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Chemical Engineering Hand Written_Notes_Part_52

Chemical Engineering Hand Written_Notes_Part_52 - 106 3...

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106 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES is not a contraction mapping near (1,1) and the iterations do not converge even if we start from a value close to the solution. On the other hand, the rearrangement x ( k + 1) = q ( y ( k ) + 2) / 3 (6.66) y ( k +1) = q ( x ( k ) + 1) / 2 (6.67) is a contraction mapping and solution converges if the starting guess is close to the solution. 6.3.2. Convergence Criteria for Iteration Schemes. De fi ning error (6.68) e ( k +1) = x ( k +1) x = G ( x ( k ) ) G ( x ) and using Taylor series expansion, we can write G ( x ) = G [ x ( k ) ( x ( k ) x )] (6.69) ' G ( x ( k ) ) ∂G x ¸ x = x ( k ) ( x ( k ) x ) (6.70) Substituting in (6.68) (6.71) e ( k +1) = ∂G x ¸ x = x ( k ) e ( k ) where e ( k ) = x ( k ) x and using de fi nition of induced matrix norm, we can write (6.72) || e ( k +1) || || e ( k ) || < ° ° ° ° ∂G x ¸ x = x ( k ) ° ° ° ° It is easy to see that the successive errors will reduce in magnitude if the fol- lowing condition is satis fi ed at each iteration i.e.
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