Complex System Of Linear Equations Research Articles

Gröbner Basis Approach for Solving Fuzzy Complex System of Linear Equations

The foundation of Gröbner’s work is a potent approach for separating polynomial systems of equations. In this study, we solve complex fuzzy systems of linear equations utilizing Gröbner’s basic technique. First, a system of explicit polynomial equations is generated by applying fuzzy computations and fuzzy complex computations to the fundamental system. Taking into account the lexicographical order, Gröbner’s basis is then determined for the ideal given by a definite equivalent polynomial system. The Gröbnerbase that was obtained has a high triangular structure. Thus, the system solution may be readily implemented by proceeding. Hence, all of the system’s answers are readily attainable. Next, a criterion is established to identify whether or not a solution exists for the primary system. The advantage of the proposed approach is that it acquires all of the problem’s solutions simultaneously. Using this technique, a mathematical example and an application issue based on an electrical circuit are resolved. Comparing the generated findings to the known outcomes reveals a high level of congruence.

Parameter-by-parameter method for steric mass action model of ion exchange chromatography: Simplified estimation for steric shielding factor

Mechanistic models play a crucial role in the process development and optimization of ion-exchange chromatography (IEC). Recent researches in steric mass action (SMA) model have heightened the need for better estimation of nonlinear parameter, steric shielding factor σ. In this work, a straightforward approach combination of simplified linear approximation (SLA) and inverse method (IM) was proposed to initialize and further determine σ, respectively. An existed, unique, and positive σ can be derived from SLA. Compared with linear approximation (LA) developed in our previous study, σ of the multi-component system can be calculated easily without solving the complex system of linear equations, leading to a time complexity reduction from O(n3) to O(n). The proposed method was verified first in numerical experiments about the separation of three charge variants. The calculated σ was more reasonable than that of LA, and the error of elution profiles with the parameters estimated by SLA+IM was only one-sixth of that by LA in numerical experiments. Moreover, the error accumulation effect could also be reduced. The proposed method was further confirmed in real-world experiments about the separation of monomer-dimer mixtures of monoclonal antibody. The results gave a lower error and better physical understanding compared to LA. In conclusion, SLA+IM developed in the present work provides a novel and straightforward way to determine σ. This simplification would help to save the effort of calibration experiments and accelerate the process development for the multi-component IEC separation.

A new iterative method for solving the systems arisen from finite element discretization of a time-harmonic parabolic optimal control problems

In this paper, we focus on solving a class of two-by-two block complex system of linear equations arising from finite element discretization of a distributed optimal control constrained by a time-harmonic parabolic equations. We propose a new iterative method for solving the obtained system. In each iteration of the method a two-by-two block system of linear equations with real coefficient matrix should be solved. We solve this system inexactly using the generalized minimal residual (GMRES) and the Chebyshev acceleration methods in conjunction with the real-valued preconditioned square block (PRESB) preconditioner. The convergence and spectral properties of the method are discussed. Numerical results in 2-dimensional case are presented to demonstrate the efficiency of the method.

Exciting Fixed Point Results on a Novel Space with Supportive Applications

The goal of this paper is to present a new space, a complex valued controlled rectangular b -metric space (for short, υ ℂ -metric space). Some examples and topological properties of υ ℂ -metric spaces are given. Also, some related common fixed point results are discussed. Our results generalize a lot of works in this direction. Moreover, we apply the theoretical results to find a unique solution of a complex valued Atangana-Baleanu fractional integral operator and a system of complex linear equations. Finally, a numerical example to find the current that passes through the RLC circuit is illustrated.

Matrix method of formulating rations for dairy cows

The objective of this work was to disclose the use of operations with matrices in the formulation of rations for dairy cows, considering the great potential that this methodology presents to solve pre-defined parameters and unknowns in systems of linear equations. With the use of matrices it is possible to solve systems of complex linear equations that would be difficult to solve by other mathematical means. Thus, using the matrix theory, depending on the number of parameters and unknowns, the determinants calculation, the Gauss elimination method and the Gauss-Jordan scaling can be used as mathematical tools for solving the systems. These methods, besides finding the value of the variables, also allow fixing ingredients in order to obtain in the final formula foods in a pre-defined percentage, thus guaranteeing the fulfilment of the nutritional requirements required by the animal category.

A discussion on “solving fuzzy complex system of linear equations”

Recently, Behera and Chakraverty [Solving fuzzy complex system of linear equations, Information Sciences 277 (2014) 154–162] proposed a new method for solving general fuzzy complex linear systems. However, in this note, we show that there are some problems in their method. Also, we present a modified version of their method which can avoid these problems for solving a fuzzy complex linear system.

Mehar and Keerat method for solving system of fuzzy complex linear equations

In this paper, it is pointed out that in the existing methods (Sarangam Majumdar, Numerical solutions of fuzzy complex system of linear equations, German Journal of Advanced Mathematical Sciences 1 (2013), 20-26) for solving fuzzy complex system of linear equations in which all the parameters except variables are represented by triangular/trapezoidal fuzzy complex numbers as well as the fuzzy complex system of linear equations in which all the parameters except coefficients are represented by triangular/trapezoidal fuzzy complex numbers, Majumdar has assumed that on subtracting a fuzzy number from itself, the obtained value is a zero fuzzy number as well as on dividing a fuzzy number by itself, the obtained value is a unit fuzzy number. To resolve these flaws of the existing methods, two new methods (named as Mehar method and Keerat method) are proposed for solving the same systems of fuzzy complex linear equations.

Solving fuzzy complex system of linear equations using eigenvalue method

The occurrence of imprecision in the real world is inevitable due to some unexpected situations. The imprecision is often involved in any engineering design process. The imprecision and uncertainty are often interpreted as fuzziness. Fuzzy systems have an essential role in the uncertainty modeling, which can formulate the uncertainty in the actual environment. In this paper, a new method based on linear algebra for resolution of a fuzzy complex system of linear equations is presented. Moreover, a new algorithm is designed to find all the solutions of the system based on the eigenvalue method. Thanks to our approach, we can find all the solutions of the system at a time. Also, we are able to determine when the solution sets of the system are nonempty sets.To illustrate the easy application of the proposed method, numerical examples are given and the obtained results are discussed.

Performance of preconditioned iterative and multigrid solvers in solving the three-dimensional magnetotelluric modeling problem using the staggered finite-difference method: a comparative study

An effective solver for the large complex system of linear equations is critical for improving the accuracy of numerical solutions in three-dimensional (3D) magnetotelluric (MT) modeling using the staggered finite-difference (SFD) method. In electromagnetic modeling, the formed system of linear equations is commonly solved using preconditioned iterative relaxation methods. We present 3D MT modeling using the SFD method, based on former work. The multigrid solver and three solvers preconditioned by incomplete Cholesky decomposition—the minimum residual method, the generalized product bi-conjugate gradient method and the bi-conjugate gradient stabilized method—are used to solve the formed system of linear equations. Divergence correction for the magnetic field is applied. We also present a comparison of the stability and convergence of these iterative solvers if divergence correction is used. Model tests show that divergence correction improves the convergence of iterative solvers and the accuracy of numerical results. Divergence correction can also decrease the number of iterations for fast convergence without changing the stability of linear solvers. For consideration of the computation time and memory requirements, the multigrid solver combined with divergence correction is preferred for 3D MT field simulation.

On preconditioned iteration methods for complex linear systems

A complex system of linear equations arises in many important applications. We further explore algebraic and convergence properties and present analytical and numerical comparisons among several available iteration methods such as C-to-R and PMHSS for solving such a class of linear systems. Theoretical analyses and computational results show that reformulating a complex linear system into an equivalent real form is a feasible and effective approach, for which we can construct, analyze, and implement accurate, efficient, and robust preconditioned iteration methods.

Solving fuzzy complex system of linear equations

This paper proposes a new and simple method for solving general fuzzy complex system of linear equations. In the original system, the elements of unknown variable vector and right hand side vector are considered as complex fuzzy number. Initially the general system is solved by adding and subtracting the left and right bounds of the fuzzy complex unknown and right hand side fuzzy complex vector respectively. Subsequently above obtained solutions are used to get the final solution of the general fuzzy complex system of linear equations. Two theorems are stated and proved related to the proposed method. A mathematical example and an application problem based on electrical circuit are solved using the present method. Results obtained are compared with the known results and are found to be in good agreement.

The block preconditioned LSQR and GL-LSQR algorithms for the block partitioned matrices

In this paper, a new block preconditioner is proposed for the block partitioned matrices. This preconditioner is based on the block C-orthogonalization, where C is a symmetric positive definite matrix. The block preconditioned least squares (BPLS) and block preconditioned global least squares (BPGLS) algorithms are presented to solve the linear system of equations with block partioned coefficient matrix and multiple linear system of equations, respectively. The BPLS algorithm is applied to solve the complex linear system of equations and also BPLS and classical preconditioned least squares (PLS) algorithms are compared. Finally, some numerical experiments are given to show the efficiency of the new block preconditioner.

The Optimal Filter Recurrence Calculating Method on a System of Equations Pseudo Solution in the Large Capacity of Computing Technology

In the large capacity of computing technology, this article sets up the statistic recurrence calculating method of linear algebra pseudosolution by making use of the optimal filter theory. So we can get recurrence form of pseudosolution under the linear simultanuity and linear non-simultanuity instead of using the complex calculating pseudo-inverse matrix. This method is reliable and its effect is excellent. The results of the discussion show that the method of this article will provide an effective mathematic processing method in order to realize the optimal calculating of complex system of linear equations in the large capacity of calculating.

FUZZY CENTRE BASED SOLUTION OF FUZZY COMPLEX LINEAR SYSTEM OF EQUATIONS

A new approach to solve Fuzzy Complex System of Linear Equations (FCSLE) based on fuzzy complex centre procedure is presented here. Few theorems related to the investigation are stated and proved. Finally the presented procedure is used to analyze an example problem of linear time invariant electric circuit with complex crisp coefficient and fuzzy complex sources. The results obtained are also compared with the known solutions and are found to be in good agreement.

B-splines approach of the Sturmian basis: Atomic and scattering calculations

We test a new methodology to obtain a Sturmian basis set using B-spline expansion of arbitrary order and imposing the correct boundary conditions for both negative and positive energy domains. The Sturmians can be applied to solve both atomic systems, which are transformed into an eigenvalue problem, and scattering processes, which leads to a complex system of linear equations. We show how our methodology converges rapidly and give accurate results compared with other techniques

Distributed Response Time Analysis of GSPN Models with MapReduce

Generalized stochastic Petri nets (GSPNs) are widely used in the performance analysis of computer and communications systems. Response time densities and quantiles are often key outputs of such analysis. These can be extracted from a GSPN’s underlying semi-Markov process using a method based on numerical Laplace transform inversion. This method typically requires the solution of thousands of systems of complex linear equations, each of rank n, where n is the number of states in the model. For large models substantial processing power is needed and the computation must therefore be distributed. In this paper we describe the implementation of a response time analysis module for the Platform Independent Petri net Editor (PIPE2) which interfaces with Hadoop, an open-source implementation of Google’s MapReduce distributed programming environment, to provide distributed calculation of response time densities in GSPN models. The software is validated with results calculated analytically as well as simulated results for larger models. Excellent scalability is shown.

Fuzzy Complex System of Linear EquationsApplied to Circuit Analysis

In this paper, application of fuzzy system of linear equations (FSLE) is investigated in the circuit analysis (CA). In the field of CA, each circuit includes linear resistance, inductance and capacitance are modeled to complex linear equation. A fuzzy current or voltage in the circuit is more intellectual relative to crisp value because of some reasons as environmental conditions, tolerance in the elements, and noise (leakage of power harmonics,). This scheme forces for presentation of fuzzy complex system of linear equations (FCSLE). Derivation is presented for solving FCSLE and obtained results show ability of this method in CA.

Preconditioned Iterative Solvers for Inductance Extraction of VLSI Circuits

Parasitic extraction techniques are used to estimate signal delay in VLSI circuits. Inductance extraction is a critical component of the parasitic extraction process that involves estimation of on‐chip inductive effects with high accuracy. The problem requires the solution of a large, dense, complex linear system of equations, where the unknown current must satisfy the constraint imposed by Kirchoff’s current law. In this paper, we describe a solenoidal basis method to transform the constrained linear system into an unconstrained one. The method uses discrete local solenoidal flows to represent the unknown current, and to obtain a reduced linear system. The paper proposes preconditioning techniques that do not require explicit construction of the reduced system. Numerical experiments are presented to illustrate the effectiveness of the preconditioning approach. Comparisons with a well‐known inductance extraction package are provided to highlight the advantages of the proposed scheme.

Finite-difference simulation of borehole EM measurements in 3D anisotropic media using coupled scalar-vector potentials

This paper describes the implementation and successful validation of a new staggered-grid, finite-difference algorithm for the numerical simulation of frequency-domain electromagnetic borehole measurements. The algorithm is based on a coupled scalar-vector potential formulation for arbitrary 3D inhomogeneous electrically anisotropic media. We approximate the second-order partial differential equations for the coupled scalar-vector potentials with central finite differences on both Yee’s staggered and standard grids. The discretization of the partial differential equations and the enforcement of the appropriate boundary conditions yields a complex linear system of equations that we solve iteratively using the biconjugate gradient method with preconditioning. The accuracy and efficiency of the algorithm is assessed with examples of multicomponent-borehole electromagnetic-induction measurements acquired in homogeneous, 1D anisotropic, 2D isotropic, and 3D anisotropic rock formations. The simulation examples consider vertical and deviated wells with and without borehole and mud-filtrate invasion regions. Simulation results obtained with the scalar-vector coupled potential formulation favorably compare in accuracy with results obtained with 1D, 2D, and 3D benchmarking codes in the dc to megahertz frequency range for large contrasts of electrical conductivity. Our numerical exercises indicate that the coupled scalar-vector potential equations provide a general and consistent algorithmic formulation to simulate borehole electromagnetic measurements from dc to megahertz in the presence of large conductivity contrasts, dipping wells, electrically anisotropic media, and geometrically complex models of electrical conductivity.

Performance of discrete dipole approximation for prediction of amplitude and phase of electromagnetic scattering by particles

Electromagnetic scattering provides useful signatures for nonintrusive particle characterization. Scattered wave which carries characteristic information about particles is identified completely by its intensity, polarization state and phase. Recent developments in measurement techniques have enabled measurement of phase of the scattered wave which is a source of additional information about particles. In the present study, accuracy of discrete dipole approximation (DDA) in predicting amplitude and phase of scattered wave is investigated via publicly available DDSCAT code by Draine and Flatau, which is a well-established tool for DDA and has found wide range of applications in the literature due to its flexibility. DDSCAT routine is modified to enable accurate computation of phase of complex amplitude scattering matrix (ASM) elements as well as their magnitude. DDA method was implemented by using lattice dispersion relation for dipole polarizabilities, generalized prime factor algorithm for fast-Fourier transformation and pre-conditioned bi-conjugate gradient method with stabilization for the solution of the complex linear system of equations. Accuracy of ASM elements predicted by DDA is assessed on single sphere problems with various size parameters and refractive indices by validation against Mie theory solutions. Excellent agreement between predictions and exact solutions proves the reliability of the modified DDSCAT code for prediction of amplitude and phase of scattered electromagnetic wave. Applicability conditions and requirements of the present DDA application to ensure accurate prediction of complete set of scattering parameters are mapped for single spheres, on an extensive domain of size parameters and refractive indices. A correlation is presented to estimate the magnitude and phase errors associated with given size parameter, refractive index and cubic lattice subdivision. Assessment of computational time requirements for different optical constants shows that implementation of DDA with the present specifications is unfeasible for size parameters larger than 4 when Re( m)>2 and Im( m)<0.1 at the same time, due to slow convergence rate.