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Chemical Engineering Hand Written_Notes_Part_84

# Chemical Engineering Hand Written_Notes_Part_84 - 170 5...

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5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES Figure 1 where Ψ = h v (1) v (2) .... v ( n ) i represents unitary matrix with eigenvectors of the Hessian as its columns and Λ is a diagonal matrix with eigenvalues on the diagonal. Using this transforma- tion, we have q ( z )=( Ψ T z ) T Λ ¡ Ψ T z ¢ =( y ) T Λ ( y ) (2.28) = λ 1 ( y 1 ) 2 + λ 2 ( y 2 ) 2 + ....λ n ( y n ) 2 =1 (2.29) From the above expression, it is easy to see that q ( y ) is an ellipsoid in n dimensions with its major axis aligned along the eigenvector v ( n ) and the minor axis along the eigenvector v (1) . Thus, function F ( z ) looks like an ellipsoid in the neighborhood of z = z when z is a minimum. Remark 8 . Whether a matrix is positive (semi)de f nite, negative (semi) de f nite or inde f n i t ec anb ee s ta b l i

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Chemical Engineering Hand Written_Notes_Part_84 - 170 5...

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