数学建模与数学实验 非线性迭代(7)

100 0.9 1000 0.5 1000 0.9 实验观察：在迭代过程中它们逐渐分开。

有序中包含了无序，无序中包含着有序。 3.

方程求根

实验3.1

方程x3?2x?1?0的根为：

{{x?-1.61803},{x?0.618034},{x?1.}}

x3?12x?13首先分别考虑函数，2，2x?1，取不同的初值x0（例如-9,-5,-2,-1,-0.5,0,0.5,1,

x22 ,8,10 等），生成相应的数列。

函数 初值 迭代序列 25

x3?1 2-2 ?

2,3.5,20.9375,4588.78,4.83127?10,5.63838?10,8.96259?103.59973?10284103194,2.332276783776881?102559853,6.34322727025716?10 ?,1.27614734162579 2x?1 x2-1 -0.5 0 0.5 1 -2 -1 -0.5 {-1, 0., 0.5, 0.5625, 0.588989, 0.602163, 0.609172, 0.613029, 0.61519, 0.616412, 0.617107} {-0.5, 0.4375, 0.54187, 0.579553, 0.597331, 0.606565, 0.611584, 0.614377, \\ 0.615951, 0.616845, 0.617354} {0, 0.5, 0.5625, 0.588989, 0.602163, 0.609172, 0.613029, 0.61519, 0.616412, \\ 0.617107, 0.617504} {0.5, 0.5625, 0.588989, 0.602163, 0.609172, 0.613029, 0.61519, 0.616412, \\ 0.617107, 0.617504, 0.61773} {1, 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.} {-2, -1.25, -2.24, -1.09216, -2.6696, -0.889491, -3.51239, -0.650471, \\ -5.43813, -0.401588, -11.1809} {-1, -3., -0.777778, -4.22449, -0.529464, -7.3446, -0.290847, -18.6979, \\ -0.109824, -101.121, -0.0198761} ? ?0.5,8.,0.265625,21.7024,0.0942788,133.719,0.0150127,4 10.00043767,5.225?106,3.82775? 31 2x?1 {1, 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.} {1, 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.} ?1 实验观察：1为该方程的根。

实验3.2

用牛顿切线法求根，取不同的初值x0（例如-9,-5,-2,-1,-0.5,0,0.5,1, 2,8,10等） 初图形 迭代序列 值 -9 2015105-1.61803-5-10-15-20-1.61803-1.61{-9,-6.05394,-6.05394,-6.05394,-6.05394, -6.05394,-6.05394,-6.05394,-6.05394,-6.05394, -6.05394} 26

-5 2015105-1.61803-5-10-15-20-1.61803{-5, -3.43836, -3.43836, -3.43836, -3.43836, -3.43836,-3.43836,-3.43836,-3.43836,-3.43836, -3.43836} -1.61-2 2015105-1.61803-5-10-15-20-1.61803{-2, -1.7, -1.7, -1.7, -1.7, -1.7, -1.7, -1.7, -1.7, -1.7, -1.7} -1.61-1 2015105-1.61803-5-10-15-20-1.61803{-1, -3., -3., -3., -3., -3., -3., -3., -3., -3., -3.} -1.61 27

-0.5 20151051-5-10-15-2023{-0.5, 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.} 0 20151050.618034-5-10-15-200.618034{0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5} 0.60.5 20151052-5-10-15-203{0.5, 0.6, 0.6, 0.6, 0.6, 0.6, 0.6, 0.6, 0.6, 0.6, 0.6} 28

1 20151052-5-10-15-203{1, 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.} 2 20151052-5-10-15-203{2, 1.5, 1.5, 1.5, 1.5, 1.5, 1.5, 1.5, 1.5, 1.5, 1.5} 8 20151051-5-10-15-20111{8,5.38421,5.38421,5.38421,5.38421, 5.38421, 5.38421, 5.38421, 5.38421, 5.38421, 5.38421} 29