Lecture note 2 Numerical Analysis Mean Value Theorem:Given x E [p-6,p+6],There exist a number between x and p. 9)=9)-9包 T-p lg(x)-gp川=lg(ξ)llz-pl≤klz-pl≤k6≤6. This implies that g(r)∈[p-6,p+],x∈p-6p+. ▣ .Newton's method.Find p,s.t.f(p)=0.Iteration:Pn+1=Pnm f(pn) ·Use a graph to illustrate it！ It converges only when po is close enough to p. Use graph to give some examples that Newton's method doesnot converge. Newton's method converges fast if it converges. ●Drawbacks 1.Need a good po.Converges only when po is close enough to p. -Use,e.g.,the bisection method to find a good po. 2.Use some examples to illustrate it. 3.Need f'(x). Use the graph to review Newton's method -Use the graph to illustrate Secant method Secant method:Use difference to replace the derivative. 1.3.2 Secant Method f'(pn)=lim f()-f(pn) x→PmnE-pn f'n)≈fn--f Pn-1-Pn ·Algorithm: Pn-1-Pn Pati =Pa-T(pn-i)-f)(Pa). ·Need po and p1 to work. Converges only slightly slower than Newton's method in practice. ·Use graph to illustrate it！ 14Lecture note 2 Numerical Analysis • Mean Value Theorem: Given x ∈ [p−δ, p+δ], There exist a number ξ between x and p. g(ξ) = g(x) − g(p) x − p . |g(x) − g(p)| = |g(ξ)||x − p| ≤ k|x − p| ≤ kδ ≤ δ. This implies that g(x) ∈ [p − δ, p + δ], ∀x ∈ [p − δ, p + δ]. • Newton’s method. Find p, s.t. f(p) = 0. Iteration: pn+1 = pn − f(pn) f′(pn) . • Use a graph to illustrate it! • It converges only when p0 is close enough to p. • Use graph to give some examples that Newton’s method doesnot converge. • Newton’s method converges fast if it converges. • Drawbacks 1. Need a good p0. Converges only when p0 is close enough to p. – Use, e.g., the bisection method to find a good p0. 2. Use some examples to illustrate it. 3. Need f ′ (x). – Use the graph to review Newton’s method – Use the graph to illustrate Secant method – Secant method: Use difference to replace the derivative. 1.3.2 Secant Method f ′ (pn) = limx→pn f(x) − f(pn) x − pn . f ′ (pn) ≈ f(pn−1) − f(pn) pn−1 − pn . • Algorithm: pn+1 = pn − pn−1 − pn f(pn−1) − f(pn) f(pn). • Need p0 and p1 to work. • Converges only slightly slower than Newton’s method in practice. • Use graph to illustrate it! 14