Solving Large Systems of Linear Simultaneous Equations

Solving hulky Systems of one-dimensional synchronal EquationsNICOLE LESIRIMAMETHODS OF SOLVING LARGE SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONSPROJECT DESCRIPTIONLinear systems simulate realistic- military man problems utilise use numerical procedure. The main use up of this roll is to consider what factors affect the efficiency of the non-homogeneous manners of understand elongated synchronic equations. So far, one of the main factors is rounding errors that skunk rise a focussing solutions. Moreover, MATLAB programs have been produced to date the computer science speed to desex the efficiency of the regularitys. Gener exclusivelyy, these methods atomic number 18 subdivided into ii direct and iterative methods. Direct methods be commonly employ to solve small systems of equations. The iterative methods are used to solve real-world problems that produce systems of equations for which the co high-octane matrices are sparse.The relevance of studying these methods h ave its real world applications. The real world applications john be seen in various fields much(prenominal)(prenominal) as science and engineering, accounting and finance, business management and in operable research. The move up provides a logical framework for solving complex decisions in a wide range of industries. The advantage is that, decisions are founded on entropy analysis.Environmentalists and meteorologists may use banging systems of synchronic elongated equations to predict coming(prenominal) outcomes. For instance, to predict weather patterns or temper change, a tumid stack of data is collected over a long span of age on many variables including, solar radiation, carbon emissions and ocean temperatures. Queen bloody shame University of London (2015). This data is represented in the take of a change ground substance that has to be row reduced into a probability intercellular substance that can then be used in the prediction of climate change.The objectiv e of an enterprise is to maximize returns while maintaining minimum costs. Whereas the use of large systems of coinciding linear equations may provide a basis for shew based business decision making in an enterprise, it is important to make love which linear systems are nigh appropriate in order to decrease undesirable outcomes for an enterprise.PROJECT REPORT OUTLINEChapter 1 conceptionLarge systems of linear simultaneous equations are used to simulate real-world problems using applied numerical procedure. The real world applications can be seen in various fields such as science and engineering, accounting and finance, business management. The approach provides a logical framework for solving complex decisions in a wide range of industries. The advantage is that decisions are founded on data analysis. The aim of this project is to explore the efficiency of a large systems of linear simultaneous equations in the optimal decision making of an enterprise.Chapter 2Direct Methods Ga ussian Elimination and LU factorizationDirect methods of solving linear simultaneous equations are put ind. This chapter allow look at the Gaussian Elimination and LU Factorisation methods. Gaussian Elimination involves representing the simultaneous equations in an increase mannikin, playing uncomplicated row operations to reduce the upper triangular get up and finally choke substituting to phase the solution vector. LU Factorisation on the other hand is where a hyaloplasm A finds a set down triangular intercellular substance L and an upper triangular intercellular substance U such that A = LU. The purpose of this lower triangular intercellular substance and upper triangular matrix is so that the forward and self-referent substitutions can be directly applied to these matrices to obtain a solution to the linear system. An operation count and computing times using MATLAB is calculated so as to determine the best method to use.Chapter 3Cholesky Factorisation instauration t o the Cholesky method. This is a procedure whereby the matrix A is factorised into the product of a lower triangular matrix and its transpose the forward and backward substitutions can be directly applied to these matrices to obtain a solution. A MATLAB program is compose to compute timings. A conclusion can be drawn by comparing the three methods and determining which is the most suitable method that testament produce the most accu tread result as well as take the shortest computing time.Chapter 4Iterative Methods Jacobi Method and Gauss-SeidelThis chapter will introduce the iterative methods that are used to solve linear systems with coefficient matrices that are large and sparse. Both methods involve splitting the matrix A into lower triangular, accident and upper triangular matrices L, D, U respectively. The main difference comes down to the way the x pass judgment are calculated. The Jacobi method uses the previous x values (n) to calculate the next ite arranged x values (n +1). The Gauss-Seidel uses the new x value (n+1) to calculate the x2 value.Chapter 5Successive everyplace Relaxation and Conjugate slopeOther iterative methods are introduced. The Successive Over Relaxation method over relaxes the solution at each iteration. This method is calculated using the weighted sum of the values from the previous iteration and the values form the Gauss-Seidel method at the current iteration. The Conjugate Gradient method involves meliorate the approximated value of xk to the exact solution which may be reached after a finite number of iterations usually smaller than the size of the matrix.Chapter 6 demonstrationAll the project purposes and results are summarised in this chapter. Conclusion can be made from both direct methods and iterative methods whereby the most accurate method with the shortest computing time can be found. Drawbacks from each method will be mentioned as well its suitability for solving real world problems.PROGRESS TO DATEThe project to date has covered the direct methods of solving simultaneous equations.Gaussian EliminationThis involves representing the simultaneous equations in an increase form, perform elementary row operations to reduce the upper triangular form and finally back substituting to form the solution vector. For example, to solve an mxn matrixAx = bThe aim of the Gaussian elimination is to manipulate the augmented matrix Ab using elementary row operations by adding a multiple of the pivot rows to the rows beneath the pivot row i.e. Ri Ri +kRj. Once the augmented matrix is in the row echelon form, the solution is found using back substitution.The following general matrix equation has been reduced to row echelon formThis corresponds to the linear systemRearranging the final solution is given byFor all other equations i = n 1, . . .,The operation count and timing the Gaussian Elimination was performed. The total number of operations for an nxn matrix using the Gaussian elimination is with O(N3).LU FactorisationThis is where a matrix A finds a lower triangular matrix L and an upper triangular matrix U such that A = LU. The purpose of this lower triangular matrix and upper triangular matrix is so that the forward and backward substitutions can be directly applied to these matrices to obtain a solution to the linear system.In general,L and U is an m x n matrixL = U = For higher order matrices, we can derive the calculation of the L and U matrices. Given a set of n elementary matrices E1, E2,, Enapplied to matrix A, row reduce in row echelon form without permuting rows such that A can be written as the product of two matrices L and U that isA = LU,WhereU = EnE2E1A,L = E1-1 E2-1En-1 For a general nxn matrix, the total number of operations is O(N3). A Matlab program has been produced to time the LU Factorisation. So far, this method has proven more efficient than the Gaussian Elimination.Cholesky FactorisationThis is a procedure whereby the matrix A is factorised into the product of a lower triangular matrix and its transpose i.e. A = LLT or = The Cholesky factorisation is only possible if A is a positive definite. Forward and backward substitution is employed in finding the solutions.The method was besides time at it can be concluded that it is the most effective and efficient direct method for solving simultaneous equations.The indirect methods have been introduced with a short outline of what each method entails.Work Still to be Completed As from the objectives layed out from the terms of reference, the following are the objectives that are yet to be completed.Week 13 16 Evaluating the convergence rate of the iterative methods in detail as well as finding out which method improves the solution efficiency. Production of MATLAB programs analysing the different methods and other methods. Over the next 3 weeks, the bods for convergence will be analysed. One of the most important conditions that will be studied is the spectral radius. This is a conditio n applied on the indirect methods to determine how fast or loath a method takes to achieve the state of convergence. Moreover, the project will also produce Matlab programs for the iterative methods and employ the spectral radius on these programs to determine the speed of convergence for large sparse matrices.Weeks 17 19 Introduction to the Successive Over-Relaxation (SOR) method and the Conjugate Gradient method. Successive Over-Relaxation method improves the rate of convergence of the Gauss-Siedel method by over-relaxing the solution at every iteration. bit the Conjugate Gradient improves the approximated value of x to the exact solution. Matlab programs will be produced for the two methods together with the speed of convergence of different sizes of matrices.Week 20 24Writing the findings and conclusions of the report, finalising on the bibliography and doing a review of the project as a whole. Preparing oral and poster presentation.