Lecture note 2 Numerical Analysis Convergence of the algorithm? Weakness of Newton's method po must be close to po,otherwise diverge.Plot a graph. Combine with other methods.such as bisection method 1.3.3 Newton's method with f(p)=0 Definition:zero of multiplicity m.A solution p of f(x)=0 is a zero of multiplicity m of f if for p, f(x)=(x-p)"h(x), where limh(x)≠0. 工+D Use figures to illustrate this. How to characterize the multiplicity of a zero? Theorem 7 (Theorem 4.2)The function fE Cm has a zero of multiplicity of m if and only if fp)=f'(p)=.fm(p)=0,fm)≠0. (1.2) Proof. 1.“multiplicity→(1.2)” f(x) f四=r-pm,h四=-ccm f(p)=0,f(p)=limz-pf()-lim (-p)mh()+m(-p)m-1h()=0: 2.“(1.2)→multiplicity” Taylor's expansion: f)=)+P(p)(c((-) 21 (m-1)！ (m-1)1 f(m-(E((p)m. (m-1)！ fm-》=fm-9imx→p-fm-卫）， ho=四m- (m-1)！ m-≠0. □ 15Lecture note 2 Numerical Analysis • Convergence of the algorithm? Weakness of Newton’s method • p0 must be close to p0, otherwise diverge. Plot a graph. • Combine with other methods, such as bisection method 1.3.3 Newton’s method with f ′ (p) = 0 • Definition: zero of multiplicity m. A solution p of f(x) = 0 is a zero of multiplicity m of f if for x 6= p, f(x) = (x − p) mh(x), where limx→p h(x) 6= 0. • Use figures to illustrate this. • How to characterize the multiplicity of a zero? Theorem 7 (Theorem 4.2) The function f ∈ C m has a zero of multiplicity of m if and only if f(p) = f ′ (p) = . . . f(m) (p) = 0, f(m) 6= 0. (1.2) Proof. 1. “multiplicity =⇒ (1.2)” f(x) = (x − p) mh(x), h(x) = f(x) (x − p)m ∈ C m f(p) = 0, f′ (p) = limx→pf ′ (x) = limx→p (x−p) mh ′ (x)+m(x−p) m−1h(x) = 0 . . . 2. “(1.2) =⇒ multiplicity” Taylor’s expansion: f(x) = f(p) + f ′ (p)(x − p) + f ′′(p) 2! (x − p) 2 + . . . + f (m−1)(p) (m − 1)! (x − p) m−1 + f (m−1)(ξ(x)) (m − 1)! (x − p) m = f (m−1)(ξ(x)) (m − 1)! (x − p) m. limx→p h(x) = limx→p f (m−1)(ξ(x)) (m − 1)! = f (m−1)(limx→p ξ(x)) (m − 1)! = f (m−1)(p) (m − 1)! 6= 0. 15