Chemical Engineering Hand Written_Notes_Part_51

Chemical Engineering Hand Written_Notes_Part_51 - 104 3...

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104 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES 6.3.1. Contraction Mapping Principle. Contraction mapping theorem devel- ops su cient conditions for convergence of general nonlinear iterative equation (6.5). Unlike the eigenvalue analysis used for analysis of its linear counterpart, this theorem is more useful for developing understanding of convergence process. Consider general nonlinear iteration equation (6.5) which de f nes a mapping from a Banach space X into itself. Definition 28 . (Contraction Mapping): An operator G : X X given by equation (6.5), mapping a Banach space X into itself, is called a contraction mapping of closed ball U ( x (0) ,r )= © x X : ° ° x x (0) ° ° r ª if there exists a real number θ (0 θ< 1) such that ° ° G ¡ x (1) ¢ G ¡ x (2) ¢° ° θ ° ° x (1) x (2) ° ° for all x (1) , x (2) U ( x (0) ,r ) . The quantity θ is called contraction constant of G on U ( x (0) ,r ) . When the map
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Chemical Engineering Hand Written_Notes_Part_51 - 104 3...

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