Chemical Engineering Hand Written_Notes_Part_42

# Chemical Engineering Hand Written_Notes_Part_42 - 86 3...

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

Chemical Engineering Hand Written_Notes_Part_42 - 86 3...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online