Chemical Engineering Hand Written_Notes_Part_102

Chemical Engineering Hand Written_Notes_Part_102 - 206 5....

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206 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES Then approximate error vector at k th iteration can be de f ned as (8.27) e ( k ) = y e y ( k ) = £ y F ( k 1) ¤ J ( k 1) 4 θ ( k ) and k th linear sub-problem is de f ned as (8.28) min 4 θ ( j ) £ e ( k ) ¤ T W e ( k ) The least square solution to above sub problem can be obtained by solving the normal equation (8.29) ¡ J ( k 1) ¢ T WJ ( k 1) 4 θ ( k ) = ¡ J ( k 1) ¢ T W £ y F ( k 1) ¤ (8.30) 4 θ ( k ) = h ¡ J ( k 1) ¢ T WJ ( k 1) i 1 ¡ J ( k 1) ¢ T W £ y F ( k 1) ¤ andanimprovedguesscanbeobta inedas (8.31) θ ( k ) = θ ( k 1) + 4 θ ( k ) Termination criterion : De f ning e ( k ) = y F ( k ) and (8.32) Φ ( k ) = £ e ( k ) ¤ T We ( k ) terminate iterations when Φ ( k ) changes only by a small amount, i.e. (8.33) | Φ ( k ) Φ ( k 1) | | Φ ( k ) | Gauss Newton Algorithm: INITIALIZE: θ (0) , ε 1 , ε 2 , α, k, δ 1
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Chemical Engineering Hand Written_Notes_Part_102 - 206 5....

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