BSR is a general program to calculate atomic continuum processes using the B-spline R-matrix method, including electron-atom and electron-ion scattering, and radiative processes such as bound-bound transitions, photoionization and polarizabilities. The calculations can be performed in LS-coupling or in an intermediate-coupling scheme by including terms of the Breit-Pauli Hamiltonian.

New version program summary

Title of program: BSRCatalogue identifier: ADWYProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADWYProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandComputers on which the program has been tested: Microway Beowulf cluster; Compaq Beowulf cluster; DEC Alpha workstation; DELL PCOperating systems under which the new version has been tested: UNIX, Windows XPProgramming language used: FORTRAN 95Memory required to execute with typical data: Typically 256-512 Mwords. Since all the principal dimensions are allocatable, the available memory defines the maximum complexity of the problemNo. of bits in a word: 8No. of processors used: 1Has the code been vectorized or parallelized?: noNo. of lines in distributed program, including test data, etc.: 69 943No. of bytes in distributed program, including test data, etc.: 746 450Peripherals used: scratch disk store; permanent disk storeDistribution format: tar.gzNature of physical problem: This program uses the R-matrix method to calculate electron-atom and electron-ion collision processes, with options to calculate radiative data, photoionization, etc. The calculations can be performed in LS-coupling or in an intermediate-coupling scheme, with options to include Breit-Pauli terms in the Hamiltonian.Method of solution: The R-matrix method is used [P.G. Burke, K.A. Berrington, Atomic and Molecular Processes: An R-Matrix Approach, IOP Publishing, Bristol, 1993; P.G. Burke, W.D. Robb, Adv. At. Mol. Phys. 11 (1975) 143; K.A. Berrington, W.B. Eissner, P.H. Norrington, Comput. Phys. Comm. 92 (1995) 290].     

Resolution of singularities for multi-loop integrals

We report on a program for the numerical evaluation of divergent multi-loop integrals. The program is based on iterated sector decomposition. We improve the original algorithm of Binoth and Heinrich such that the program is guaranteed to terminate. The program can be used to compute numerically the Laurent expansion of divergent multi-loop integrals regulated by dimensional regularisation. The symbolic and the numerical steps of the algorithm are combined into one program.

Program summary

Program title: sector_decompositionCatalogue identifier: AEAG_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAG_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.: 47 506No. of bytes in distributed program, including test data, etc.: 328 485Distribution format: tar.gzProgramming language: C++Computer: allOperating system: UnixRAM: Depending on the complexity of the problemClassification: 4.4External routines: GiNaC, available from http://www.ginac.de, GNU scientific library, available from http://www.gnu.org/software/gslNature of problem: Computation of divergent multi-loop integrals.Solution method: Sector decomposition.Restrictions: Only limited by the available memory and CPU time.Running time: Depending on the complexity of the problem.     

FRODO: a MuPAD program to calculate matrix elements between contracted wavefunctions

A symbolic program performing the Formal Reduction of Density Operators (FRODO) has been developed in the MuPAD computer algebra system with the purpose of evaluating the matrix elements of the electronic Hamiltonian between internally contracted functions in a complete active space (CAS) scheme. The program is illustrated making use of two meaningful examples.

Program summary

Title of program:FRODOCatalogue identifier:ADVYProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADVYProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandComputer:Any computer on which the MuPAD computer algebra system can be installedOperating systems under which the program has been tested:LinuxProgramming language used:MuPAD vs. 2.5.3 for LinuxNo. of lines in distributed program, including test data, etc.:3939No. of bytes in distributed program, including test data, etc.:19 661Distribution format:tar.gzNature of physical problem: In order to improve on the CAS-SCF wavefunction one can resort to multireference perturbation theory or configuration interaction based on internally contracted functions (ICF) which are obtained by application of the excitation operators to the reference CAS-SCF wavefunction. The formulation of such matrix elements is quite cumbersome and a computer algebra system like MuPAD appears ideally suited to perform such a task.Method of solution: The method adopted consists in successively eliminating all occurrences of inactive orbital indices (core and virtual) from the products of excitation operators which appear in the definition of the ICF's and in the electronic Hamiltonian expressed in the second quantization formalism.Restrictions due to the complexity of the problem: The program is limited to no more than doubly excited ICF's.     

Classification of integrable super-systems using the SsTools environment

A classification problem is proposed for supersymmetric evolutionary PDE that satisfy the assumptions of nonlinearity, nondegeneracy, and homogeneity. Four classes of nonlinear coupled boson-fermion systems are discovered under the weighting assumption . The syntax of the Reduce package SsTools, which was used for intermediate computations, and the applicability of its procedures to the calculus of super-PDE are described.

Program summary

Program title:SsToolsCatalogue identifier:ADYY_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADYY_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.:89 178No. of bytes in distributed program, including test data, etc.:869 212Distribution format:tar.gzProgramming language:REDUCE 3.7, REDUCE 3.8Computer:(i) IBM PC, (ii) clusterOperating system:LINUXRAM:problem dependent (10 Mb-1 Gb), typical working size <100 MbWord size:32, 64 bitsNature of problem:The program allows the classification of N?1 supersymmetric nonlinear scaling-invariant evolution equations that admit infinitely many local symmetries propagated by recursion operators; here b(x,t;θ) is the set of bosonic super-fields and f(x;t;θ) are fermionic super-fields.Solution method:First, (half-)integer weights |f|,|b|,…,|Dt|,|Dx|≡1 are assigned to all variables and derivatives and then pairs of commuting flows that are homogeneous w.r.t. these weights are constructed. Secondly, the seeds of higher symmetry sequences [P.J. Olver, Applications of Lie Groups to Differential Equations, second ed., Springer, Berlin, 1993] for the systems are sorted out, and finally the recursion operators that generate the symmetries are obtained [I.S. Krasil'shchik, P.H.M. Kersten, Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations, Kluwer, Dordrecht, 2000]. The intermediate algebraic systems upon the undetermined coefficients are solved by using [T. Wolf, Applications of Crack in the classification of integrable systems, in: CRM Proc. Lecture Notes, vol. 37, 2004, pp. 283-300].Restrictions:Computation of symmetries of high differential order for very large evolutionary systems may cause memory restrictions. Additional size/time restrictions may occur if the homogeneity weights of some super-fields are non-positive, see Section 1.2Unusual features:SsTools has been extensively tested using hundreds of PDE systems within three years on UNIX-based PC-machines. SsTools is applicable to the computation of symmetries, conservation laws, and Hamiltonian structures for N?1 evolutionary super-systems with any N. SsTools is also useful for performing extensive arithmetic of general nature including differentiations of super-field expressions.Running time:Depends on the size and complexity of the input system and varies between seconds and minutes.References:[1] http://lie.math.brocku.ca/crack/susy/sstools.red; The support package Crack is obtained from http://lie.math.brocku.ca/crack/src/crack.tar.gz     

A program for accurate solutions of two-electron atoms

We present a comprehensible computer program capable of treating non-relativistic ground and excited states for a two-electron atom having infinite nuclear mass. An iterative approach based on the implicitly restarted Arnoldi method (IRAM) is employed. The Hamiltonian matrix is never explicitly computed. Instead the action of the Hamiltonian operator on discrete pair functions is implemented. The finite difference method is applied and subsequent extrapolations gives the continuous grid result. The program is written in C and is highly optimized. All computations are made in double precision. Despite this relatively low degree of floating point precision (48 digits are not uncommon), the accuracy in the results can reach about 10 significant figures. Both serial and parallel versions are provided. The parallel program is particularly suitable for shared memory machines such as the Sun Starcat series. The serial version is simple to compile and should run on any platform.

Program summary

Title of program: corr2elCatalogue identifier: ADUXProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADUXProgram obtainable from:CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputer for which the program is designed and others on which it has been tested:Computers: Sun Fire 15K StarCat, Sun Ultra SPARC III, PCOperating systems or monitors under which the program has been tested: Sun Solaris 9, LinuxProgramming language used: ANSI CMemory required to execute with typical data: 3 Mwords or moreNo. bits in a word: 32No. processors used: arbitraryHas the code been vectorized or parallelized: parallelizedNumber of lines in distributed program, including test data, etc.:5885Number of bytes in distributed program, including test data, etc.: 26 199Nature of physical problem: The Schrödinger equation for two-electron atoms is solved using finite differences.Method of solution: An iterative eigenvalue-solver that requires only the action of the Hamiltonian on a trial function is applied. The two-electron wave function is expanded in a sum of partial waves. The finite difference method is then applied to approximate the derivatives of the pair functions. The total action of the Hamiltonian on the partial waves, including correlation effects, is computed using highly optimized routines.Restriction on the complexity of the problem: The Hamiltonian employed here does not take relativistic or finite nuclear mass effects into account. The amount of computing time may become unreasonable for excited states far above the ground state. The use of double precision puts a limit on the accuracy obtainable.Typical running time: This ranges from half a minute (to obtain 10 significant figures for the s-limit of the Helium ground state) to perhaps a day for advanced examples depending on the level of parallelization.Unusual features of the program: The implicitly restarted Arnoldi method used to obtain the eigenvalues is implemented by using the ARPACK/PARPACK program library [http://www.netlib.org/arpack]. This package also depends on the standard numerical libraries BLAS and LAPACK [http://www.netlib.org/lapack]. Good performance is obtained by using Sun's optimized performance libraries [http://www.sun.com]. By using a 64-bit environment (Ultra SPARC III and Solaris 9), memory limitations are non-problematic. Shared memory is used in the parallel version. Fast communication between the nodes is made over shared memory using Sun's implementation of MPI.     

A survey of strategies in rule-based program transformation systems

Program transformation is the mechanical manipulation of a program in order to improve it relative to some cost function and is understood broadly as the domain of computation where programs are the data. The natural basic building blocks of the domain of program transformation are transformation rules expressing a ‘one-step’ transformation on a fragment of a program. The ultimate perspective of research in this area is a high-level, language parametric, rule-based program transformation system, which supports a wide range of transformations, admitting efficient implementations that scale to large programs. This situation has not yet been reached, as trade-offs between different goals need to be made. This survey gives an overview of issues in rule-based program transformation systems, focusing on the expressivity of rule-based program transformation systems and in particular on transformation strategies available in various approaches. The survey covers term rewriting, extensions of basic term rewriting, tree parsing strategies, systems with programmable strategies, traversal strategies, and context-sensitive rules.     

A numerical algorithm for efficiently obtaining a Feynman parameter representation of one-gluon loop QCD Feynman diagrams for a large number of external gluons

A numerical program is presented which facilitates a computation pertaining to the full set of one-gluon loop diagrams (including ghost loop contributions), with M attached external gluon lines in all possible ways. The feasibility of such a task rests on a suitably defined master formula, which is expressed in terms of a set of Grassmann and a set of Feynman parameters. The program carries out the Grassmann integration and performs the Lorentz trace on the involved functions, expressing the result as a compact sum of parametric integrals. The computation is based on tracing the structure of the final result, thus avoiding all intermediate unnecessary calculations and directly writing the output. Similar terms entering the final result are grouped together. The running time of the program demonstrates its effectiveness, especially for large M.

Program summary

Program title:DILOG2Program identifier:ADXN_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADXN_v1_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandProgramming language:FORTRAN 90Computer(s) for which the program has been designed:Personal ComputerOperating system(s) for which the program has been designed: Windows 98, XP, LINUXNumber of processors used:oneNo. of lines in distributed program, including test data, etc.:2000No. of bytes in distributed program, including test data, etc.:16 249Distribution format:tar.gzExternal routines/libraries used:noneCPC Program Library subprograms used:noneNature of problem:The computation of one gluon/ghost loop diagrams in QCD with many external gluon lines is a time consuming task, practically beyond reasonable reach of analytic procedures. We apply recently proposed master formulas towards the computation of such diagrams with an arbitrary number (M) of external gluon lines, achieving a final result which reduces the problem to one involving integrals over the standard set, for given M, of Feynman parameters.Solution method:The structure of the master expressions is analyzed from a numerical computation point of view. Using the properties of Grassmann variables we identify all the different forms of terms that appear in the final result. Each form is called “structure”. We calculate theoretically the number of terms belonging to every “structure”. We carry out the calculation organizing the whole procedure into separate calculations of the terms belonging to every “structure”. Terms which do not contribute to the final result are thereby avoided. The final result, extending to large values of M, is also presented with terms belonging to the same “structure” grouped together.Restrictions:M is coded as a 2-digit integer. Overflow in the dimension of used array is expected to appear for M?20 in a processor that uses 4-bytes integers or for M?34 in a processor with 8-bytes integers.Running time:Depends on M, see enclosed figures.     

PSEUDO: applications of streams and lazy evaluation to integrable models

Procedures to manipulate pseudo-differential operators in MAPLE are implemented in the program PSEUDO to perform calculations with integrable models. We use lazy evaluation and streams to represent and operate with pseudo-differential operators. No order of truncation is needed since terms are produced on demand. We give a series of concrete examples.

Program summary

Title of program: PSEUDOCatalogue identifier: ADUOProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADUOProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions: NoneComputers: IBM PCOperating systems under which the program has been tested: Windows systemsProgramming language used: MAPLE V Release 8Memory required to execute with typical data: Depends strongly on the problemNo. of lines in distributed program, including test data, etc.: 737No. of bytes in distributed program, including test data, etc.: 8822Distribution format: tar.gzNature of mathematical problem: Determination of equations of motion and conserved charges in the theory of integrable modelsMethods of solution: Pseudo-differential Lax operatorsRestrictions on the complexity of the problem: Handles only one-dimensional pseudo-differential operators with scalar coefficientsTypical running time: This depends strongly on the problem to be solved, usually taking from a few seconds to a few minutesUnusual features of the program: Use of delayed evaluation and streams     

Four-index integral transformation exploiting symmetry

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

Program summary

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