Chemical Engineering Hand Written_Notes_Part_92

Chemical Engineering Hand Written_Notes_Part_92 - 186 5....

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186 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES vector Y from a subspace S . The sub-space involved is nothing but the column space of Φ . Let Φ be represented as (5.18) Φ = h a (1) a (2) .... a ( m ) i where a ( i ) R N are columns of matrix Φ . Let us de f ne the inner product for any u , v R N as (5.19) h u , v i = u T Wv Let the vector p such that (5.20) p = b θ 1 a (1) + b θ 2 a (2) + .... + b θ m a ( m ) = Φ b θ Then, using geometric arguments, we have D a (1) , y Φ b θ E = £ a (1) ¤ T Wy £ a (1) ¤ T W h Φ b θ i =0 (5.21) D a (2) , y Φ b θ E = £ a (2) ¤ T Wy £ a (2) ¤ T W h Φ b θ i (5.22) .................... = ........................................... D a ( m ) , y Φ b θ E = £ a ( m ) ¤ T Wy £ a ( m ) ¤ T W h Φ b θ i (5.23) Rearranging above equations, we have £ a (1) ¤ T £ a (2) ¤ T .... £ a ( m ) ¤ T W Φ b θ 1 b θ 2 .... b θ m = £ a (1) ¤ T Wy £ a (2) ¤ T Wy .... £ a ( m ) ¤ T Wy = £ a (1) ¤ T £ a (2) ¤ T ....
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Chemical Engineering Hand Written_Notes_Part_92 - 186 5....

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