Chemical Engineering Hand Written_Notes_Part_42

Chemical Engineering Hand Written_Notes_Part_42 - 86 3....

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86 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES Thus, the error norm at each iteration is reduced by factor of 0.5 For Gauss-Seidel method S 1 T = " 01 / 2 / 4 # (4.90) ρ ( S 1 T )=1 / 4 (4.91) Thus, the error norm at each iteration is reduced by factor of 1/4. This implies that, for the example under consideration (4.92) 1 Gauss Seidel iteration 2 Jacobi iterations For relaxation method, S 1 T = " 20 ω 2 # 1 " 2(1 ω ) ω 02 ( 1 ω ) # (4.93) = (1 ω )( ω/ 2) ( 2)(1 ω 1 ω + ω 2 4 ) (4.94) λ 1 λ 2 =d e t ( S 1 T )=(1 ω ) 2 (4.95) λ 1 + λ 2 = trace ( S 1 T ) (4.96) =2 2 ω + ω 2 4 (4.97) Now, if we plot ρ ( S 1 T ) v/s ω , then it is observed that λ 1 = λ 2 at ω = ω opt . From equation (4.95), it follows that (4.98) λ 1 = λ 2 = ω opt 1 at optimum ω. Now, λ 1 + λ 2 ( ω opt 1) (4.99) 2 ω opt + ω 2 opt 4 (4.100) ω opt =4(2 3) = 1 . 07 (4.101) ρ ( S 1 T )= λ 1 = λ 2 = 0 . 07 (4.102) This is a major reduction in spectral radius when compared to Gauss-Seidel method. Thus, the error norm at each iteration is reduced by factor of 1/16
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Chemical Engineering Hand Written_Notes_Part_42 - 86 3....

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