Chemical Engineering Hand Written_Notes_Part_100

Chemical Engineering Hand Written_Notes_Part_100 - 202 5....

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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)
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 2

Chemical Engineering Hand Written_Notes_Part_100 - 202 5....

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online