Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 4679

Search results for: linear matrix inequalities

4679 Robust Control of a Dynamic Model of an F-16 Aircraft with Improved Damping through Linear Matrix Inequalities

Authors: J. P. P. Andrade, V. A. F. Campos

Abstract:

This work presents an application of Linear Matrix Inequalities (LMI) for the robust control of an F-16 aircraft through an algorithm ensuring the damping factor to the closed loop system. The results show that the zero and gain settings are sufficient to ensure robust performance and stability with respect to various operating points. The technique used is the pole placement, which aims to put the system in closed loop poles in a specific region of the complex plane. Test results using a dynamic model of the F-16 aircraft are presented and discussed.

Keywords: F-16 aircraft, linear matrix inequalities, pole placement, robust control

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4678 Parameterized Lyapunov Function Based Robust Diagonal Dominance Pre-Compensator Design for Linear Parameter Varying Model

Authors: Xiaobao Han, Huacong Li, Jia Li

Abstract:

For dynamic decoupling of linear parameter varying system, a robust dominance pre-compensator design method is given. The parameterized pre-compensator design problem is converted into optimal problem constrained with parameterized linear matrix inequalities (PLMI); To solve this problem, firstly, this optimization problem is equivalently transformed into a new form with elimination of coupling relationship between parameterized Lyapunov function (PLF) and pre-compensator. Then the problem was reduced to a normal convex optimization problem with normal linear matrix inequalities (LMI) constraints on a newly constructed convex polyhedron. Moreover, a parameter scheduling pre-compensator was achieved, which satisfies robust performance and decoupling performances. Finally, the feasibility and validity of the robust diagonal dominance pre-compensator design method are verified by the numerical simulation of a turbofan engine PLPV model.

Keywords: linear parameter varying (LPV), parameterized Lyapunov function (PLF), linear matrix inequalities (LMI), diagonal dominance pre-compensator

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4677 Heinz-Type Inequalities in Hilbert Spaces

Authors: Jin Liang, Guanghua Shi

Abstract:

In this paper, we are concerned with the further refinements of the Heinz operator inequalities in Hilbert spaces. Our purpose is to derive several new Heinz-type operator inequalities. First, with the help of the Taylor series of some hyperbolic functions, we obtain some refinements of the ordering relations among Heinz means defined by Bhatia with different parameters, which would be more suitable in obtaining the corresponding operator inequalities. Second, we present some generalizations of Heinz operator inequalities. Finally, we give a matrix version of the Heinz inequality for the Hilbert-Schmidt norm.

Keywords: Hilbert space, means inequality, norm inequality, positive linear operator

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4676 Reliable Consensus Problem for Multi-Agent Systems with Sampled-Data

Authors: S. H. Lee, M. J. Park, O. M. Kwon

Abstract:

In this paper, reliable consensus of multi-agent systems with sampled-data is investigated. By using a suitable Lyapunov-Krasovskii functional and some techniques such as Wirtinger Inequality, Schur Complement and Kronecker Product, the results of this systems are obtained by solving a set of Linear Matrix Inequalities(LMIs). One numerical example is included to show the effectiveness of the proposed criteria.

Keywords: multi-agent, linear matrix inequalities (LMIs), kronecker product, sampled-data, Lyapunov method

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4675 Sampled-Data Control for Fuel Cell Systems

Authors: H. Y. Jung, Ju H. Park, S. M. Lee

Abstract:

A sampled-data controller is presented for solid oxide fuel cell systems which is expressed by a sector bounded nonlinear model. The sector bounded nonlinear systems, which have a feedback connection with a linear dynamical system and nonlinearity satisfying certain sector type constraints. Also, the sampled-data control scheme is very useful since it is possible to handle digital controller and increasing research efforts have been devoted to sampled-data control systems with the development of modern high-speed computers. The proposed control law is obtained by solving a convex problem satisfying several linear matrix inequalities. Simulation results are given to show the effectiveness of the proposed design method.

Keywords: sampled-data control, fuel cell, linear matrix inequalities, nonlinear control

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4674 Formation Control for Linear Multi-Robot System with Switched Directed Topology and Time-Varying Delays

Authors: Yaxiao Zhang, Yangzhou Chen

Abstract:

This study investigate the formation problem for high-order continuous-time multi-robot with bounded symmetric time-varying delay protocol under switched directed communication topology. By using a linear transformation, the formation problem is transformed to stability analysis of a switched delay system. Under the assumption that each communication topology has a directed spanning tree, sufficient conditions are presented in terms of linear matrix inequalities (LMIs) that the multi-robot system can achieve a desired formation by the trade-off among the pre-exist topologies with the help of the scheme of average dwell time. A numeral example is presented to illustrate the effectiveness of the obtained results.

Keywords: multi-robot systems, formation, switched directed topology, symmetric time-varying delay, average dwell time, linear matrix inequalities (lmis)

Procedia PDF Downloads 445
4673 Online Robust Model Predictive Control for Linear Fractional Transformation Systems Using Linear Matrix Inequalities

Authors: Peyman Sindareh Esfahani, Jeffery Kurt Pieper

Abstract:

In this paper, the problem of robust model predictive control (MPC) for discrete-time linear systems in linear fractional transformation form with structured uncertainty and norm-bounded disturbance is investigated. The problem of minimization of the cost function for MPC design is converted to minimization of the worst case of the cost function. Then, this problem is reduced to minimization of an upper bound of the cost function subject to a terminal inequality satisfying the l2-norm of the closed loop system. The characteristic of the linear fractional transformation system is taken into account, and by using some mathematical tools, the robust predictive controller design problem is turned into a linear matrix inequality minimization problem. Afterwards, a formulation which includes an integrator to improve the performance of the proposed robust model predictive controller in steady state condition is studied. The validity of the approaches is illustrated through a robust control benchmark problem.

Keywords: linear fractional transformation, linear matrix inequality, robust model predictive control, state feedback control

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4672 Stability of Hybrid Stochastic Systems

Authors: Manlika Ratchagit

Abstract:

This paper is concerned with robust mean square stability of uncertain stochastic switched discrete time-delay systems. The system to be considered is subject to interval time-varying delays, which allows the delay to be a fast time-varying function and the lower bound is not restricted to zero. Based on the discrete Lyapunov functional, a switching rule for the robust mean square stability for the uncertain stochastic discrete time-delay system is designed via linear matrix inequalities. Finally, some examples are exploited to illustrate the effectiveness of the proposed schemes.

Keywords: robust mean square stability, discrete-time stochastic systems, hybrid systems, interval time-varying delays, Lyapunov functional, linear matrix inequalities

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4671 New Results on Stability of Hybrid Stochastic Systems

Authors: Manlika Rajchakit

Abstract:

This paper is concerned with robust mean square stability of uncertain stochastic switched discrete time-delay systems. The system to be considered is subject to interval time-varying delays, which allows the delay to be a fast time-varying function and the lower bound is not restricted to zero. Based on the discrete Lyapunov functional, a switching rule for the robust mean square stability for the uncertain stochastic discrete time-delay system is designed via linear matrix inequalities. Finally, some examples are exploited to illustrate the effectiveness of the proposed schemes.

Keywords: robust mean square stability, discrete-time stochastic systems, hybrid systems, interval time-varying delays, lyapunov functional, linear matrix inequalities

Procedia PDF Downloads 349
4670 Regional Pole Placement by Saturated Power System Stabilizers

Authors: Hisham M. Soliman, Hassan Yousef

Abstract:

This manuscript presents new results on design saturated power system stabilizers (PSS) to assign system poles within a desired region for achieving good dynamic performance. The regional pole placement is accomplished against model uncertainties caused by different load conditions. The design is based on a sufficient condition in the form of linear matrix inequalities (LMI) which forces the saturated nonlinear controller to lie within the linear zone. The controller effectiveness is demonstrated on a single machine infinite bus system.

Keywords: power system stabilizer, saturated control, robust control, regional pole placement, linear matrix inequality (LMI)

Procedia PDF Downloads 481
4669 Fault-Tolerant Predictive Control for Polytopic LPV Systems Subject to Sensor Faults

Authors: Sofiane Bououden, Ilyes Boulkaibet

Abstract:

In this paper, a robust fault-tolerant predictive control (FTPC) strategy is proposed for systems with linear parameter varying (LPV) models and input constraints subject to sensor faults. Generally, virtual observers are used for improving the observation precision and reduce the impacts of sensor faults and uncertainties in the system. However, this type of observer lacks certain system measurements which substantially reduce its accuracy. To deal with this issue, a real observer is then designed based on the virtual observer, and consequently a real observer-based robust predictive control is designed for polytopic LPV systems. Moreover, the proposed observer can entirely assure that all system states and sensor faults are estimated. As a result, and based on both observers, a robust fault-tolerant predictive control is then established via the Lyapunov method where sufficient conditions are proposed, for stability analysis and control purposes, in linear matrix inequalities (LMIs) form. Finally, simulation results are given to show the effectiveness of the proposed approach.

Keywords: linear parameter varying systems, fault-tolerant predictive control, observer-based control, sensor faults, input constraints, linear matrix inequalities

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4668 Robust H∞ State Feedback Control for Discrete Time T-S Fuzzy Systems Based on Fuzzy Lyapunov Function Approach

Authors: Walied Hanora

Abstract:

This paper presents the problem of robust state feedback H∞ for discrete time nonlinear system represented by Takagi-Sugeno fuzzy systems. Based on fuzzy lyapunov function, the condition ,which is represented in the form of Liner Matrix Inequalities (LMI), guarantees the H∞ performance of the T-S fuzzy system with uncertainties. By comparison with recent literature, this approach will be more relaxed condition. Finally, an example is given to illustrate the proposed result.

Keywords: fuzzy lyapunov function, H∞ control , linear matrix inequalities, state feedback, T-S fuzzy systems

Procedia PDF Downloads 204
4667 A Robust Model Predictive Control for a Photovoltaic Pumping System Subject to Actuator Saturation Nonlinearity and Parameter Uncertainties: A Linear Matrix Inequality Approach

Authors: Sofiane Bououden, Ilyes Boulkaibet

Abstract:

In this paper, a robust model predictive controller (RMPC) for uncertain nonlinear system under actuator saturation is designed to control a DC-DC buck converter in PV pumping application, where this system is subject to actuator saturation and parameter uncertainties. The considered nonlinear system contains a linear constant part perturbed by an additive state-dependent nonlinear term. Based on the saturating actuator property, an appropriate linear feedback control law is constructed and used to minimize an infinite horizon cost function within the framework of linear matrix inequalities. The proposed approach has successfully provided a solution to the optimization problem that can stabilize the nonlinear plants. Furthermore, sufficient conditions for the existence of the proposed controller guarantee the robust stability of the system in the presence of polytypic uncertainties. In addition, the simulation results have demonstrated the efficiency of the proposed control scheme.

Keywords: PV pumping system, DC-DC buck converter, robust model predictive controller, nonlinear system, actuator saturation, linear matrix inequality

Procedia PDF Downloads 107
4666 Advanced Stability Criterion for Time-Delayed Systems of Neutral Type and Its Application

Authors: M. J. Park, S. H. Lee, C. H. Lee, O. M. Kwon

Abstract:

This paper investigates stability problem for linear systems of neutral type with time-varying delay. By constructing various Lyapunov-Krasovskii functional, and utilizing some mathematical techniques, the sufficient stability conditions for the systems are established in terms of linear matrix inequalities (LMIs), which can be easily solved by various effective optimization algorithms. Finally, some illustrative examples are given to show the effectiveness of the proposed criterion.

Keywords: neutral systems, time-delay, stability, Lyapnov method, LMI

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4665 H∞ Sampled-Data Control for Linear Systems Time-Varying Delays: Application to Power System

Authors: Chang-Ho Lee, Seung-Hoon Lee, Myeong-Jin Park, Oh-Min Kwon

Abstract:

This paper investigates improved stability criteria for sampled-data control of linear systems with disturbances and time-varying delays. Based on Lyapunov-Krasovskii stability theory, delay-dependent conditions sufficient to ensure H∞ stability for the system are derived in the form of linear matrix inequalities(LMI). The effectiveness of the proposed method will be shown in numerical examples.

Keywords: sampled-data control system, Lyapunov-Krasovskii functional, time delay-dependent, LMI, H∞ control

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4664 Parallel Computation of the Covariance-Matrix

Authors: Claude Tadonki

Abstract:

We address the issues related to the computation of the covariance matrix. This matrix is likely to be ill conditioned following its canonical expression, thus consequently raises serious numerical issues. The underlying linear system, which therefore should be solved by means of iterative approaches, becomes computationally challenging. A huge number of iterations is expected in order to reach an acceptable level of convergence, necessary to meet the required accuracy of the computation. In addition, this linear system needs to be solved at each iteration following the general form of the covariance matrix. Putting all together, its comes that we need to compute as fast as possible the associated matrix-vector product. This is our purpose in the work, where we consider and discuss skillful formulations of the problem, then propose a parallel implementation of the matrix-vector product involved. Numerical and performance oriented discussions are provided based on experimental evaluations.

Keywords: covariance-matrix, multicore, numerical computing, parallel computing

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4663 On Direct Matrix Factored Inversion via Broyden's Updates

Authors: Adel Mohsen

Abstract:

A direct method based on the good Broyden's updates for evaluating the inverse of a nonsingular square matrix of full rank and solving related system of linear algebraic equations is studied. For a matrix A of order n whose LU-decomposition is A = LU, the multiplication count is O (n3). This includes the evaluation of the LU-decompositions of the inverse, the lower triangular decomposition of A as well as a “reduced matrix inverse”. If an explicit value of the inverse is not needed the order reduces to O (n3/2) to compute to compute inv(U) and the reduced inverse. For a symmetric matrix only O (n3/3) operations are required to compute inv(L) and the reduced inverse. An example is presented to demonstrate the capability of using the reduced matrix inverse in treating ill-conditioned systems. Besides the simplicity of Broyden's update, the method provides a mean to exploit the possible sparsity in the matrix and to derive a suitable preconditioner.

Keywords: Broyden's updates, matrix inverse, inverse factorization, solution of linear algebraic equations, ill-conditioned matrices, preconditioning

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4662 Anti-Phase Synchronization of Complex Delayed Networks with Output Coupling via Pinning Control

Authors: Chanyuan Gu, Shouming Zhong

Abstract:

Synchronization is a fundamental phenomenon that enables coherent behavior in networks as a result of interactions. The purpose of this research had been to investigate the problem of anti-phase synchronization for complex delayed dynamical networks with output coupling. The coupling configuration is general, with the coupling matrix not assumed to be symmetric or irreducible. The amount of the coupling variables between two connected nodes is flexible, the nodes in the drive and response systems need not to be identical and there is not any extra constraint on the coupling matrix. Some pinning controllers are designed to make the drive-response system achieve the anti-phase synchronization. For the convenience of description, we applied the matrix Kronecker product. Some new criteria are proposed based on the Lyapunov stability theory, linear matrix inequalities (LMI) and Schur complement. Lastly, some simulation examples are provided to illustrate the effectiveness of our proposed conditions.

Keywords: anti-phase synchronization, complex networks, output coupling, pinning control

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4661 Some Inequalities Related with Starlike Log-Harmonic Mappings

Authors: Melike Aydoğan, Dürdane Öztürk

Abstract:

Let H(D) be the linear space of all analytic functions defined on the open unit disc. A log-harmonic mappings is a solution of the nonlinear elliptic partial differential equation where w(z) ∈ H(D) is second dilatation such that |w(z)| < 1 for all z ∈ D. The aim of this paper is to define some inequalities of starlike logharmonic functions of order α(0 ≤ α ≤ 1).

Keywords: starlike log-harmonic functions, univalent functions, distortion theorem

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4660 System of Linear Equations, Gaussian Elimination

Authors: Rabia Khan, Nargis Munir, Suriya Gharib, Syeda Roshana Ali

Abstract:

In this paper linear equations are discussed in detail along with elimination method. Gaussian elimination and Gauss Jordan schemes are carried out to solve the linear system of equation. This paper comprises of matrix introduction, and the direct methods for linear equations. The goal of this research was to analyze different elimination techniques of linear equations and measure the performance of Gaussian elimination and Gauss Jordan method, in order to find their relative importance and advantage in the field of symbolic and numeric computation. The purpose of this research is to revise an introductory concept of linear equations, matrix theory and forms of Gaussian elimination through which the performance of Gauss Jordan and Gaussian elimination can be measured.

Keywords: direct, indirect, backward stage, forward stage

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4659 Robust Control of a Single-Phase Inverter Using Linear Matrix Inequality Approach

Authors: Chivon Choeung, Heng Tang, Panha Soth, Vichet Huy

Abstract:

This paper presents a robust control strategy for a single-phase DC-AC inverter with an output LC-filter. An all-pass filter is utilized to create an artificial β-signal so that the proposed controller can be simply used in dq-synchronous frame. The proposed robust controller utilizes a state feedback control with integral action in the dq-synchronous frame. A linear matrix inequality-based optimization scheme is used to determine stabilizing gains of the controllers to maximize the convergence rate to steady state in the presence of uncertainties. The uncertainties of the system are described as the potential variation range of the inductance and resistance in the LC-filter.

Keywords: single-phase inverter, linear matrix inequality, robust control, all-pass filter

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4658 Approximations of Fractional Derivatives and Its Applications in Solving Non-Linear Fractional Variational Problems

Authors: Harendra Singh, Rajesh Pandey

Abstract:

The paper presents a numerical method based on operational matrix of integration and Ryleigh method for the solution of a class of non-linear fractional variational problems (NLFVPs). Chebyshev first kind polynomials are used for the construction of operational matrix. Using operational matrix and Ryleigh method the NLFVP is converted into a system of non-linear algebraic equations, and solving these equations we obtained approximate solution for NLFVPs. Convergence analysis of the proposed method is provided. Numerical experiment is done to show the applicability of the proposed numerical method. The obtained numerical results are compared with exact solution and solution obtained from Chebyshev third kind. Further the results are shown graphically for different fractional order involved in the problems.

Keywords: non-linear fractional variational problems, Rayleigh-Ritz method, convergence analysis, error analysis

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4657 Kalman Filter Gain Elimination in Linear Estimation

Authors: Nicholas D. Assimakis

Abstract:

In linear estimation, the traditional Kalman filter uses the Kalman filter gain in order to produce estimation and prediction of the n-dimensional state vector using the m-dimensional measurement vector. The computation of the Kalman filter gain requires the inversion of an m x m matrix in every iteration. In this paper, a variation of the Kalman filter eliminating the Kalman filter gain is proposed. In the time varying case, the elimination of the Kalman filter gain requires the inversion of an n x n matrix and the inversion of an m x m matrix in every iteration. In the time invariant case, the elimination of the Kalman filter gain requires the inversion of an n x n matrix in every iteration. The proposed Kalman filter gain elimination algorithm may be faster than the conventional Kalman filter, depending on the model dimensions.

Keywords: discrete time, estimation, Kalman filter, Kalman filter gain

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4656 The Second Smallest Eigenvalue of Complete Tripartite Hypergraph

Authors: Alfi Y. Zakiyyah, Hanni Garminia, M. Salman, A. N. Irawati

Abstract:

In the terminology of the hypergraph, there is a relation with the terminology graph. In the theory of graph, the edges connected two vertices. In otherwise, in hypergraph, the edges can connect more than two vertices. There is representation matrix of a graph such as adjacency matrix, Laplacian matrix, and incidence matrix. The adjacency matrix is symmetry matrix so that all eigenvalues is real. This matrix is a nonnegative matrix. The all diagonal entry from adjacency matrix is zero so that the trace is zero. Another representation matrix of the graph is the Laplacian matrix. Laplacian matrix is symmetry matrix and semidefinite positive so that all eigenvalues are real and non-negative. According to the spectral study in the graph, some that result is generalized to hypergraph. A hypergraph can be represented by a matrix such as adjacency, incidence, and Laplacian matrix. Throughout for this term, we use Laplacian matrix to represent a complete tripartite hypergraph. The aim from this research is to determine second smallest eigenvalues from this matrix and find a relation this eigenvalue with the connectivity of that hypergraph.

Keywords: connectivity, graph, hypergraph, Laplacian matrix

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4655 BIASS in the Estimation of Covariance Matrices and Optimality Criteria

Authors: Juan M. Rodriguez-Diaz

Abstract:

The precision of parameter estimators in the Gaussian linear model is traditionally accounted by the variance-covariance matrix of the asymptotic distribution. However, this measure can underestimate the true variance, specially for small samples. Traditionally, optimal design theory pays attention to this variance through its relationship with the model's information matrix. For this reason it seems convenient, at least in some cases, adapt the optimality criteria in order to get the best designs for the actual variance structure, otherwise the loss in efficiency of the designs obtained with the traditional approach may be very important.

Keywords: correlated observations, information matrix, optimality criteria, variance-covariance matrix

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4654 Effects of the Mass and Damping Matrix Model in the Non-Linear Seismic Response of Steel Frames

Authors: Alfredo Reyes-Salazar, Mario D. Llanes-Tizoc, Eden Bojorquez, Federico Valenzuela-Beltran, Juan Bojorquez, Jose R. Gaxiola-Camacho, Achintya Haldar

Abstract:

Seismic analysis of steel buildings is usually based on the use of the concentrated mass (ML) matrix and the Rayleigh damping matrix (C). Similarly, the initial stiffness matrix (KO) and the first two modes associated with lateral vibrations are commonly used to develop matrix C. The evaluation of the accuracy of these practices for the particular case of steel buildings with moment-resisting steel frames constitutes the main objective of this research. For this, the non-linear seismic responses of three models of steel frames, representing low-, medium- and high-rise steel buildings, are considered. Results indicate that if the ML matrix is used, shears and bending moments in columns are underestimated by up to 30% and 65%, respectively when compared to the corresponding results obtained with the consistent mass matrix (MC). It is also shown that if KO is used in C instead of the tangent stiffness matrix (Kt), axial loads in columns are underestimated by up to 80%. It is concluded that the consistent mass matrix should be used in the structural modelling of moment-resisting steel frames and that the tangent stiffness matrix should be used to develop the Rayleigh damping matrix.

Keywords: moment-resisting steel frames, consistent and concentrated mass matrices, non-linear seismic response, Rayleigh damping

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4653 Effects of Matrix Properties on Surfactant Enhanced Oil Recovery in Fractured Reservoirs

Authors: Xiaoqian Cheng, Jon Kleppe, Ole Torsæter

Abstract:

The properties of rocks have effects on efficiency of surfactant. One objective of this study is to analyze the effects of rock properties (permeability, porosity, initial water saturation) on surfactant spontaneous imbibition at laboratory scale. The other objective is to evaluate existing upscaling methods and establish a modified upscaling method. A core is put in a container that is full of surfactant solution. Assume there is no space between the bottom of the core and the container. The core is modelled as a cuboid matrix with a length of 3.5 cm, a width of 3.5 cm, and a height of 5 cm. The initial matrix, brine and oil properties are set as the properties of Ekofisk Field. The simulation results of matrix permeability show that the oil recovery rate has a strong positive linear relationship with matrix permeability. Higher oil recovery is obtained from the matrix with higher permeability. One existing upscaling method is verified by this model. The study on matrix porosity shows that the relationship between oil recovery rate and matrix porosity is a negative power function. However, the relationship between ultimate oil recovery and matrix porosity is a positive power function. The initial water saturation of matrix has negative linear relationships with ultimate oil recovery and enhanced oil recovery. However, the relationship between oil recovery and initial water saturation is more complicated with the imbibition time because of the transition of dominating force from capillary force to gravity force. Modified upscaling methods are established. The work here could be used as a reference for the surfactant application in fractured reservoirs. And the description of the relationships between properties of matrix and the oil recovery rate and ultimate oil recovery helps to improve upscaling methods.

Keywords: initial water saturation, permeability, porosity, surfactant EOR

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4652 On the Construction of Lightweight Circulant Maximum Distance Separable Matrices

Authors: Qinyi Mei, Li-Ping Wang

Abstract:

MDS matrices are of great significance in the design of block ciphers and hash functions. In the present paper, we investigate the problem of constructing MDS matrices which are both lightweight and low-latency. We propose a new method of constructing lightweight MDS matrices using circulant matrices which can be implemented efficiently in hardware. Furthermore, we provide circulant MDS matrices with as few bit XOR operations as possible for the classical dimensions 4 × 4, 8 × 8 over the space of linear transformations over finite field F42 . In contrast to previous constructions of MDS matrices, our constructions have achieved fewer XORs.

Keywords: linear diffusion layer, circulant matrix, lightweight, maximum distance separable (MDS) matrix

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4651 On the Algorithmic Iterative Solutions of Conjugate Gradient, Gauss-Seidel and Jacobi Methods for Solving Systems of Linear Equations

Authors: Hussaini Doko Ibrahim, Hamilton Cyprian Chinwenyi, Henrietta Nkem Ude

Abstract:

In this paper, efforts were made to examine and compare the algorithmic iterative solutions of the conjugate gradient method as against other methods such as Gauss-Seidel and Jacobi approaches for solving systems of linear equations of the form Ax=b, where A is a real n×n symmetric and positive definite matrix. We performed algorithmic iterative steps and obtained analytical solutions of a typical 3×3 symmetric and positive definite matrix using the three methods described in this paper (Gauss-Seidel, Jacobi, and conjugate gradient methods), respectively. From the results obtained, we discovered that the conjugate gradient method converges faster to exact solutions in fewer iterative steps than the two other methods, which took many iterations, much time, and kept tending to the exact solutions.

Keywords: conjugate gradient, linear equations, symmetric and positive definite matrix, gauss-seidel, Jacobi, algorithm

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4650 On Algebraic Structure of Improved Gauss-Seide Iteration

Authors: O. M. Bamigbola, A. A. Ibrahim

Abstract:

Analysis of real life problems often results in linear systems of equations for which solutions are sought. The method to employ depends, to some extent, on the properties of the coefficient matrix. It is not always feasible to solve linear systems of equations by direct methods, as such the need to use an iterative method becomes imperative. Before an iterative method can be employed to solve a linear system of equations there must be a guaranty that the process of solution will converge. This guaranty, which must be determined a priori, involve the use of some criterion expressible in terms of the entries of the coefficient matrix. It is, therefore, logical that the convergence criterion should depend implicitly on the algebraic structure of such a method. However, in deference to this view is the practice of conducting convergence analysis for Gauss-Seidel iteration on a criterion formulated based on the algebraic structure of Jacobi iteration. To remedy this anomaly, the Gauss-Seidel iteration was studied for its algebraic structure and contrary to the usual assumption, it was discovered that some property of the iteration matrix of Gauss-Seidel method is only diagonally dominant in its first row while the other rows do not satisfy diagonal dominance. With the aid of this structure we herein fashion out an improved version of Gauss-Seidel iteration with the prospect of enhancing convergence and robustness of the method. A numerical section is included to demonstrate the validity of the theoretical results obtained for the improved Gauss-Seidel method.

Keywords: linear algebraic system, Gauss-Seidel iteration, algebraic structure, convergence

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