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Chemical Engineering Hand Written_Notes_Part_57

# Chemical Engineering Hand Written_Notes_Part_57 - 116 3...

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116 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES As matrix A is diagonally dominant, we have n X j =1( j 6 = i ) | a ij | < | a ii | for i = 1 , 2 ., ...n (9.3) n X j =1( j 6 = i ) ¯ ¯ ¯ ¯ a ij a ii ¯ ¯ ¯ ¯ < 1 for i = 1 , 2 ., ...n (9.4) ° ° S 1 T ° ° = max i n X j =1( j 6 = i ) ¯ ¯ ¯ ¯ a ij a ii ¯ ¯ ¯ ¯ < 1 (9.5) Thus, Jacobi iteration converges if A is diagonally dominant. For Gauss Seidel iterations, the iteration equation for i’th component of the vector is given as (9.6) x ( k +1) i = μ 1 a ii " b i i 1 X j =1 a ij x ( k +1) j n X j = i +1 a ij x ( k ) j # Let x denote the true solution of A x = b . Then, we have (9.7) x i = μ 1 a ii " b i i 1 X j =1 a ij x j n X j = i +1 a ij x j # Subtracting (9.7) from (9.6), we have (9.8) x ( k +1) i x i = μ 1 a ii " i 1 X j =1 a ij ³ x j x ( k +1) j ´ + n X j = i +1 a ij ³ x j x ( k ) j ´ # or (9.9) ¯ ¯ ¯ x ( k +1) i x i ¯ ¯ ¯ " i 1 X j =1 ¯ ¯ ¯ ¯ a ij a ii ¯ ¯ ¯ ¯ ¯ ¯ ¯ ³ x j x ( k +1) j ´¯ ¯ ¯ + n X j = i +1 ¯ ¯ ¯

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Chemical Engineering Hand Written_Notes_Part_57 - 116 3...

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