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Chemical Engineering Hand Written_Notes_Part_40

Chemical Engineering Hand Written_Notes_Part_40 - 82 3...

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82 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES where Λ is the diagonal matrix (4.63) Λ = λ 1 0 ..... 0 0 λ 2 0 ... .... .... ..... .... 0 .... 0 λ n Now, consider set of n equations (4.64) Bv ( i ) = λ i v ( i ) for ( i = 1 , 2 , .... n ) which can be rearranged as Ψ = h v (1) v (2) .... v ( n ) i B Ψ = λ 1 0 ..... 0 0 λ 2 0 ... .... .... ..... .... 0 .... 0 λ n Ψ (4.65) or B = ΨΛΨ 1 (4.66) Using above identity, it can be shown that (4.67) B k = ¡ ΨΛΨ 1 ¢ k = Ψ ( Λ ) k Ψ 1 and the solution of equation (6.54) reduces to (4.68) z ( k ) = B k z (0) and z ( k ) 0 as k → ∞ if and only if ρ ( B ) < 1 . The largest magnitude eigen value, i.e., ρ ( B ) will eventually dominate and determine the rate at which z ( k ) ¯ 0 . The result proved in this section can be summarized as follows: Theorem 3 . A sequence of vectors © z ( k ) : k = 0 , 1 , , 2 , .... ª generated by the iteration scheme z ( k +1) = Bz ( k ) where z R n and B R n × R n , starting from any arbitrary initial condition z (0) will converge to limit z = ¯ 0 if and only if ρ ( B ) < 1 Note that computation of eigenvalues is a computationally intensive task.
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