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Chemical Engineering Hand Written_Notes_Part_100

# Chemical Engineering Hand Written_Notes_Part_100 - 202 5...

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202 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES Thus, we can approximate ( Φ T Φ ) matrix by the Hilbert matrix (7.23) ( Φ T Φ )= N ( H N 11 / 21 / 3 ... ... 1 /m 1 / / 31 / 4 ... ... 1 / ( m +1) ... ... ... ... ... ... 1 /m ... ... ... ... 1 / (2 m 1) which is highly ill- conditioned for large m . Thus, whether we have a continuous function or numerical data over interval [0 , 1] , the numerical di culties persists as the Hilbert matrix appears in both the cases. Similar to the previous case, modelling in terms of orthogonal polynomials can considerably improve numerical accuracy. The orthogonal set under con- sideration is now di f erent. Let p m ( z i ) denote a orthogonal polynomial of order m . The inner product is now de f ned as (7.24) h p j ( z ) , p k ( z ) i = N X i =1 w i p j ( z i ) p k ( z i ) By de f nition, a set of polynomials { p j ( z i ) } are orthogonal over a set of points { z i } with weights w i , if N X i =1 w i p j ( z i ) p k ( z i )=0 ; j, k =1 ............... m and ( j 6 = k ) Let a linear polynomial model be de f ned as (7.25) y i = m X j =1 α j p j ( z i )+ e i ; i , ................... N Then the normal equation becomes (7.26)

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Chemical Engineering Hand Written_Notes_Part_100 - 202 5...

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