A Study of Numerical Methods for Simultaneous Equations

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A Study of Numerical Methods for Simultaneous Equations Er. Chandan Krishna Mukherjee B.Sc.Engg., ME, MBA Asstt. Prof. ( Mechanical ), SSBT s College of Engg. & Tech.

1 A Study of Numerical Methods for Simultaneous Equations Er. Chandan Krishna Mukherjee B.Sc.Engg., ME, MBA Asstt. Prof. ( Mechanical ), SSBT s College of Engg. & Tech., Jalgaon, Maharashtra Abstract: - Engineers are often confronted with the problem of finding a solution for a set of simultaneous equations involving multiple variables. Traditional mathematical methods are not amenable to computerization & hence too cumbersome to use in case of large number of variables. Numerical methods are programmable, can deliver solutions rapidly with reasonable accuracy & hence provide a viable alternative for handling such problems. In this paper the working procedure of different numerical methods have been amply described, using simple illustration and compared. Keywords:-, Gauss Elimination, Gauss-Jordan, Doo Little, Crout, Cholesky, Jacobi, Gauss-Seidel methods I. INTRODUCTION A set of simultaneous equations involves n algebraic equations in n unknown variables. In cases where n 3, traditional mathematical methods like elimination & re-substitution method or Kramer s rule, taught in UG level, may be conveniently used. However these methods are not programmable & is tedious to use for solving cases wherein n > 3. Engineers are frequently confronted with the problem of solving simultaneous equations involving large number of variables. Finding solutions by use of traditional mathematical techniques is daunting & tedious. In these cases numerical methods, which are programmable, can be much more convenient applied to find the solution rapidly. Numerical methods available may be classified according to the approach used as follows a. Direct approach ( ie. Gauss Elimination & Gauss Jordan methods) b. Factorization approach ( ie. LU decomposition & Cholesky methods ) and c. Iterative approach ( ie. Jacobi & Gauss Seidel methods ) The working procedure of these methods are described & compared below. 1. DIRECT APPROACH METHODS This approach may be considered as a programmable version of the elimination & re-substitution method. The set of simultaneous equations is first represented as an augmented matrix. Eg. Equations X + Y = 6 X Y = 4 may be represented by the augmented matrix [ ] Please note that the LHS elements of the n th row are the co-efficients of the n variables of the n th Eq n, the constant term being placed in RHS of the augmented matrix. This augmented matrix is then reduced, using basic matrix transformation operation, to either an upper triangular matrix ( Gauss DOI : /IJRTER WNISA 208

2 elimination method ) or an unit matrix ( Gauss Jordan method ). The working procedure of these 2 methods is described below using the above example. Gauss Elimination method :- [1],[2],[3] On row 2 of the above augmented matrix, the operation [ R2 ( a 21 a11 ) R1 ] is performed to obtain this upper triangular matrix [ ] Please note that if the pivot element a11 is zero the program will stall & the situation has to be corrected by interchanging rows if necessary. From row 2 we get - 2 Y = - 2 or, Y = 1 Back Substituting value of Y in row 1, we have X + 1 = 6 or, X = 5 Gauss Jordan method :- [1],[2],[3] On row 1 of the above upper triangular matrix, another operation [ R1 ( a 12 a22 ) R2 ] is performed to obtain this unit matrix [ ] From row 1 we directly get X = 5. Additional computation in back substitution is not required, from row 2 we directly get Y = FACTORIZATION APPROACH METHODS The general procedure in this approach involves a finite no. of steps described below - Step 1 - The system is represented as [ A ] [ X ] = [ B ]. ( 1 ) where [ A ] is a n x n coefficient matrix, [ B ] is a n x 1 R.H.S. constant matrix. [ X ] is a n x 1 matrix having elements X1, X2,, Xn variables Step 2 - The coefficient matrix [ A ] is decomposed into an n x n lower triangular matrix [ L ] and an n x n upper triangular matrix [ U ], such that [ L ] [ U ] = [ A ] ( 2 ) Equation ( 1 ) can hence be re written as [ L ] [ U ] [ X ] = [ B ] which can be further represented as [ L ] [ V ] = [ B ].. ( 3 ) where dummy variable matrix [ V ] is given by [ U ] [ X ] = [ V ]. ( 4 ) The elements of [ L ] and [ U ] are determined from eqn ( 2 ) using elementary rules of matrix multiplication. Step 3 - Using elementary rules of matrix multiplication on relation ( 3 ), elements of the n x 1 dummy variable matrix [ V ] is determined. Step 4 - Using elementary rules of matrix multiplication on relation ( 4 ) the elements of [ X ], is derived, which is the required All Rights Reserved 209

3 Different forms of lower & upper triangular matrices used gives rise to different methods - 1. LU Decomposition method a) Doo Little method [1],[2] b) Crout method [3] 2. Cholesky method [1] Doo Little method :- Here [ L ] = [ L 11 0 ] and [ U ] = [ 1 U 12 L 21 L ] In step 2, from Eqn ( 2 ), we have [ L 11 0 L 21 L 22 ] [ 1 U ] = [ ] Row 1 of [ L ] x Column 1 of [ U ] yields 1 st. row 1 st. column element of [ A ], giving L11 = 1. Row 1 of [ L ] x Column 2 of [ U ] yields L11 * U12 = 1, giving U12 = 1 Row 2 of [ L ] x Column 1 of [ U ] yields L21 = 1 Row 2 of [ L ] x Column 2 of [ U ] yields L21 * U12 + L22 = - 1, giving L22 = - 2 In step 3, from Eqn ( 3 ), we have [ ] [ V 1 V 2 ] = [ 6 4 ] Row 1 of [ L ] x Column 1 of [ V ] yields V1 = 6 and Row 2 of [ L ] x Column 1 of [ V ] yields V1 2 V2 = 4, giving V2 = 1 In step 4, from Eqn ( 4 ), we have [ ] [ X 1 X 2 ] = [ 6 1 ] Row 2 of [ U ] x Column 1 of [ X ] yields X2 = 1 Row 1 of [ U ] x Column 1 of [ X ] yields X1 + X2 = 6, giving X1 = 5 Crout method :- Here [ L ] = [ In step 2, from eqn ( 2 ), we have [ 1 0 L 11 1 ] and [ U ] = [ U 11 U 12 0 U 22 ] 1 0 L 21 1 ] [ U 11 U 12 0 U 22 ] = [ ] Row 1 of [ L ] x Col 1 of [ U ] yields U11 = 1, Row 1 of [ L ] x Col 2 of [ U ] yields U12 = 1, Row 2 of [ L ] x Column 1 of [ U ] yields L21 * U11 = 1, giving L21 = 1, Row 2 of [ L ] x Column 2 of [ U ] yields L21 * U12 + U22 = -1, giving U22 = - 2 In step 3, from Eqn ( 3 ), we have [ ] [ V 1 V 2 ] = [ 6 4 All Rights Reserved 210

4 Row 1 of [ L ] x Column 1 of [ V ] gives V1 = 6 Row 2 of [ L ] x Column 1 of [ V ] yields V1 + V2 = 4, giving V2 = - 2 In step 4, from Eqn ( 4 ), we have [ ] [ X 1 6 ] = [ X 2 2 ] Row 2 of [ U ] x Column 1 of [ X ] gives -2 X2 = -2, ie. X2 = 1 Row 1 of [ U ] x Column 1 of [ X ] yields X1 + X2 = 6, giving X1 = 5 Cholesky Method :- [1] This method may be considered as a special type of LU decomposition method, applicable for cases where the coefficient matrix [ A ] is a symmetrical positive definite matrix. For a symmetrical matrix [ A ] = [ A ] T or a ij = a ji for all i, j For a positive definite matrix [ X ] T [ A ] [ X ] > 0 for all non zero vectors [ X ] General procedure of Cholesky method is similar to that of factorization method except for the fact that we consider [ U ] = [ L ] T or U ij = L ji for all i, j. The method is illustrated below Eg. For Equations X 1 + X 2 = 2 X X 2 = 6 ; [ A ] = [ ] which is symmetrical Here we take [ L ] = [ L 11 0 L 21 L 22 ] and [ U ] = [ L 11 L 21 0 L 22 ] In step 2, from Eqn ( 2 ), we have [ L 11 0 L 21 L 22 ] [ L 11 L 21 0 L 22 ] = [ ] Row 1 of [ L ] x Col 1 of [ U ] yields L = 1 or L 11 = 1. Since [ A ] is positive definite, L11 will never be imaginary. Either value may be taken. Row 1 of [ L ] x Col 2 of [ U ] or Row 2 of [ L ] x Col 1 of [ U ] gives L11 * L21 = 1. If L11 is taken positive, L21 = 1 if L11 is taken negative, L21 = - 1. Row 2 of [ L ] x Column 2 of [ U ] yields L L = 5 giving L 22 = 2 For { L11, L21, L22 } the values { 1, 1, 2 }, { 1, 1, -2 }, { -1, -1, 2 }, { -1, -1, -2 } are obtainable, any of which may be taken for proceeding to next step, giving same end result. In step 3, from Eqn ( 3 ), using 4 th set of values, we have [ ] [ V 1 ] = [ 2 V 2 6 ] Row 1 of [ L ] x Column 1 of [ V ] gives - V1 = 2 or V1 = - 2 Row 2 of [ L ] x Column 1 of [ V ] yields V1 2 V2 = 6 or V2 = - All Rights Reserved 211

5 1 1 In step 4, from Eqn ( 4 ), we have [ 0 2 ] [ X 1 ] = [ 2 X 2 2 ] Row 2 of [ U ] x Column 1 of [ X ] gives 2 X2 = - 2 or X2 = 1 Row 1 of [ U ] x Column 1 of [ X ] yields X1 X2 = - 2 or X1 = 1 II. ITERATIVE APPROACH METHODS In an iterative approach, starting from an initial trial solution, a better solution is expected after each iteration, ultimately converging to a reasonably close approximate solution. To ensure convergence, the equations are first re-arranged so that all the large coefficients of each variable lie on the leading diagonal of the co efficient matrix [ A ]. Subsequently, relation for finding a variable is obtained from that equation having largest co-efficient for that variable, as illustrated below [1],[2],[3] Eg. Equations 4 X 10 Y + 3 Z = 3 10 X 5 Y 2 Z = 3 X + 6 Y + 10 Z = 3 re arranged 10 X 5 Y 2 Z = 3 4 X 10 Y + 3 Z = 3 X + 6 Y + 10 Z = Check co efficient matrix [ A ] = [ ] All large values are on leading diagonal X 5 Y 2 Z = 3 X = [ Y + 2 Z ] / 10.. ( 4 ) Obtain relation 4 X 10 Y + 3 Z = 3 Y = [ X + 3 Z ] / 10.. ( 5 ) X + 6 Y + 10 Z = 3 Z = [ 3 X 6 Y ] / 10. ( 6 ) Jacobi Method :- The iteration is started with an initial solution X0 = Y0 = Z0 = 0 being substituted in the relations 4, 5 & 6 respectively to obtain X1 = 0.3, Y1 = 0.3, Z1 = which are re - substituted in the relations 4, 5 & 6 to obtain even better values X 2 = 0.39, Y2 = 0.33, Z2 = , etc. Iteration is continued until no significant improvement is achieved. After 9 iterations we have X9 = 0.342, Y9 = 0.285, Z9 = Gauss Seidel Method :- The iteration is started with initial solution Y0 = Z0 = 0 being substituted in relation 4 to obtain X1 = 0.3. X1 & Z0 is substituted in relation 5, to obtain Y1 = X1 & Y1 is substituted in relation 6 to obtain Z1 = Iteration is continued in the same sequence to obtain even better values till no further improvement is achieved. After 5 iterations we have X 5 = 0.342, Y 5 = 0.285, Z5 = Thus this method is more efficient than Jacobi method. III. CONCLUSION In this paper, the working procedures of all the numerical methods have been amply described using simple illustration. While comparing the different numerical methods, it is found that iterative methods usually give fast, reasonably accurate results with minimum computational efforts but sometimes convergence may be slow or may not occur. Direct methods give more accurate results but are plagued by division by zero situations, require more computational efforts & memory space. Factorization methods give most accurate results, with less computational efforts & memory space. All the above methods are programmable & hence can be run on a computer. It is hoped that this paper will help generate interest in the subject as well as act as a ready reference for engineering students & All Rights Reserved 212

6 REFERENCE 1. S.C. Chapra & Raymond P. Canale : Numerical Methods for Engineers ; Tata McGraw Hill publication 2. B.S. Grewal : Higher Engineering Mathematics, Numerical Methods ; Khanna publication 3. P. Kandasamy, K. Thilagavathy, K. Gunavathy : Numerical Methods ; S. Chand All Rights Reserved 213

3. Replace any row by the sum of that row and a constant multiple of any other row.

3. Replace any row by the sum of that row and a constant multiple of any other row. Math Section. Section.: Solving Systems of Linear Equations Using Matrices As you may recall from College Algebra or Section., you can solve a system of linear equations in two variables easily by applying