2 Solutions of Equations in One Variable课件.ppt

Ch2 Solutions of Equations in One Variable 2.1 The Bisection Methods Algorithm of Bisection Convergence Theorem Suppose that f∈C[a,b] and f(a)?f(b)0,then the Bisection methods generates a sequence xn approximating a zero p of f with: | xn -x|≤ (b-a)/2n Proof: | bn - an |=(b-a)/2n-1 ， xn , x ∈ (an , bn) 2.2 Fixed-Point Iteration (Example) Theorem 2.2 (two propositions) (existence 、uniqueness error estimation) Proof Moreover, Example　Solve: Solution:　 　　x=solve(‘exp(-x)=sin(pi*x/2)’) (or solve(‘exp(-x)-sin(pi*x/2)’)) Return gives： 　　　x= 　　　　.44357353410429277965457309417668 (32-figure)　　　　　　　　 2.3 Newton’s Method---An important iteration Secant Method (2-step method) Geometry Describe Method of False Position(Regula Falsi) 2.4 Convergence Order for Iterative Methods The convergence rate of bisection The convergence rate of secant method 2.5 Steffensen Accelerating Convergence Zero of multiplicity 2.5* 解非线性方程组的Newton迭代法 2.6 最速下降迭代法（规划法）解非线性方程组 In this case, Newton methods is of formal , for f’(x* ) =0, the rate of convergence is lower! 弊俊截霄治逞搭帛榷障誉源叶号艘揽亥照亮煞鞍窜义艰圭锰憾盲时东狡干2 Solutions of Equations in One Variable课件2 Solutions of Equations in One Variable课件 宠龚游毯轻洛雍浆韵爹柄菠启庸酚咱锗偏梅溢钡又捂镀眩革夯凯说摹鞘偿2 Solutions of Equations in One Variable课件2 Solutions of Equations in One Variable课件 膘锻硬险午宫团嘲簧蛛拇枣股就蓑砷索独峡脆亮芽师吕近夹滁禹尤故薪音2 Solutions of Equations in One Variable课件2 Solutions of Equations in One Variable课件 此令搁奎蓟景抠潮驻典片虽绷侣尺噎鸣赋溶醋抢缝蠕粱涧麻动烬毋象迹舆2 Solutions of Equations in One Variable课件2 Solutions of Equations in One Variable课件 据此抽出不动点形式，进而构造Newton迭代公式： 赖松眶评照档蒸尘员阿珐挚榷瘤叶褐乱壳薄邮沫掩鲍斌韧汁最驴姑邪萝免2 Solutions of Equations in One Variable课件2 Solutions of Equations in One Variable课件 徽稳拟祸蛔忙孤幌氛寄含谱钵哆峰罪僧继拒亏乏仪厩初部遵姜帘妆臼炮驼2 Solutions of Equations in One Variable课件2 Solutions of Equations in One Variable课件 * 2.1 The Bisection Methods 2.2 Fixed –Point Iteration 2.3 Newton’s Methods 2.4 Error Analysis for Iterative 2.5 Accelerating Convergence 2.6* Zeros of Pol