Lecture note 2 Numerical Analysis y=g(x) y=T Figure 1.2:Fixed point The advantage of using fixed point. 1.Fixed point form is easier to analyze; 2.certain fix-point iterations lead to powerful root-finding techniques.(We have many freedoms to choose h():e.g.,h()) Example 1:Find the fixed point of g(x)=z2-2. g()=x÷x2-2=x分x2-x-2=0÷x=-1,x=2. Not all function exists a fixed point. Example 2:Find the fixed point of g(x)=2+z2. g(x)=x÷x2+2=x÷x2-x+2=0.No solution! No fixed points. From the examples.we see that the fixed points may not necessarily exist and be unique. Questions:When there exists a fixed point?When the fixed point is unique? Existence and uniqueness are two important aspects in numerical analysis. Theorem 2 (Theorem 2.1)(Eristence)If 1.g∈C[a,b 2.g(x)∈[a, Then g has a fired point in (a,b]. 6Lecture note 2 Numerical Analysis p y = g(x) x y −2 −1 1 2 y = x b b Figure 1.2: Fixed point • The advantage of using fixed point. 1. Fixed point form is easier to analyze; 2. certain fix-point iterations lead to powerful root-finding techniques. (We have many freedoms to choose h(x): e.g., h(x) = 1 f ′(x) ) • Example 1: Find the fixed point of g(x) = x 2 − 2. g(x) = x ⇔ x 2 − 2 = x ⇔ x 2 − x − 2 = 0 ⇔ x = −1, x = 2. • Not all function exists a fixed point. Example 2: Find the fixed point of g(x) = 2 + x 2 . g(x) = x ⇔ x 2 + 2 = x ⇔ x 2 − x + 2 = 0. No solution! No fixed points. • From the examples, we see that the fixed points may not necessarily exist and be unique. • Questions: When there exists a fixed point? When the fixed point is unique? • Existence and uniqueness are two important aspects in numerical analysis. Theorem 2 (Theorem 2.1) (Existence) If 1. g ∈ C[a, b] 2. g(x) ∈ [a, b] Then g has a fixed point in [a, b]. 6