Solution Of Simultaneous Linear Equations Research Articles

Riemannian Trust Region Method for Minimization of the Fourth Central Moment for Localized Molecular Orbitals.

The importance of localized molecular orbitals (MOs) in correlation treatments beyond mean-field calculation and in the illustration of chemical bonding (and antibonding) can hardly be overstated. However, the generation of orthonormal localized occupied MOs is significantly more straightforward than obtaining orthonormal localized virtual MOs. Orthonormal MOs allow facile use of highly efficient group theoretical methods (e.g., graphical unitary group approach) for calculation of Hamiltonian matrix elements in multireference configuration interaction calculations (such as MRCISD) and in quasi-degenerate perturbation treatments, such as the Generalized Van Vleck Perturbation Theory. Moreover, localized MOs can elucidate qualitative understanding of bonding in molecules, in addition to high-accuracy quantitative descriptions. We adopt the powers of the fourth moment cost function introduced by Jørgensen and coworkers. Because the fourth moment cost functions are prone to having multiple negative Hessian eigenvalues when starting from easily available canonical (or near-canonical) MOs, standard optimization algorithms can fail to obtain the orbitals of the virtual or partially occupied spaces. To overcome this drawback, we applied a trust region algorithm on an orthonormal Riemannian manifold with an approximate retraction from the tangent space built into the first and second derivatives of the cost function. Moreover, the Riemannian trust region outer iterations were coupled to truncated Conjugate Gradient inner loops, which avoided any costly solutions of simultaneous linear equations or eigenvector/eigenvalue solutions. Numerical examples are provided on model systems, including the high-connectivity H10 set in 1-, 2-, and 3-dimensional arrangements, and on a chemically realistic description of cyclobutadiene (c-C4H4) and the propargyl radical (C3H3). In addition to demonstrating the algorithm on occupied and virtual blocks of orbitals, the method is also shown to work on the active space at the MCSCF level of theory.

Implicit dynamic system for online simultaneous linear equations solving

A novel implicit dynamic system is exploited and investigated for online simultaneous linear equations solving. Compared to the classic gradient-based dynamic system and the recently-proposed Zhang dynamic system, the proposed implicit dynamic system can achieve superior global exponential convergence performance. Theoretical analysis and computer simulation results substantiate the effectiveness and efficiency of the proposed model for the real-time solution of simultaneous linear equations.

Using gauss - Jordan elimination method with CUDA for linear circuit equation systems

Many scientific and engineering problems can use a system of linear equations. In this study, solution of Linear Circuit Equation System (LCES) for an nxn matrix using Compute Unified Device Architecture (CUDA) is described. Solution of LCES is realized on Graphics Processing Unit (GPU) instead of Central Processing Unit (CPU). CUDA is a parallel computing architecture developed by NVIDIA. Linear Circuits include resistance, impedance, capacitance, dependent - independent current sources and DC, AC voltage source. In this study, solutions of circuits that include resistance, independent current sources and DC voltage source have analyzes. Circuit analysis frequently involves solution of linear simultaneous equations that are solved Gauss-Jordan Elimination Method in this study. Gauss-Jordan Elimination is a variant of Gaussian Elimination that a method of solving a linear system equations (Ax=B). Gauss-Jordan Elimination is an algorithm for getting matrices in reduced row echelon form using elementary row operations. Gaussian Elimination has two parts. The first part (Forward Elimination) reduces a given system to triangular form. The second step uses back substitution to find the solution of the triangular echelon form system Because of elements of unknowns column matrix are dependent on each other, second step algorithm is not appropriate for parallel programming. Two parts of Gauss–Jordan Elimination are not like Gaussian Elimination's part so it is preferred. GPU implementation is more faster than solution of linear equation systems on CPU

GPU-acceleration for Moving Particle Semi-Implicit method

The MPS (Moving Particle Semi-implicit) method has been proven useful in computation free-surface hydrodynamic flows. Despite its applicability, one of its drawbacks in practical application is the high computational load. On the other hand, Graphics Processing Unit (GPU), which was originally developed for acceleration of computer graphics, now provides unprecedented capability for scientific computations. The main objective of this study is to develop a GPU-accelerated MPS code using CUDA (Compute Unified Device Architecture) language. Several techniques have been shown to optimize calculations in CUDA. In order to promote the acceleration by GPU, particular attentions are given to both the search of neighboring particles and the iterative solution of simultaneous linear equations in the Poisson Pressure Equation. In this paper, 2-dimensional calculations of elliptical drop evolution and dam break flow have been carried out by the GPU-accelerated MPS method, and the accuracy and performance of GPU-based code are investigated by comparing the results with those by CPU. It is shown that results of GPU-based calculations can be obtained much faster with the same reliability as the CPU-based ones.

Derivation of potential model for LiAlO 2 by simple and effective optimization of model parameters

Interatomic potentials of LiAlO 2 were constructed by a simple and effective method. In this method, the model function consists of multiple inverse polynomial functions with an exponential truncation function, and parameters in the potential model can be optimized as a solution of simultaneous linear equations. Potential energies obtained by ab initio calculation are used as fitting targets for model parameter optimization. Lattice constants, elastic properties, defect-formation energy, thermal expansions and the melting point were calculated under the constructed potential models. The results showed good agreement with experimental values and ab initio calculation results, which underscores the validity of the presented method.

Application Of A Balanced Adder-Subtractor In The Solution Of Simultaneous Linear Equations

A balanced adder-subtractor employing operational amplifier is used to solve simultaneous linear equations. A pair of simultaneous linear equation with two unknowns have been considered for the design of the balanced adder-subtractor operated from a 6- volts d.c power source. A general purpose operational amplifier, μA 741 of the 8- lead mini DIP type is chosen for the design. It's low power requirement, high gain, wide range of operating voltage, absence of “ latch-up” tendencies and high common mode voltage range are some of the prime advantages of the μA 741 considered for it's choice in this work. The performance of the designed circuit has been evaluated for input voltages, 20V and 5V respectively. The output obtained from the system, x o = 0.623V and y o = 0.754V on the scale of 0.1V = 1 confirmed the solution of the pair of simultaneous equation. Keywords : Balanced adder-subtractor, linear equations and operational amplifier Nigerian Journal of Physics Vol. 19 (1) 2007: pp. 115-120

Sojourn time distribution in the M/ M/1 queue with discriminatory processor-sharing

In this paper, we consider a queue with multiple K job classes, Poisson arrivals, exponentially distributed required service times in which a single processor serves according to the DPS discipline. More precisely, if there are n i class i jobs in the system, i=1,…, K, each class j job receives a fraction α j /∑ i=1 K α i n i of the processor capacity. For this queue, we obtain a system of equations for joint transforms of the sojourn time and the number of jobs. Using this system of equations we find the moments of the sojourn time as a solution of linear simultaneous equations, which solves an open problem.

Application of Pseudospectral Methods for Receding Horizon Control

A pseudospectral method for solving nonlinear receding-horizon control problems is presented. The method depends on solutions of simultaneous linear equations. The algorithm is demonstrated to be practical for online implementation in a tether system. The closed-loop performance is found to be excellent in the face of unmodeled disturbances due to tether mass and flexibility.

The C-S-H gel of Portland cement mortars: Part I. The interpretation of energy-dispersive X-ray microanalyses from scanning electron microscopy, with some observations on C-S-H, AFm and AFt phase compositions

Scanning electron microscopy (SEM) microanalyses of the calcium-silicate-hydrate (C-S-H) gel in Portland cement pastes rarely represent single phases. Essential experimental requirements are summarised and new procedures for interpreting the data are described. These include, notably, plots of Si/Ca against other atom ratios, 3D plots to allow three such ratios to be correlated and solution of linear simultaneous equations to test and quantify hypotheses regarding the phases contributing to individual microanalyses. Application of these methods to the C-S-H gel of a 1-day-old mortar identified a phase with Al/Ca=0.67 and S/Ca=0.33, which we consider to be a highly substituted ettringite of probable composition C 6A 2S̄ 2H 34 or {Ca 6[Al(OH) 6] 2·24H 2O}(SO 4) 2[Al(OH) 4] 2. If this is true for Portland cements in general, it might explain observed discrepancies between observed and calculated aluminate concentrations in the pore solution. The C-S-H gel of a similar mortar aged 600 days contained unsubstituted ettringite and an AFm phase with S/Ca=0.125.

Parallelization of a Lagrange–Euler-model for three-dimensional free surface flow and transport processes

A three-dimensional model for free surface flow and transport processes was parallelized for computer systems with distributed memory. This involved the development of numerical algorithms for an existing serial software system which uses operator- splitting for solving the basic equations. As such it combines Lagrangian and Eulerian concepts through the coupling of Characteristics (explicit) and Finite- Element Methods (implicit). A modular communication interface offering the choice between machine-dependent and portable routines was implemented. The unstructured meshes are partitioned and assigned to the parallel processors applying the Recursive Spectral Bisection method which can be combined with the Kernighan– Lin heuristic. Several problem- independent parallel basic tasks were newly developed. For the iterative solution of simultaneous linear equations, different Conjugate Gradient methods were parallelized for symmetric and non-symmetric matrices using preconditioners specially adapted to the parallel hardware. The Characteristics method is based on an algorithm for tracing path lines which was applied to a parallel system with distributed memory. A field case study of the Weser estuary (North Sea) was chosen to demonstrate the performance of the algorithm on different parallel platforms.

Numerical solution of fuzzy linear equations in engineering analysis

A computational methodology is presented for the solution of simultaneous linear equations that arise in the analysis of engineering systems involving fuzzy input parameters. In addition to a discussion on the existence of solution to the problem, the numerical solution to the fuzzy linear systems is defined and formalized. The framework of the proposed methodology consists of three major aspects involving (1) computerized selection of fuzziness, (2) implementation of fuzzy operations and (3) development and execution of a search-based algorithm. Two numerical examples are presented to illustrate the computational aspects of the proposed methodology. The present results are compared with those given by the combinatorial approach. © 1998 John Wiley & Sons, Ltd.

Numerical solution of fuzzy linear equations in engineering analysis

A computational methodology is presented for the solution of simultaneous linear equations that arise in the analysis of engineering systems involving fuzzy input parameters. In addition to a discussion on the existence of solution to the problem, the numerical solution to the fuzzy linear systems is defined and formalized. The framework of the proposed methodology consists of three major aspects involving (1) computerized selection of fuzziness, (2) implementation of fuzzy operations and (3) development and execution of a search-based algorithm. Two numerical examples are presented to illustrate the computational aspects of the proposed methodology. The present results are compared with those given by the combinatorial approach. © 1998 John Wiley & Sons, Ltd.

Queue-Length Distribution for the Discriminatory Processor-Sharing Queue

In this paper, we study a multiple class discriminatory processor-sharing queue. The queue is assumed to have Poisson input and exponentially distributed service times. In this discipline there are K classes of customers. When there are ni customers present in the system of class i(i = 1, …, K), each member of class j receives a fraction of the server's capacity given by αj/∑i=1Kniαi. Thus, associated with class i customers is a weight αi which determines the level of service discrimination. For this problem, we find the moments of the queue-length distribution as a solution of linear simultaneous equations. We also prove a heavy traffic limit theorem for the joint queue-length distribution for this queue.

A nondiagonal quasidegenerate fourth-order perturbation theory

A canonical quasidegenerate Rayleigh-Schrodinger perturbation theory, correct through fourth order in the energy, is explored for a block-diagonal unperturbed Hamiltonian. The theory is developed completely within a Lie Algebra in Hilbert space. Explicit equations forn-particle transition elements in terms of solutions of simultaneous linear equations are presented. A two-dimensional anisotropic anharmonic oscillator is used to provide numerical results. The perturbation theory is shown to be stable under small separation of model and complement spaces. An iterative variant of the fourth order perturbation theory is introduced; the iterative variant is related to the non-iterative one in much the same way as nondegenerate coupled cluster theories are related to nondegenerate perturbation theory. The quasidegenerate coupled cluster theory appears to be stable in the presence of multiple intruder states.

Symbolic computation using L-systems II: Extensions

An approach, based on Lindenmayer systems (L-systems), to symbolic computation presented earlier [1] is extended to allow differentiation and integration of various functions and solution of simultaneous linear equations. All the rules presented earlier and the new ones are incorporated in a symbolic L-system software (SLMATH) which is implemented on a UNIX based system. Sample results are presented.

Parallel-Vector Algorithms for Analysis of Large Structures

In analysis of large space structures, the major computational steps are evaluation and assembly of the structure stiffness matrix and solution of the resulting simultaneous linear equations. In this paper we present efficient parallel-vector algorithms for these steps of structural analysis. The goal is to optimize the performance of the algorithms through judicious combination of parallel processing and vectorization. Parallel-vector algorithms are presented for solution of linear simultaneous equations using Cholesky decomposition and preconditioned conjugate gradient approaches. The algorithms are applied to three large space structures modeling the exterior envelope of high-rise and super high-rise building structures in the range of 50–162 stories with up to 6,136 members. Performance results are presented in terms of central-processing-unit time on a Cray Y-MP8/864 supercomputer, MFLOPS (millions of floating point operations per second), and speedup.

離散時間ニューラルネットワークに基づく連立1次方程式求解並列処理システム

離散時間ニューラルネットワークに基づく連立1次方程式求解並列処理システム

Computation of analytical expressions for transfer functions

A new method for computing transfer functions of linear time-invariant multi-variable control systems is presented. This method has the following attractive features: (i) it produces analytical expressions for transfer functions, (ii) computations are inherently parallel and can be implemented, for instance, on parallel processors with a fine-grain architecture, (iii) calculations of transfer functions are reduced to two well-studied computational problems: solution of simultaneous linear equations and the fast Fourier transform (FFT), and (iv) state-variable formulations can be avoided. Theoretical conclusions are demonstrated with numerical examples.

Algorithm for Mixing Problems in Water Systems

The “Solver” algorithm, developed as an outgrowth of work on cost allocation in water distribution systems, is a simple technique that solves a number of interesting problems in water distribution system analysis. Problems related to mixing water from different sources within the distribution network, travel time from any source to any node of the network, and development of the cost of service to any node in the network can be solved assuming steady‐state conditions, given a prior solution of the hydraulics (flow in each link) of the network. All these problems are formulated as the solution of simultaneous linear equations. The Solver algorithm has been coded in Fortran and incorporated within the structure of a multi‐purpose system of computer programs that allows for the storage, display and manipulation of data associated with node‐link networks describing water distribution systems. These programs are identified by the name Water Supply Simulation Model. The formulation of the Solver algorithm is de...

Parallel iterative methods for solving linear equations

In this paper a form of matrix splitting into interlocking quadrants is introduced which leads to a new class of iterative methods for the numerical solution of linear simultaneouse quations which are applicable for use on parallel computers.