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The Islamic University of GazaFaculty of EngineeringCivil Engineering DepartmentNumerical AnalysisChapter 11.Special Matrices and Gauss Siedel Symmetric Matrices Certain matrices have particular structuresthat can be exploited to develop efficientsolution schemes . Symmetric matrix 5 1 2 16 If aij aji A nxn is a 1 3 7 39 symmetric matrix A . 2 7 9 6 16 39 6 88 Cholesky Decompositionmethod is suitable for only symmetric systems l11 0 0 .a ij a ji and A A T L l 21 l 22 0 l 31 l 32 l 33 A L L a11 a12 a13 l11 0 0 l11 l 21 l 31 a a a23 l 21 l 22 0 0 l 22 l 32 . 21 22 a31 a32 a33 l 31 l 32 l 33 0 0 l 33 Cholesky DecompositionL11 a11 L22 a22 L21For 3x3 a21 a32 L21 L31.matrix L21 L32 a31 L33 a33 L232 L231 Once the matrix L has been calculated LT can easily be found and theforward backward substitution steps can be followed to find thesolution X for any right hand side B . Pseudocode for Cholesky s LUDecomposition algorithm cont d Gauss Siedel Iterative or approximate methods provide analternative to the elimination methods The Gauss .Seidel method is the most commonly used iterative The system A X B is reshaped by solving thefirst equation for x1 the second equation for x2 andthe third for x3 and nth equation for xn We willlimit ourselves to a 3x3 set of equations . Gauss Siedela11 x1 a12 x2 a13 x3 b1 b1 a12 x2 a13 x3 b2 a21 x1 a23 x3a21 x1 a22 x2 a23 x3 b2 x2 b3 a31 x1 a32 x2.a31 x1 a32 x2 a33 x3 b3 a33Now we can start the solution process by choosingguesses for the x s A simple way to obtain initialguesses is to assume that they are zero Thesezeros can be substituted into x1 equation to.calculate a new x1 b1 a11 Gauss Siedel New x1 is substituted to calculate x2 and x3 Theprocedure is repeated until the convergencecriterion is satisfied . a i new100 s Gauss Seidal Graphical Interpretation Jacobi iteration Method.An alternative approach called Jacobi iteration utilizes a somewhat different technique Thistechnique includes computing a set of new x s on thebasis of a set of old x s Thus as the new values aregenerated they are not immediately used but are.retained for the next iteration Gauss SiedelThe Gauss Seidel method The Jacobi iteration method Convergence Criterion for Gauss Seidel Method The gauss siedel method is similar to the technique of.fixed point iteration The Gauss Seidel method has two fundamentalproblems as any iterative method 1 It is sometimes non convergent and2 If it converges converges very slowly . Sufficient conditions for convergence of two linearequations u x y and v x y are Convergence Criterion for Gauss SeidelMethod cont d Similarly in case of two simultaneous equations the.Gauss Seidel algorithm can be expressed as u x 1 x 2 x2v x 1 x 2 x1 u u a12 0 . x 1 x 2 a11 v a21 v 0 x 1 a22 x 2 Convergence Criterion for Gauss Seidel.Method cont d Substitution into convergence criterion of two linearequations yield a12 a21 1 1In other words the absolute values of the slopes must be.less than unity for convergence For n equationsa11 a12 naii ai ja22 a21 j 1.That is the diagonal element must be greater than the sumof off diagonal element for each row Gauss Siedel Method Example 17 85 0 1x2 0 2 x3 3 0 1 0 2 x1 7 85 x1 3. 0 1 19 3 0 1x1 0 3 x3 7 0 3 x2 19 3 x2 7 0 3 0 2 10 x3 71 4 x3 71 4 0 3 x1 0 2 x2 Guess x1 x2 x3 zero for the first guess.Iter x1 x2 x3 a 1 a 2 a 3 0 0 0 0 1 2 6167 2 7945 7 005610 100 100 1002 2 990557 2 499625 7 000291 12 5 11 8 0 076 Improvement of Convergence Using.Relaxationnew new oldx i x i 1 x i Where is a weighting factor that is assigned a valuebetween 0 2 . If 1 the method is unmodified If is between 0 and 1 under relaxation this isemployed to make a non convergent system to If is between 1 and 2 over relaxation this isemployed to accelerate the convergence . Gauss Siedel Method Example 2The system can be re arranged to the following6x1 x2 x3 36x1 9x2 x3 40 3x1 x2 12x3 50. Gauss Siedel Method Example 23 x2 x340 6 x1 x350 3 x1 x2Without Relaxation. Gauss Siedel Method Example 2Faculty of Engineering Civil Engineering Department Numerical Analysis ECIV 3306 Chapter 11 Special Matrices and Gauss-Siedel Symmetric Matrices Certain matrices have particular structures that can be exploited to develop efficient solution schemes.