# Chapter3 - University of Houston Department of Mathematics...

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University of 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 ) = 0 . 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. A vector x lR n is called a fixed point of Φ , if Φ ( x ) = x . Moreover, given a start vector x ( 0 ) lR n , the iteration x ( k + 1 ) = Φ ( x ( k ) ) , k lN 0 is said to be a fixed point iteration (method of successive approximations) .

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University of 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 show that a fixed point iteration may be convergent as well as divergent. In these cases, the fixed point is called attractive resp. repulsive. Definition 3.4 Contraction A mapping Φ : ¯ D lR n lR n is said to be a contraction on ¯ D , if there exists 0 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 .
University of 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(x-3/2) 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 ] .

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University of Houston, Department of Mathematics Numerical Analysis, Fall 2005 Theorem 3.7 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 , 0 κ < 1 . Then, there holds: (i) The mapping Φ has a unique fixed point x ¯ D . (ii) For any start vector x ( 0 ) ¯ D , the fixed point iteration converges satisfying the a priori estimate ( ) bardbl x ( k ) x bardbl ≤ κ k 1 κ bardbl x ( 1 ) x ( 0 ) bardbl , k lN , and the a posteriori estimate ( ∗∗ ) bardbl x ( k ) x bardbl ≤ κ 1 κ bardbl x ( k ) x ( k 1 ) bardbl , k lN .
University of Houston, Department of Mathematics Numerical Analysis, Fall 2005 Proof: We have: bardbl x ( k + 1 ) x ( k ) bardbl ≤ bardbl Φ ( x ( k ) Φ ( x ( k 1 ) bardbl ≤ κ bardbl x ( k ) x ( k 1 ) bardbl ≤ ... κ k bardbl x ( 1 ) x ( 0 ) bardbl .

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