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Chemical Engineering Hand Written_Notes_Part_60

Chemical Engineering Hand Written_Notes_Part_60 - 122 3...

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122 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES (c) rounding o ff each number to 3 fi gures. Perform 4 iterations each by (a) Jacobi method (b) Gauss- Seidel method (c) Successive over-relaxation method with ω = 1 . 5 Use initial guess x (0) = h 1 1 1 i T and compare in each case how close to the x (4) is to the exact solution. (Use 2-norm for comparison). Analyze the convergence properties of the above three iterative processes using eigenvalues of the matrix ( S 1 T ) in each case. Which iteration will converge to the true solution? (16) The Jacobi iteration for a general 2 by 2 matrix has A = " a b c d # ; D = " a 0 0 d # If A is symmetric positive de fi nite, fi nd the eigenvalues of J = S 1 T = D 1 ( D A ) and show that Jacobi iterations converge. (17) It is desired to solve Ax = b using Jacobi and Gauss-Seidel iteration scheme where A = 4 2 1 1 5 3 2 4 7 ; A = 1 2 2 1 1 1 2 2 1 ; A = 7 1 2 3 1 8 1 3 2 1 5 1 1 0 1 3
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