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A New Coding Method in MATLAB Used for Solving a System of N Linear Equations by LU Decomposition

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A New Coding Method in MATLAB Used for Solving a System of N Linear Equations by LU Decomposition International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 02 Issue: 02 | May-2015 www.irjet.net p-ISSN: 2395-0072 A new Coding method in MATLAB used for solving a System of n Linear Equations by LU Decomposition Digvijay Singh1, Dr L. N. Das2 1 2Department of Applied Mathematics,Delhi Technological University,Shahabad Daulatpur, Delhi,INDIA [email protected] ---------------------------------------------------------------------***--------------------------------------------------------------------- Abstract - MATLAB coding which we have executed is 1.2 Required Concepts not restricted to LU Decomposition only; rather it can be 2.1Rank -The rank of a matrix A is the size of the largest used for the matrix computation operations such as collection of linearly independent columns of Cholesky method etc. In the existing MATLAB coding, to 2.2Coefficient Matrix - The coefficient matrix refers to a solve a system of n linear equations using LU Decomposition, matrix consisting of the coefficients of the variables in a a pivot matrix is used to reassign the largest element of each set of linear equation. column to the principal diagonal in the coefficient matrix. 2.3Augmented Matrix -An augmented matrix is a matrix Whereas in our MATLAB coding the pivot matrix is not obtained by appending the columns of coefficient matrix required. In this paper we have executed a MATLAB coding with the constant matrix, usually for the purpose of for the solution of a system of n linear equations using LU performing the same elementary row operations on each Decomposition of the given matrices. Key Words: LU decomposition 2.4Inverse of a Matrix -For a square matrix A, the inverse 1.1 Introduction is written A-1. When A is multiplied by A-1 the result is the MATLAB is a highly resourceful and useful tool used identity matrix I. Non-square matrices do not have by a majority of the academic community to create inverses. Not all square matrices have inverses. models, simulations and solve many problems from 2.5Zero Matrix -A zero matrix is the additive identity of the fields of engineering, science and mathematics. It the additive group of matrices. is particularly known for its ease in handling arrays and matrices. A lot of in-built functions are provided 2.6Identity Matrix -A square matrix in which all the to make computation easy. elements of the principal diagonal are ones and all other elements are zeros. The effect of multiplying a given In this paper, we have defined a MATLAB coding that matrix by an identity matrix is to leave the given matrix is useful to solve a system of n linear equations with unchanged. n variables. In the literature[1], we find that a matrix called pivot matrix is used to obtain the solution 2.7 Positive Definite Matrix - Asymmetricn accordingly. In the pivot matrix multiplication, the xnrealmatrixA is said to be positive definite if zTAz is main aim is to rearrange the columns elements of the positive for every non-zero column vectorz of n real coefficient matrix, so that the largest element of each numbers. Here zT denotes the transpose of z. column falls on the principal diagonal. This pivotization is more time taking. We suggest a new 2.8Matrix Multiplication - method to overcome these difficulties. In this paper, we have presented the brief definition and MATLAB Let Am x p and Bp x n be two matrices whose elements are coding of the LU Decomposition of the coefficient represented by, matrix of the linear equations, and using this decomposition have found the solution of the system A = [aij] & B = [bij] for i = 1,2,3,… m and j = 1,2,3,... n. of equations. Then their matrix product is given by[2], C = a * b ij ik kj © 2015, IRJET.NET- All Rights Reserved Page 626 International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 02 Issue: 02 | May-2015 www.irjet.net p-ISSN: 2395-0072 2.9System of n Linear Equations – A system of linear equations is a set or collection of equations that are dealt with all together at once. A system of m equations in n variables is defined as, The sufficient condition for a matrix A to be decomposable is positive definiteness. Now, for forming system of equations solvable for unique solution, one of the upper or lower triangular matrices should have 1’s in the diagonal elements. whereaij, bi, are real numbers and xj are variables for i = 1,2,…,m, j = 1,2,…,n. In matrix notation it is represented as, AX = b If upper triangular matrix, U has diagonal elements as 1, it is called Crout’s Method. Three cases arise to the solution of this system of equations: If lower triangular matrix, L has diagonal elements as 1, it is called Doolittle’s Method. AX = B 1. Unique solution 2. Infinite solutions Let A = LU 3. No solution LUX = B The uniqueness of the solution ofthissystem can be seen Let Z = UX using matrix algebra[3] by comparing the rank of the coefficient matrix A and augmented matrix K as follows LZ = B Z = L-1B (where n is the number of variables/unknowns) Here, A is the coefficient matrix, B is the constant matrix, 1. If rank(A) = rank(K) = n, there exists a unique X is the variable matrix (whose elements are to be solution. determined), L is the lower triangular matrix (whose 2. If rank(A) = rank(K) < n, there exist infinitely elements are to be determined) and U is the upper many solutions. triangular matrix (whose elements are to be determined). There are various methods to solve this system of 2.1 MATLAB Functions equations like substitution, cross-multiplication, matrix The following MATLAB functions[5] have been used in algebra methods etc. We concentrate here on LU the below defined coding: Decomposition method. size(X)returns the row and column size of matrix X. 1.3 LU Decomposition Method In LU Decomposition[4], every square matrix A can be zeros(m)defines a square zero matrix of size m. decomposed into a product of a lower triangular matrix L eye(m)defines an identity matrix of size m. and an upper triangular matrix U. A = LU solve(f, x)solves an equation f for variable x. 2.2 MATLAB Codes Below is provided the MATLAB R2014b (version 8.4.0)code written for Crout’s Method. © 2015, IRJET.NET- All Rights Reserved Page 627 International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 02 Issue: 02 | May-2015 www.irjet.net p-ISSN: 2395-0072 1function [ L, U, X ] = LU_Crout( A, b ) 30 end 2 31end 3% Function to find solution of AX = b by Crout's 32 33 4% LU Decomposition method. 34 symstemp; 5% Input = A, b 35 6% Output = L, U, X 36 L = sym(zeros(n(1))); 7 37 U = sym(eye(n(1))); 8 n = size(A); 38 9 39 T = sym(zeros(n(1))); 10 flag = 1; 40 11 41fori = 1:n(1) 12if( n(1) ~= n(2) ) 42 for j = 1:n(1) 13 fprintf('\nNot square matrix\n'); 43 if( i>= j ) 14 flag = 0; 44 L(i,j) = temp; 15 return; 45 elseU(i,j) = temp; 16end 46 end 17 47 for k = 1:n(1) 18if( flag == 1 ) 48 T(i,j) = T(i,j) + L(i,k)*U(k,j); 19 K = [A b]; 49 end 20end 50 eq_ans = solve(T(i,j) == A(i,j), temp); 21 51 T(i,j) = eq_ans; 22if( flag == 1 ) 52 if( i>= j ) 23 if( rank(A) ~= rank(K) ) 53 L(i,j) = T(i,j); 24 fprintf('\nNo solution'); 54 elseU(i,j) = T(i,j); 25 return; 55 end 26 elseif( rank(A) == rank(K) && rank(A) < n(1) ) 56 end 27 fprintf('\nInfinite solutions') 57end 28 return; 58 29 end © 2015, IRJET.NET- All Rights Reserved Page 628 International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 02 Issue: 02 | May-2015 www.irjet.net p-ISSN: 2395-0072 59 Z = L\b; % Z = inv(L)*b The condition in line 43-46 checks which variable is unknown in the current multiplication iteration. 60 This unknown is replaced by temp in the product equation. 61 X = U\Z; % X = inv(U)*Z Using solve(f, x) where x = temp, we get value of unknown. 62 At each iterative step, the calculated value of unknown is replaced in the L or U matrix. 63end After the final iteration, the unknown elements of L and U are replaced, and they can be used to find 2.3 Explanation of the codes the solution vector X. Line 1 defines a function LU_Crout that takes A In line 59, we find out Z = L-1B using the command Z = (coefficient matrix) and b (constant column vector) as L\B. Similarly, we find out X = U-1Z using the command X inputs, and gives decomposed lower and upper triangular = U\Z. matrices (L&U respectively), as well as X, solution to the system AX = b as output. Thus, solution to the system of equations is obtained.. Lines 8-31 are measures taken to check whether A is a 2.4. Advantages of Proposed method The main advantage of the written coding is to overcome square matrix. If not, then LU Decomposition cannot be the problems existing in the given MATLAB coding. used. These measures also check if the system has a MATLAB cannot currently handle symbolic matrices; unique solution or not, by comparing rank of coefficient rather the coding is supported by symbolic variables.
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