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Chemical Engineering Hand Written_Notes_Part_39

# Chemical Engineering Hand Written_Notes_Part_39 - 80 3...

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80 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES 4.4.1. Eigenvalue Analysis. To begin with, let us consider scalar linear iter- ation scheme (4.45) z ( k +1) = b z ( k ) where z ( k ) R and b is a real scalar. It can be seen that (4.46) z ( k ) = ( b ) k z (0) 0 as k → ∞ if and only if | b | < 1 . To generalize this notation to a multidimensional case, consider equation of type (4.44) where z ( k ) R n . Taking motivation from the scalar case, we propose a solution to equation (4.44) of type (4.47) z ( k ) = λ k v where λ is a scalar and v R n is a vector. Substituting equation (4.47) in equation (6.54), we get λ k +1 v = B ( λ k v ) (4.48) or λ k ( λI B ) v = 0 (4.49) Since we are interested in a non-trivial solution, the above equation can be reduced to (4.50) ( λI B ) v = 0 where v 6 = 0 . Note that the above set of equations has n equations in ( n + 1) unknowns ( λ and n elements of vector v ) . Moreover, these equations are non- linear. Thus, we need to generate an additional equation to be able to solve the above set exactly. Now, the above equation can hold only when the columns of

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