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 fi 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 fi 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 W J ( k 1) 4 θ ( k ) = ¡ J ( k 1) ¢ T W £ y F ( k 1) ¤ (8.30) 4 θ ( k ) = h ¡ J ( k 1) ¢ T W J ( k 1) i 1 ¡ J ( k 1) ¢ T W £ y F ( k 1) ¤ and an improved guess can be obtained as (8.31) θ ( k ) = θ ( k 1) + 4 θ ( k ) Termination criterion : De fi ning e ( k ) = y F ( k ) and (8.32) Φ ( k ) = £ e ( k ) ¤ T W e ( k ) terminate iterations when Φ ( k ) changes only by a small amount, i.e. (8.33) | Φ ( k ) Φ ( k 1) | | Φ ( k ) | < ε Gauss Newton Algorithm: INITIALIZE: θ (0) , ε 1 , ε 2 , α, k, δ 1 , δ 2 , k max e (0) = y F [ X , θ (0) ] δ 1 = e (0) T W e (0) WHILE [( δ 1 > ε 1 ) AND ( δ 2 > ε 2 ) AND ( k < k max )] e ( k ) = y F [ X , θ ( k ) ] £ J ( k ) T WJ ( k ) ¤ 4 θ ( k ) = J ( k ) T W e ( k ) ) λ (0) = 1 , j = 0 θ ( k +1) = θ ( k ) + λ (0) 4 θ ( k ) δ 0 = e ( k

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