研一上學期課程課件數(shù)值分析

1、7.1 方程求根與二分法第7章 非線性方程、方程組的數(shù)值解法一、引言非線性方程的分兩類： 則可用搜索法求有根區(qū)間. x 1 0 1 2f(x)的符號 + +求根問題的三個方面：存在性，分布，精確化。二、二分法二分法簡述.圖7-1k ak bk xkf(xk)符號0123456 1.0 1.25 1.31251.3203 1.5 1.3751.34381.3281 1.25 1.375 1.3125 1.3438 1.3281 1.3203 1.3242 + + + 二分法優(yōu)、缺點； 用途。7.2 不動點迭代法及其收斂性一、不動點與不動點迭代法圖7-2kxk012345671.51.357211

2、.330861.325881.324941.324761.324731.32472二、不動點的存在性與迭代法的收斂性三、局部收斂性與收斂階kxk迭代法(1)迭代法(2)迭代法(3)迭代法(4)0123 x0 x1 x2 x3 2398721.521.521.751.734751.73263121.751.7321431.732051作業(yè): P238, 2,4.7.3 迭代收斂的加速方法一、埃特金加速收斂方法二、斯蒂芬森迭代法kxkykzk0123451.51.416291.355651.329851.324801.324722.375001.840921.491401.347101.32518

3、12.39655.238882.317281.444351.32714說明: (2.2)不收斂,(3.3)可能收斂; (2.2)線性收斂,(3.3)平方收斂!kxkykzk0123.53.734443.733073.604143.733813.662023.733477.4 牛頓法一、牛頓法及其收斂性二、牛頓法應用舉例kxk01230.50.571020.567160.56714x=0.5,for k=1:3x=x-(x-exp(-x)/(1+x),endkxk012341010.75000010.72383710.72380510.723805三、簡化牛頓法與牛頓下山法kxkxkxk f(x

4、k)012341.51.347831.325201.324720.617.9發(fā)散0.6 -1.3841.140625 -0.6566431.36181 0.18661.32628 0.006671.32472 0.0000086四、重根情形kxk(1)(2)(3)0123x0 x1x2x31.51.4583333331.4366071431.4254976191.51.4166666671.4142156861.4142135621.51.4117647061.4142114381.414213562format long;x=1.5for k=1:3x=x-(x2-2)/4/x,%x=x-(x2-2)/2/x,%x=x-x*(x2-2)/(x2+2),end作業(yè): P239, 12,15.7.5 弦截法與拋物線法kxk012340.50.60.565320.567090.567147.6* 解非線性方程組的迭代法kx (k)0123(1.5, 1.0)