Chemical Engineering Hand Written_Notes_Part_99

Chemical Engineering Hand Written_Notes_Part_99 - 200...

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200 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES where (7.7) H = 11 / 21 / 3 ... ... 1 /m 1 / / 31 / 4 ... ... 1 / ( m +1) ... ... ... ... ... ... 1 /m ... ... ... ... 1 / (2 m 1) The matrix H is known as Hilbert matrix and this matrix is highly ill-conditioned for m> 3 . The following table shows condition numbers for a few values of m. (7.8) m 3 4 5 6 7 8 c ( H ) 524 1.55e4 4.67e5 1.5e7 4.75e8 1.53e10 Thus, for polynomial models of small order, say m =3 we obtain good sit- uation, but beyond this order, what ever be the method of solution, we get approximations of less and less accuracy. This implies that approximating a continuous function by polynomial of type (7.1) with the choice of basis vectors as (7.3) is extremely ill-conditioned problem from the viewpoint of numerical computations. Also, note that if we want to increase the degree of polynomial to say ( m +1 )from m , then we have to recompute θ 1 , .... , θ m along with θ m +1 . On the other hand, consider the model (7.9) b y ( z )= α 1 p 1 ( z )+ α 2 p 2 ( z α 3 p 3 ( z ............. + α m p m ( z ) where p i ( z )
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Chemical Engineering Hand Written_Notes_Part_99 - 200...

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