Efficient Matrix Method Research Articles

An efficient matrix method for coupled systems of variable fractional order differential equations

We establish a powerful numerical algorithm to compute numerical solutions of coupled system of variable fractional order differential equations. Our numer?ical procedure is based on Bernstein polynomials. The mentioned polynomials are non-orthogonal and have the ability to produce good numerical results as compared to some other numerical method like wavelet. By variable fractional order differentiation and integration, some operational matrices are formed. On using the obtained matrices, the proposed coupled system is reduced to a system of algebraic equations. Using MATLAB, we solve the given equation for required results. Graphical presentations and maximum absolute errors are given to illustrate the results. Some useful features of our sachem are those that we need no discretization or collocation technique prior to develop operational matrices. Due to these features the computational complexity is much more reduced. Further, the efficacy of the procedure is enhanced by increasing the scale level. We also compare our results with that of Haar wavelet method to justify the useful?ness of our adopted method.

BCR Algorithm for Solving Quadratic Inverse Eigenvalue Problems for Partially Bisymmetric Matrices

AbstractThe inverse eigenvalue problem appears repeatedly in a variety of applications. The aim of this paper is to study a quadratic inverse eigenvalue problem of the form AXΛ2 + BXΛ + CX = 0 where A, B and C should be partially bisymmetric under a prescribed submatrix constraint. We derive an efficient matrix method based on the Hestenes‐Stiefel (HS) version of biconjugate residual (BCR) algorithm for solving this constrained quadratic inverse eigenvalue problem. The theoretical results demonstrate that the matrix method solves the constrained quadratic inverse eigenvalue problem within a finite number of iterations in the absence of round‐off errors. Finally we validate the accuracy and efficiency of the matrix method through the numerical results.

A numerical approach for solving weakly singular partial integro‐differential equations via two‐dimensional‐orthonormal Bernstein polynomials with the convergence analysis

In this paper, we develop an efficient matrix method based on two‐dimensional orthonormal Bernstein polynomials (2D‐OBPs) to provide approximate solution of linear and nonlinear weakly singular partial integro‐differential equations (PIDEs). First, we approximate all functions involved in the considerable problem via 2D‐OBPs. Then, by using the operational matrices of integration, differentiation, and product, the solution of Volterra singular PIDEs is transformed to the solution of a linear or nonlinear system of algebraic equations which can be solved via some suitable numerical methods. With a small number of bases, we can find a reasonable approximate solution. Moreover, we establish some useful theorems for discussing convergence analysis and obtaining an error estimate associated with the proposed method. Finally, we solve some illustrative examples by employing the presented method to show the validity, efficiency, high accuracy, and applicability of the proposed technique.

A numerical method for solving some model problems arising in science and convergence analysis based on residual function

In this study, we solve some widely-used model problems consisting of linear, nonlinear differential and integral equations, employing Dickson polynomials with the parameter-α and the collocation points for an efficient matrix method. The convergence of a Dickson polynomial solution of the model problem is investigated by means of the residual function. We encode useful computer programs for model problems, in order to obtain the precise Dickson polynomial solutions. These solutions are plotted along with the exact solutions in figures and the numerical results are compared with other well-known methods in tables.

Numerical solution of nonlinear weakly singular partial integro-differential equation via operational matrices

In this paper, we propose and analyze an efficient matrix method based on shifted Legendre polynomials for the solution of non-linear volterra singular partial integro-differential equations(PIDEs). The operational matrices of integration, differentiation and product are used to reduce the solution of volterra singular PIDEs to the system of non-linear algebraic equations. Some useful results concerning the convergence and error estimates associated to the suggested scheme are presented. illustrative examples are provided to show the effectiveness and accuracy of proposed numerical method.

Imaging with tilted surfaces: an efficient matrix method for the generalized Scheimpflug condition and its application to rotationally symmetric triangulation

An efficient two-dimensional matrix method is presented that facilitates the design of optical systems with tilted surfaces for which the requirement or knowledge of the orientation of the image plane is necessary, i.e., for which a generalized Scheimpflug condition is needed. In more general terms, the method results in imaging properties of second-order expansion, but the method is linear. Therefore the complexity of the design process is considerably reduced. The strength of the design method is demonstrated in detail for a novel application in which a reflective optical system of several surfaces is required for rotationally symmetric triangulation.

Strongly convergent method to solve one-dimensional quantum problems.

An algorithm to solve the one-dimensional Schr\"odinger equation subject to Dirichlet boundary conditions is presented. The algorithm is based on a set of theorems that guarantee that when one solves the Schr\"odinger equation for a confined system and allows the boundaries to increase, the solutions converge strongly, in the norm of Hilbert space ${\mathit{L}}_{2}$(-\ensuremath{\infty},\ensuremath{\infty}), to the exact solutions of the unbounded problem. For the calculation of the solutions of the confined system we use a very efficient matrix method. By applying the algorithm to the harmonic oscillator and to the quartic and sextic potentials we show that with this method one can calculate the eigenvalues and eigenfunctions of a nonbounded one-dimensional problem with a high degree of accuracy and with very reasonable computational effort. We show that the eigenvalues corresponding to the sextic potential, V(x)=1/2${\mathit{x}}^{2}$+${\mathrm{\ensuremath{\alpha}}}_{2}$${\mathit{x}}^{4}$+${\mathrm{\ensuremath{\alpha}}}_{3}$${\mathit{x}}^{6}$, for different values of the parameter ${\mathrm{\ensuremath{\alpha}}}_{3}$ behave in a similar fashion as that described by Hioe et al. [Phys. Rep. 43C, 305 (1978)] for the quartic oscillator. \textcopyright{} 1996 The American Physical Society.

Systems With Infinitesimal Mobility: Part I—Matrix Analysis and First-Order Infinitesimal Mobility

Structural and mechanical assemblies with infinitesimal mobility are studied by means of analytical statics, elaborating and capitalizing on the conceptual difference between virtual and kinematic displacements. The analysis is based on the resolution of a given load into two mutually orthogonal components, which allows the introduction of a statical-kinematic stiffness matrix. The issue of statical-kinematic duality is explored and extended to second-order analysis. It leads to the complementary exclusion condition between kinematic mobility and statical possibility of stable self-stress, and the ensuing statical criterion of immobility. A computationally efficient matrix method for detecting simple first-order mechanisms is presented and compared to an existing method based on the analysis of a certain reduced quadratic form.

Quantum Monte Carlo for the Electronic Structure of Atoms and Molecules

Since the late 1960s molecular orbital (MO) theory has dominated the development of ab initio quantum chemistry. Specifically, the Hartree­ Fock-Roothaan (HFR) (1-3) formalism of the self-consistent field (SCF) method, which was greatly facilitated by the development of gaussian basis sets as suggested by Boys (4, 5), and the post-Hartree-Fock methods of configuration interaction (CI) (6-8), multiconfiguration SCF (MCSCF) (9, 10), many-body perturbation theory (MBPT) and coupled-cluster methods (CCM) (11), and Moller-Plesset perturbation theory (MPPT) (12). With the advent of high speed vector supercomputers the field has virtually exploded; larger molecules or very high accuracy for small molecules are now possible. In addition to such capability has come more accurate knowledge of the size of I-particle and N-particle basis sets required for high accuracy (13-16). Not only does the use of large basis sets require large memory and file-storage capability, but HFR and post-HFR algorithms contain significant bottlenecks to vectorization and parallelization. Although many new and efficient matrix methods have been developed

Effect Analysis in Structural Equation Models

One of the great virtues of structural equation models is that they permit the quantification of causal and noncausal sources of statistical relationship. The present article discusses efficient matrix methods of computation for effect decomposition and extends these methods to models with unstandardized variables and to nonrecursive models. An appendix includes a computer program, written in APL, which implements the techniques described in the article.

The structure of threshold graphs

Manca has derived an efficient matrix method for testing a given graph to see whether or not it is a threshold graph.

The Analysis Of Geometrical And Thermal Errors OfNon-Cartesian Structures

There is currently much interest in non-Cartesian machines in the machine tool industry. The analysis of errors arising from component variations in geometry and temperature is far more complex than in the case of the Cartesian machine. This is because the errors are no longer additive along each axis. This paper describes techniques leading to the development of general software which is capable of analysing the geometric and thermal errors in any structure consisting of measurable struts. This limitation is not so restrictive as it might appear, since solid components can usually be modelled as interconnected struts if large temperature gradients do not produce high stresses within their volume. The software also provides an animated display showing the moving structure. The techniques described constitute three efficient matrix methods which can be programmed in a general way. The approach is to break down a machine into sub-assemblies and to analyse each of these individually. The subassemblies are brought together into an overall model and the predictions of errors, such as the tilt of a spindle, are evaluated. Each sub-assembly can be assigned an average temperature or each strut can be treated separately. Hence, the errors arising from a source of heat, such as a spindle motor, can be evaluated, as can the effect of random errors in strut lengths. Three dimensional plots of errors in any plane are available.

The Backflow-Cell Model for Continuous Two-Phase Nonlinear Mass-Transfer Operations Including Nonlinear Axial Holdup and Mixing Effects

Abstract The backflow-cell model is applied to countercurrent two-phase flow processes with exchange of a single solute. The effects of nonuniform axial holdup of the phases and nonlinear equilibrium are considered. Efficient matrix methods of solving the model equations for the array of cell concentrations in each phase are developed. These profiles are compared to those of the diffusion model for some linear cases, and methods of smoothing them are discussed.