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Unformatted text preview: Universityof Houston, Department of Mathematics Numerical Analysis, Fall 2005 3 Numerical Solution of Nonlinear Equations and Systems 3.1 Fixed point iteration Reamrk 3.1 Problem Given a function F : lR n → lR n , compute x ∗ ∈ lR n such that ( ∗ ) F ( x ∗ ) = . In this chapter, we consider the iterative solution of ( ∗ ) . Definition 3.2 Fixed point, fixed point iteration x y Φ (x) y = x Let Φ : lR n → lR n be given. Avector x ∗ ∈ lR n is called a fixed point of Φ , if Φ ( x ∗ ) = x ∗ . Moreover, given a start vector x ( ) ∈ lR n , the iteration x ( k + 1 ) = Φ ( x ( k ) ) , k ∈ lN is said to be a fixed point iteration (method of successive approximations) . Universityof Houston, Department of Mathematics Numerical Analysis, Fall 2005 x (1) x x (0) (2) (x) Φ * x y x y = x * x y x y = x x (x ) Φ (2 ) x (1 ) x (0 ) Remark 3.3 Attractive and repulsive fixed points The two figures showthat a fixed point iteration may be convergent as well as divergent. Inthesecases, thefixedpointiscalled attractive resp. repulsive. Definition 3.4 Contraction Amapping Φ : ¯ D ⊂ lR n → lR n is said to be a contraction on ¯ D , if there exists ≤ κ < 1 such that bardbl Φ ( x 1 ) − Φ ( x 2 ) bardbl ≤ κ bardbl x 1 − x 2 bardbl , x 1 , x 2 ∈ ¯ D . The number κ is referred to as the contraction number . Remark 3.5 Sufficient conditions for a contraction In case n = 1 and ¯ D =[ a , b ] ⊂ lR assume Φ ∈ C 1 ([ a , b ]) with κ := max z ∈ [ a , b ]  Φ ′ ( z )  < 1 . Then, Φ is a contraction on [ a , b ] with contraction number κ . Universityof Houston, Department of Mathematics Numerical Analysis, Fall 2005 y = x x y 1 2 4 1 10 = x 2 1/2 Φ (x) 8 2 4 8 3 3 6 6 y = x x y 1 6 6 8 10 Φ (x)=2(x3/2) 1/2 3 2 4 1 2 4 8 3 Remark 3.5 Existence of fixed points In general, the property to be a contraction is not sufficient for the existence of fixed points: (i) The mapping Φ ( x ) = 2 √ x has a fixed poinr in x = 4. We have  Φ ′ ( x )  < 1 , x ∈ [ 3 , 6 ] . Moreover, we have Φ ( 3 ) > 3 , Φ ( 6 ) < 6 = ⇒ Φ ([ 3 , 6 ]) ⊂ [ 3 , 6 ] . (ii) The mapping Φ ( x ) = 2 radicalbigg x − 3 / 2 does not have a fixed point. Here, we also have  Φ ′ ( x )  < 1 , x ∈ [ 3 , 6 ] . On the other hand, we have Φ ( 3 ) < 3 = ⇒ Φ ([ 3 , 6 ]) negationslash⊂ [ 3 , 6 ] . Universityof Houston, Department of Mathematics Numerical Analysis, Fall 2005 Theorem3.7 The Banach fixed point theorem Let D ⊂ lR n and Φ : ¯ D → lR n a mapping such that ( 1 ) Φ ( ¯ D ) ⊂ ¯ D , ( 2 ) bardbl Φ ( x 1 ) − Φ ( x 2 ) bardbl ≤ κ bardbl x 1 − x 2 bardbl , x 1 , x 2 ∈ ¯ D , ≤ κ < 1 . Then, there holds: (i) The mapping Φ has a unique fixed point x ∗ ∈ ¯ D ....
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This note was uploaded on 02/21/2012 for the course ENG 3000 taught by Professor Staff during the Spring '12 term at University of Houston.
 Spring '12
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