Chemical Engineering Hand Written_Notes_Part_49

Chemical Engineering Hand Written_Notes_Part_49 - 100 3....

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100 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES f i ( x ) is continuously di f erentiable, then, in the neighborhood of x we can approximate its behavior by Taylor series, as F ( x )= F £ x ( k ) + ¡ x x ( k ) ¢¤ (6.22) = F ( x ( k ) )+ ∂F ∂x ¸ x = x ( k ) £ x ( k ) ¤ + .... (6.23) Since x ( k ) is assumed to be close to x , second and higher order terms can be neglected. De f ning J ( k ) = ∂F ∂x ¸ x = x ( k ) (6.24) F ( k ) = F ( x ( k ) ) (6.25) we can solve for (6.26) F ( x ) e = F ( k ) + J ( k ) 4 x ( k ) = 0 Now 4 x ( k ) can be interpreted as the error committed in approximating x by x ( k ) . We can obtain an improved approximations x ( k +1) of x as J ( k ) 4 x ( k ) = F ( k ) (6.27) x ( k +1) = x ( k ) + 4 x ( k ) (6.28) Alternatively, iterations can be formulated by solving £ J ( k ) T J ( k ) ¤ 4 x ( k ) = J ( k ) T F ( k ) (6.29) x ( k +1) = x ( k ) + 4 x ( k ) (6.30) where £ J ( k ) T J
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Chemical Engineering Hand Written_Notes_Part_49 - 100 3....

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