Chemical Engineering Hand Written_Notes_Part_58

Chemical Engineering Hand Written_Notes_Part_58 - 118 3...

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118 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES 9.2. Proof of Theorem 3. For Gauss-Seidel method, when matrix A is symmetric, we have (9.26) S 1 T = ( L + D ) 1 ( U ) = ( L + D ) 1 ( L T ) Now, let e represent unit eigenvector of matrix S 1 T corresponding to eigen- value λ , i.e. ( L + D ) 1 ( L T ) e = λ e (9.27) or L T e = λ ( L + D ) e (9.28) Taking inner product of both sides with e , we have ­ L T e , e ® = λ h ( L + D ) e , e i (9.29) λ = ­ L T e , e ® h D e , e i + h L e , e i = h e ,L e i h D e , e i + h L e , e i (9.30) De fi ning α = h L e , e i = h e ,L e i (9.31) σ = h D e , e i = n X i =1 a ii ( e i ) 2 > 0 (9.32) we have (9.33) λ = α α + σ | λ | = ¯ ¯ ¯ ¯ α α + σ ¯ ¯ ¯ ¯ Note that σ > 0 follows from the fact that trace of matrix A , is positive as eigenvalues of A are positive. Using positive de fi niteness of matrix A, we have h A e , e i = h L e , e i + h D e , e i + ­ L T e , e ® (9.34) = σ + 2 α > 0 (9.35) This implies (9.36) α < ( σ + α ) Since σ > 0 , we can say that (9.37) α < ( σ + α ) i.e.
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