Chemical Engineering Hand Written_Notes_Part_52

Chemical - 106 3 LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES is not a contraction mapping near(1,1 and the iterations do not converge

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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 f 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 f 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-
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This note was uploaded on 11/26/2011 for the course EGN 3840 taught by Professor Mr.shaw during the Fall '11 term at FSU.

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Chemical - 106 3 LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES is not a contraction mapping near(1,1 and the iterations do not converge

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