An iterative algorithm for least squares problem in quaternionic quantum theory

Quaternionic least squares (QLS) problem is one method of solving overdetermined sets of quaternion linear equations AXB=E that is appropriate when there is error in the matrix E. In this paper, by means of real representation of a quaternion matrix, we introduce a concept of norm of quaternion matrices, which is different from that in [T. Jiang, L. Chen, Algebraic algorithms for least squares problem in quaternionic quantum theory, Comput. Phys. Comm. 176 (2007) 481-485; T. Jiang, M. Wei, Equality constrained least squares problem over quaternion field, Appl. Math. Lett. 16 (2003) 883-888], and derive an iterative method for finding the minimum-norm solution of the QLS problem in quaternionic quantum theory.     

Markov property in quantum logic: A reflection

A possible way of the Markov property introduction within the framework of the orthomodular quantum logic, which is commonly used as the calculus model for quantum mechanics is presented in this paper. The presented work follows the logical line rather than any physical interpretation in the framework of quantum mechanics. The basic algebraic structure, which is used as a model for noncompatible random events is an orthomodular lattice. On the orthomodular lattice, a dynamical structure is introduced coupled with mappings which have similar properties as s_n-maps on the orthomodular lattice. This construction leads to the definition of an L-process with the Markov property on the orthomodular lattice.     

Algorithm Q-LSQR for the least squares problem in quaternionic quantum theory

In this paper, an iterative algorithm for the standard quaternionic least squares problem is proposed without using the real (complex) representation. Our algorithm is implemented in the quaternion field and by means of direct quaternion arithmetic and is a natural generalization of the LSQR algorithm for the real least squares problem.     

An object-oriented C++ implementation of Davidson method for finding a few selected extreme eigenpairs of a large, sparse, real, symmetric matrix

A C++ class named Davidson is presented for determining a few eigenpairs with lowest or alternatively highest values of a large, real, symmetric matrix. The algorithm described by Stathopoulos and Fischer is used. The exception mechanism is involved to report the errors. The class is written in ANSI C++, so it is fully portable. In addition a console program as well as a program with graphical user interface for Microsoft Windows is attached, which allow one to calculate the lowest eigenstates of time-independent Schrödinger equation for a given binding potential in one, two or three spatial dimensions. The package contains the classes providing often used potential functions (model atom potential, Coulomb potential, square well potential and Kramers-Henneberger well potential) as well as a possibility to use any potential stored in a file (then any dimensionality of the problem is allowed).The described code is the subject of M.Sc. thesis of T.D. prepared under the supervision of J.M.

Program summary

Program title: DavidsonCatalogue identifier: ADZM_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZM_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.: 3 037 055No. of bytes in distributed program, including test data, etc.: 20 002 609Distribution format: tar.gzProgramming language: C++Computer: AllOperating system: AnyRAM: User's parameters dependentWord size: 32 and 64 bitsSupplementary material: Test results for the 2D and 3D cases is availableClassification: 4, 4.8Nature of problem: Finding a few extreme eigenpairs of a real, symmetric, sparse matrix. Examples in quantum optics (interaction of matter with a laser field).Solution method: Davidson algorithmRunning time: The test example included in the distribution package (1D matrix) takes approximately 30 minutes to run. 2D matrix calculations can take hours and 3D, days, to run.     

Existence of states on quantum structures

Orthomodular lattices occurred as generalized event structures in the models of probability for quantum mechanics. Here we contribute to the question of existence of states (=probability measures) on orthomodular lattices. We prove that known techniques do not allow to find examples with less than 19 blocks (=maximal Boolean subalgebras). This bound is achieved by the example by Mayet [R. Mayet, Personal communication, 1993]. Although we do not finally exclude the existence of other techniques breaking this bound, existence of smaller examples is highly unexpected.     

Algebraic systems with locally varying matrices for control in electrical energetics: A solution method

Deterministic relations are used to design a method and an algorithm for solving algebraic systems of equations with locally varying coefficient matrices encountered in operational control problems in electrical energetics.     

Matrix iterative algorithms for least-squares problem in quaternionic quantum theory

Quaternionic least squares (QLS) is an efficient method for solving approximate problems in quaternionic quantum theory. Based on Paige's algorithms LSQR and residual-reducing version of LSQR proposed in Paige and Saunders [LSQR: An algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Softw. 8(1) (1982), pp. 43–71], we provide two matrix iterative algorithms for finding solution with the least norm to the QLS problem by making use of structure of real representation matrices. Numerical experiments are presented to illustrate the efficiency of our algorithms.     

Quasilinearization approach to quantum mechanics

The quasilinearization method (QLM) of solving nonlinear differential equations is applied to the quantum mechanics by casting the Schrödinger equation in the nonlinear Riccati form. The method, whose mathematical basis in physics was discussed recently by one of the present authors (VBM), approaches the solution of a nonlinear differential equation by approximating the nonlinear terms by a sequence of the linear ones, and is not based on the existence of some kind of a small parameter. It is shown that the quasilinearization method gives excellent results when applied to computation of ground and excited bound state energies and wave functions for a variety of the potentials in quantum mechanics most of which are not treatable with the help of the perturbation theory or the 1/N expansion scheme. The convergence of the QLM expansion of both energies and wave functions for all states is very fast and already the first few iterations yield extremely precise results. The precision of the wave function is typically only one digit inferior to that of the energy. In addition it is verified that the QLM approximations, unlike the asymptotic series in the perturbation theory and the 1/N expansions are not divergent at higher orders.     

PROCRUSTES: A computer algebra package for post-Newtonian calculations in General Relativity

We report on a package of routines for the computer algebra system Maple which supports the explicit determination of the geometric quantities, field equations, equations of motion, and conserved quantities of General Relativity in the post-Newtonian approximation. The package structure is modular and allows for an easy modification by the user. The set of routines can be used to verify hand calculations or to generate the input for further numerical investigations.

Program summary

Title of the program:ProcrustesCatalogue identifier:ADYH_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/ADYH_v1_0Program obtainable from:CPC Program Library, Queen's University of Belfast, N. IrelandComputers:Platforms supported by the Maple computer algebra system (program was written under Maple 8, but also tested with Maple 9, 9.5, 10)Operating systems under which the program has been tested:Linux, Unix, Windows XPProgramming language used:Maple internal languageMemory required to execute typical problem:Dependent on problem (small ∼ couple of MBytes, large ∼ several GBytes)Classification:1.5 Relativity and Gravitation, 5 Computer AlgebraNo. bits in a word:Dependent on Maple distribution (supports 32 bit and 64 bit platforms)No. of processors used:1No. of lines in distributed program, including test data, etc.: 10 881No. of bytes in distributed program, including test data, etc.:47 743Distribution format:tar.gzNature of the physical problem:The post-Newtonian approximation represents an approximative scheme frequently used in General Relativity in which the gravitational potential is expanded into a series in inverse powers of the speed of light. Depending on the desired approximation level the field equations and equations of motion have to be determined up to given orders in the speed of light. This usually requires large algebraic computations due to the geometrical quantities entering the field equations and equations of motion.Method of solution:Automated computation using computer algebra techniques. Program has modular structure and only makes use of basic features of Maple to guarantee maximum compatibility and to allow for rapid extensions/modifications by the user.Typical running time:Dependent on problem (small ∼ couple of minutes, large ∼ couple of hours).Restrictions on the complexity of the problem:Sufficient amount of memory is the limiting factor for these calculations. The structure of the program allows one to handle large scale problems in an iterative manner to minimize the amount of memory required.     

Matrix LSQR algorithm for structured solutions to quaternionic least squares problem

In this paper, we employ matrix LSQR algorithm to deal with quaternionic least squares problem in order to find the minimum norm solutions with kinds of special structures, and propose a strategy to accelerate convergence rate of the algorithm via right–left preconditioning of the coefficient matrices. We mainly focus on analyzing the minimum norm $\eta$-Hermitian solution and the minimum norm $\eta$-biHermitian solution to the quaternionic least squares problem, $\eta \in \{\mathbf{i},\mathbf{j},\mathbf{k}\}$. Other structured solutions also can be obtained using the proposed technique. A number of numerical experiments are performed to show the efficiency of the preconditioned matrix LSQR algorithm.     

Hermitian tridiagonal solution with the least norm to quaternionic least squares problem

Quaternionic least squares (QLS) is an efficient method for solving approximate problems in quaternionic quantum theory. In view of the extensive applications of Hermitian tridiagonal matrices in physics, in this paper we list some properties of basis matrices and subvectors related to tridiagonal matrices, and give an iterative algorithm for finding Hermitian tridiagonal solution with the least norm to the quaternionic least squares problem by making the best use of structure of real representation matrices, we also propose a preconditioning strategy for the Algorithm LSQR-Q in Wang, Wei and Feng (2008) [14] and our algorithm. Numerical experiments are provided to verify the effectiveness of our method.     

Angular momentum in molecular quantum mechanical integral evaluation

Solid-harmonic derivatives of quantum-mechanical integrals over Gaussian transforms of scalar, or radial, atomic basis functions create angular momentum about each center. Generalized Gaunt coefficients limit the amount of cross differentiation for multi-center integrals to ensure that cross differentiation does not affect the total angular momentum. The generalized Gaunt coefficients satisfy a number of other selection rules, which are exploited in a new computer code for computing forces in analytic density-functional theory based on robust and variational fitting of the Kohn-Sham potential. Two-center exponents are defined for four or more solid-harmonic differentiations of matrix elements. Those differentiations can either build up angular momentum about the centers or give forces on molecular potential-energy surfaces, thus generalized Gaunt coefficients of order greater than the number of centers are considered. These 4-j generalized Gaunt coefficients and two-center exponents are used to compute the first derivatives of all integrals involving all the Gaussian exponents on a triplet of centers at once. First all angular factors are contracted with the corresponding part of the linear-combination-of-atomic-orbitals density matrix. This intermediate quantity is then reused for the nuclear attraction integral and the integrals corresponding to each basis function in the analytic fit of the Kohn-Sham potential in the muffin-tin-like, but analytic, Slater-Roothaan method that allows molecules to dissociate into atoms having any desired energy, including the experimental electronic energy. The energy is stationary in all respects and all forces precisely agree with a previous code in tests on small molecules. During geometry optimization of an icosahedral C₇₂₀ fullerene computing these angular factors and transforming them via the 4-j generalized Gaunt coefficient takes more than sixty percent of the total computer time. These same angular factors could be used in identical fashion with Gaussian transforms of Slater-type and numerical radial atomic orbitals.     

Algebraic properties and perturbation results for the indefinite least squares problem with equality constraints

In this paper, several equivalent systems without constraints of the indefinite least squares problem with equality constraints (ILSE) are established. We also derive the perturbation results for the ILSE problem and illustrate our results with numerical tests.     

A band factorization technique for transition matrix element asymptotics

A new method of evaluating transition matrix elements between wave functions associated with orthogonal polynomials is proposed. The technique relies on purely algebraic manipulation of the associated recurrence coefficients. The form of the matrix elements is perfectly suited to very large quantum number calculations by using asymptotic series expansions. In practice, this allows the accurate and fast numerical treatment of transition matrix elements in the quasi-classical limit. Examples include the matrix elements of xp in the harmonic oscillator basis, and connections with the Wigner 3j symbols.     

An efficient analytical approach for solving fourth order boundary value problems

Based on the homotopy analysis method (HAM), an efficient approach is proposed for obtaining approximate series solutions to fourth order two-point boundary value problems. We apply the approach to a linear problem which involves a parameter c and cannot be solved by other analytical methods for large values of c, and obtain convergent series solutions which agree very well with the exact solution, no matter how large the value of c is. Consequently, we give an affirmative answer to the open problem proposed by Momani and Noor in 2007 [S. Momani, M.A. Noor, Numerical comparison of methods for solving a special fourth-order boundary value problem, Appl. Math. Comput. 191 (2007) 218-224]. We also apply the approach to a nonlinear problem, and obtain convergent series solutions which agree very well with the numerical solution given by the Runge-Kutta-Fehlberg 4-5 technique.     

A direct approach with computerized symbolic computation for finding a series of traveling waves to nonlinear equations

A direct approach with computerized symbolic computation is applied to construct a series of traveling wave solutions for nonlinear equations. Compared with most existing symbolic computation methods such as tanh method and Jacobi function method, the proposed method not only gives new and more general solutions, but also provides a guideline to classify the various types of the solution according to some parameters.     

Comment on “A direct approach with computerized symbolic computation for finding a series of traveling waves to nonlinear equations” by Engui Fan and H.H. Dai: [Comput. Phys. Commun. 153 (2003) 17]

Fan and Dai [Comput. Phys. Commun. 153 (2003) 17] have found a series of traveling wave solutions for nonlinear equations by applying a direct approach with computerized symbolic computations. They have claimed that the proposed method, in comparison with most existing symbolic computation methods such as a tanh method and Jacobi function method, not only give new and more general solutions, but also provides a guideline to classify various types of the solution according to some parameters. We show that the claims by Fan and Dai are wrong since some of the solutions do not satisfy the differential equation that they have adopted for the algebraic method.     

-XSummer- Transcendental functions and symbolic summation in Form

Harmonic sums and their generalizations are extremely useful in the evaluation of higher-order perturbative corrections in quantum field theory. Of particular interest have been the so-called nested sums, where the harmonic sums and their generalizations appear as building blocks, originating for example, from the expansion of generalized hypergeometric functions around integer values of the parameters. In this paper we discuss the implementation of several algorithms to solve these sums by algebraic means, using the computer algebra system Form.

Program summary

Title of program:XSummerCatalogue identifier:ADXQ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADXQ_v1_0Program obtainable from:CPC Program Library, Queen's University of Belfast, N. IrelandLicense:GNU Public License and Form LicenseComputers:allOperating system:allProgram language:FormMemory required to execute:Depending on the complexity of the problem, recommended at least 64 MB RAMNo. of lines in distributed program, including test data, etc.:9854No. of bytes in distributed program, including test data, etc.:126 551Distribution format:tar.gzOther programs called:noneExternal files needed:noneNature of the physical problem:Systematic expansion of higher transcendental functions in a small parameter. The expansions arise in the calculation of loop integrals in perturbative quantum field theory.Method of solution:Algebraic manipulations of nested sums.Restrictions on complexity of the problem:Usually limited only by the available disk space.Typical running time:Dependent on the complexity of the problem.     

A computer program for two-particle generalized coefficients of fractional parentage

We present a FORTRAN90 program GCFP for the calculation of the generalized coefficients of fractional parentage (generalized CFPs or GCFP). The approach is based on the observation that the multi-shell CFPs can be expressed in terms of single-shell CFPs, while the latter can be readily calculated employing a simple enumeration scheme of antisymmetric A-particle states and an efficient method of construction of the idempotent matrix eigenvectors. The program provides fast calculation of GCFPs for a given particle number and produces results possessing numerical uncertainties below the desired tolerance. A single j-shell is defined by four quantum numbers, (e,l,j,t).A supplemental C++ program parGCFP allows calculation to be done in batches and/or in parallel.

Program summary

Program title:GCFP, parGCFPCatalogue identifier: AEBI_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEBI_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.: 17 199No. of bytes in distributed program, including test data, etc.: 88 658Distribution format: tar.gzProgramming language: FORTRAN 77/90 (GCFP), C++ (parGCFP)Computer: Any computer with suitable compilers. The program GCFP requires a FORTRAN 77/90 compiler. The auxiliary program parGCFP requires GNU-C++ compatible compiler, while its parallel version additionally requires MPI-1 standard librariesOperating system: Linux (Ubuntu, Scientific) (all programs), also checked on Windows XP (GCFP, serial version of parGCFP)RAM: The memory demand depends on the computation and output mode. If this mode is not 4, the program GCFP demands the following amounts of memory on a computer with Linux operating system. It requires around 2 MB of RAM for the A=12 system at Ex?2. Computation of the A=50 particle system requires around 60 MB of RAM at Ex=0 and ∼70 MB at Ex=2 (note, however, that the calculation of this system will take a very long time). If the computation and output mode is set to 4, the memory demands by GCFP are significantly larger. Calculation of GCFPs of A=12 system at Ex=1 requires 145 MB. The program parGCFP requires additional 2.5 and 4.5 MB of memory for the serial and parallel version, respectively.Classification: 17.18Nature of problem: The program GCFP generates a list of two-particle coefficients of fractional parentage for several j-shells with isospin.Solution method: The method is based on the observation that multishell coefficients of fractional parentage can be expressed in terms of single-shell CFPs [1]. The latter are calculated using the algorithm [2,3] for a spectral decomposition of an antisymmetrization operator matrix Y. The coefficients of fractional parentage are those eigenvectors of the antisymmetrization operator matrix Y that correspond to unit eigenvalues. A computer code for these coefficients is available [4]. The program GCFP offers computation of two-particle multishell coefficients of fractional parentage. The program parGCFP allows a batch calculation using one input file. Sets of GCFPs are independent and can be calculated in parallel.Restrictions:A<86 when Ex=0 (due to the memory constraints); small numbers of particles allow significantly higher excitations, though the shell with j?11/2 cannot get full (it is the implementation constraint).Unusual features: Using the program GCFP it is possible to determine allowed particle configurations without the GCFP computation. The GCFPs can be calculated either for all particle configurations at once or for a specified particle configuration. The values of GCFPs can be printed out with a complete specification in either one file or with the parent and daughter configurations printed in separate files. The latter output mode requires additional time and RAM memory. It is possible to restrict the (J,T) values of the considered particle configurations. (Here J is the total angular momentum and T is the total isospin of the system.) The program parGCFP produces several result files the number of which equals to the number of particle configurations. To work correctly, the program GCFP needs to be compiled to read parameters from the standard input (the default setting).Running time: It depends on the size of the problem. The minimum time is required, if the computation and output mode (CompMode) is not 4, but the resulting file is larger. A system with A=12 particles at Ex=0 (all 9411 GCFPs) took around 1 sec on a Pentium4 2.8 GHz processor with 1 MB L2 cache. The program required about 14 min to calculate all 1.3×¹⁰⁶ GCFPs of Ex=1. The time for all 5.5×¹⁰⁷ GCFPs of Ex=2 was about 53 hours. For this number of particles, the calculation time of both Ex=0 and Ex=1 with CompMode = 1 and 4 is nearly the same, when no other processes are running. The case of Ex=2 could not be calculated with CompMode = 4, because the RAM memory was insufficient. In general, the latter CompMode requires a longer computation time, although the resulting files are smaller in size. The program parGCFP puts virtually no time overhead. Its parallel version speeds-up the calculation. However, the results need to be collected from several files created for each configuration.References:[1] J. Levinsonas, Works of Lithuanian SSR Academy of Sciences 4 (1957) 17.[2] A. Deveikis, A. Bon?kus, R. Kalinauskas, Lithuanian Phys. J. 41 (2001) 3.[3] A. Deveikis, R.K. Kalinauskas, B.R. Barrett, Ann. Phys. 296 (2002) 287.[4] A. Deveikis, Comput. Phys. Comm. 173 (2005) 186. (CPC Catalogue ID. ADWI_v1_0)     

The analytical Scheme calculator for angular momentum coupling and recoupling coefficients

We describe a Scheme implementation of the interactive environment to calculate analytically the Clebsch-Gordan coefficients, Wigner 6j and 9j symbols, and general recoupling coefficients that are used in the quantum theory of angular momentum. The orthogonality conditions for considered coefficients are implemented. The program provides a fast and exact calculation of the coefficients for large values of quantum angular momenta.

Program summary

Title of program:Scheme2ClebschCatalogue number:ADWCProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADWCProgram obtainable from:CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions:noneComputer for which the program is designed:Any Scheme-capable platformOperating systems under which the program has been tested: Windows 2000Programming language used:SchemeMemory required to execute with typical data:50 MB (≈ size of DrScheme, version 204)No. of lines in distributed program, including test data, etc.: 2872No. of bytes in distributed program, including test data, etc.: 109 396Distribution format:tar.gzNature of physical problem:The accurate and fast calculation of the angular momentum coupling and recoupling coefficients is required in various branches of quantum many-particle physics. The presented code provides a fast and exact calculation of the angular momentum coupling and recoupling coefficients for large values of quantum angular momenta and is based on the GNU Library General Public License PLT software http://www.plt-scheme.org/.Method of solution:A direct evaluation of sum formulas. A general angular momentum recoupling coefficient for an arbitrary number of (integer or half-integer) angular momenta is expressed as a sum over products of the Clebsch-Gordan coefficients.Restrictions on the complexity of the problem:Limited only by the DrScheme implementation used to run the program. No limitation inherent in the code.Typical running time:The Clebsch-Gordan coefficients, Wigner 6j and 9j symbols, and general recoupling coefficients with small angular momenta are computed almost instantaneously. The running time for large-scale calculations depends strongly on the number and magnitude of arguments' values (i.e., of the angular momenta).