Chemical Engineering Hand Written_Notes_Part_83

Chemical Engineering Hand Written_Notes_Part_83 - 168 5...

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

View Full Document Right Arrow Icon
168 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES Thus, above equation reduces to F ( z + z ) F ( z )= 1 2! N X i =1 N X j =1 2 F ( z + λ z ) ∂z i j z i z j (2.15) (0 <λ< 1) This implies that sign of F ( a + z ) F ( a ) at extreme point z is same as sign of R.H.S. Since the 2’nd partial derivative 2 F i j ¸ is continuous in the neighborhood of z = z , its value at z = z + λ z will have same sign as its value at z = z for all su ciently small z .I fthequan t i ty (2.16) N X i =1 N X j =1 2 F ( z + λ z ) i j z i z j ' ( z ) T [ 2 F ( z )] z 0 for all z ,then z = z will be a local minimum. In other words, if Hessian ma- trix [ 2 F ( z )] positive semi-de f nite, then z = z will be a local minimum. If the quantity (2.17) N X i =1 N X j =1 2 F ( z + λ z ) i j z i z j ' ( z ) T [ 2 F ( z )] z 0 for all z z = z will be a local maximum. In other words, if Hessian matrix [ 2 F ( z )] negative semi-de f nite, then z = z
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.

This note was uploaded on 11/26/2011 for the course EGN 3840 taught by Professor Mr.shaw during the Fall '11 term at FSU.

Page1 / 2

Chemical Engineering Hand Written_Notes_Part_83 - 168 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