Chemical Engineering Hand Written_Notes_Part_41

Chemical Engineering Hand Written_Notes_Part_41 - 84 3...

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84 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES which are signi f cantly easy to evaluate than the spectral radius. Also, if the matrix A has some special properties, such as diagonal dominance or symme- try and positive de f niteness, then we can derive easily computable criteria by exploiting these properties. Definition 26 . Ama tr ix A is called diagonally dominant if (4.74) n X j =1( j 6 = i ) | a ij | < | a ii | for i =1 , 2 ., . ..n Theorem 5 . [ 5 ] Asu cient condition for the convergence of Jacobi and Gauss-Seidel methods is that the matrix A of linear system Ax = b is diagonally dominant. Proof: See Appendix. Theorem 6 . [ 4 ] The Gauss-Seidel iterations converge if matrix A an sym- metric and positive de f nite. Proof: See Appendix. Theorem 7 . [ 15 ] For an arbitrary matrix A , the necessary condition for the convergence of relaxation method is 0 <ω< 2 . Proof: The relaxation iteration equation can be given as (4.75) x ( k +1) =( D + ωL ) 1 £ [(1
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Chemical Engineering Hand Written_Notes_Part_41 - 84 3...

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