数值分析上机题(matlab版)(东南大学).doc

1、 数值分析上机报告第一章一、题目精确值为 。)123(1N1)编制按从大到小的顺序 ，计算 SN 的通用程序。113222S2)编制按从小到大的顺序 ，计算 SN 的通用程)(22NN序。3)按两种顺序分别计算 ,并指出有效位数。 （编制程序时用单精度）64210,10S4)通过本次上机题，你明白了什么？二、通用程序clearN=input(Please Input an N (N1):);AccurateValue=single(0-1/(N+1)-1/N+3/2)/2);Sn1=single(0);for a=2:N; Sn1=Sn1+1/(a2-1);endSn2=single(0);f

2、or a=2:N; Sn2=Sn2+1/(N-a+2)2-1);endfprintf(The value of Sn using different algorithms (N=%d)n,N);disp(_)fprintf(Accurate Calculation %fn,AccurateValue);fprintf(Caculate from large to small %fn,Sn1);fprintf(Caculate from small to large %fn,Sn2);disp(_)三、求解结果四、结果分析有效位数n 顺序 100 10000 1000000从大到小 6 3 3从

3、小到大 5 6 6可以得出，算法对误差的传播又一定的影响，在计算时选一种好的算法可以使结果更为精确。从以上的结果可以看到从大到小的顺序导致大数吃小数的现象，容易产生较大的误差，求和运算从小数到大数算所得到的结果才比较准确。Please Input an N (N1):102The value of Sn using different algorithms (N=100)_Accurate Calculation 0.740049Caculate from large to small 0.740049Caculate from small to large 0.740050_Please I

4、nput an N (N1):104The value of Sn using different algorithms (N=10000)_Accurate Calculation 0.749900Caculate from large to small 0.749852Caculate from small to large 0.749900_Please Input an N (N1):106The value of Sn using different algorithms (N=1000000)_Accurate Calculation 0.749999Caculate from l

5、arge to small 0.749852Caculate from small to large 0.749999_第二章一、题目（1 ）给定初值 及容许误差 ，编制牛顿法解方程 f(x)=0 的通用程序。0x（2 ）给定方程 ,易知其有三个根03)(xf 3,0,321xxa)由牛顿方法的局部收敛性可知存在 当 时，Newton 迭代序列收敛,)(0于根 x2*。试确定尽可能大的 。b)试取若干初始值，观察当 时 Newton 序),1(),(),1,(0 x列的收敛性以及收敛于哪一个根。（3 ）通过本上机题，你明白了什么？二、通用程序文件fx.m% 定义函数f(x)function F

6、x=fx(x)Fx=x3/3-x;文件dfx.m% 定义导函数df(x)function Fx=dfx(x)Fx=x2-1;文件Newton.m% Newton法求方程的根%clear%ef=10-6; %给定容许误差 10-6k=0;x0=input(Please input initial value Xo:);disp(k Xk);fprintf(0 %fn,x0); flag=1;while flag=1 endendfprintf(The maximun delta is %fn,delta); 三、求解结果1.运行 search.m 文件结果为：The maximum delta

7、is 0.774597即得最大的 为 0.774597，Newton 迭代序列收敛于根 =0 的最大区间为（-*2x0.774597，0.774597） 。2.运行 Newton.m 文件在区间 上各输入若干个数，计算结果如下：(,1)(,)(,1)区间 上取-1000，-100，-50，-30，-10，-8，-7，-5，-3 ，-1.5Please input initial value Xo:-10000k Xk0 -10000.0000001 -6666.6667332 -4444.4445893 -2962.9632094 -1975.3090315 -1316.8730256 -87

8、7.9158567 -585.2779978 -390.1864709 -260.12602210 -173.41991111 -115.61711812 -77.08384513 -51.39788014 -34.27822915 -22.87161816 -15.27694917 -10.22845918 -6.88478019 -4.68877220 -3.27480721 -2.40771422 -1.93975023 -1.76125924 -1.73276225 -1.73205126 -1.732051Please input initial value Xo:-100k Xk0

9、 -100.0000001 -66.6733342 -44.4588913 -29.6542634 -19.7920165 -13.2284476 -8.8696517 -5.9892318 -4.1073249 -2.91075510 -2.20018911 -1.84868712 -1.74223513 -1.73213914 -1.73205115 -1.732051Please input initial value Xo:-50k Xk0 -50.0000001 -33.3466722 -22.2511253 -14.8641054 -9.9544585 -6.7039606 -4.

10、5710137 -3.2005208 -2.3645159 -1.91970310 -1.75640511 -1.73254812 -1.73205113 -1.732051Please input initial value Xo:-30k Xk0 -30.0000001 -20.0222472 -13.3815443 -8.9711294 -6.0560005 -4.1505036 -2.9375247 -2.2150468 -1.8547149 -1.74323610 -1.73215811 -1.73205112 -1.732051Please input initial value

11、Xo:-10k Xk0 -10.0000001 -6.7340072 -4.5905703 -3.2128404 -2.3716535 -1.9229816 -1.7571757 -1.7325808 -1.7320519 -1.732051结果显示，以上初值迭代序列均收敛于-1.732051，即根 。*1xPlease input initial value Xo:-8k Xk0 -8.0000001 -5.4179892 -3.7393793 -2.6849344 -2.0782465 -1.8029286 -1.7360237 -1.7320648 -1.7320519 -1.73205

12、1Please input initial value Xo:-7k Xk0 -7.0000001 -4.7638892 -3.3223183 -2.4355334 -1.9529155 -1.7646306 -1.7329317 -1.7320518 -1.732051Please input initial value Xo:-5k Xk0 -5.0000001 -3.4722222 -2.5241803 -1.9960684 -1.7766185 -1.7336746 -1.7320537 -1.7320518 -1.732051Please input initial value Xo

13、:-3k Xk0 -3.0000001 -2.2500002 -1.8692313 -1.7458104 -1.7322125 -1.7320516 -1.732051Please input initial value Xo:-1.5k Xk0 -1.5000001 -1.8000002 -1.7357143 -1.7320624 -1.7320515 -1.732051在区间 即区间（-1 ，-0.774597）上取-0.774598，-0.8，-0.85，-0.9，-0.99，计算结(1,)果如下：Please input initial value Xo:-0.774598k Xk0

14、-0.7745981 0.7746052 -0.7746453 0.7748844 -0.7763245 0.7850496 -0.8406417 1.3501878 1.9938309 1.77596310 1.73362811 1.73205312 1.73205113 1.732051Please input initial value Xo:-0.8k Xk0 -0.8000001 0.9481482 -5.6253703 -3.8726254 -2.7661975 -2.1213676 -1.8182927 -1.7378228 -1.7320799 -1.73205110 -1.7

15、32051Please input initial value Xo:0.85k Xk0 0.8500001 -1.4753752 -1.8194443 -1.7379694 -1.7320815 -1.7320516 -1.732051Please input initial value Xo:-0.9k Xk0 -0.9000001 2.5578952 2.0129153 1.7816624 1.7340495 1.7320546 1.7320517 1.732051Please input initial value Xo:-0.99k Xk0 -0.9900001 32.5058292

16、 21.6910813 14.4915214 9.7072385 6.5409066 4.4649667 3.1338408 2.3260759 1.90230310 1.75247811 1.73240312 1.73205113 1.732051计算结果显示，迭代序列局部收敛于-1.732051，即根 ，局部收敛于 1.730251，即根*1x。*3x在区间 即区间（-0.774597，0.774597）上，由 search.m 的运行过程表明，在整(,)个区间上均收敛于 0，即根 。*2x在区间 即区间（0.774597，1）上取 0.774598，0.8 ，0.85，0.9 ，0.99

17、 ，计算结果如下：(,)Please input initial value Xo:0.774598k Xk0 0.7745981 -0.7746052 0.7746453 -0.7748844 0.7763245 -0.7850496 0.8406417 -1.3501878 -1.9938309 -1.77596310 -1.73362811 -1.73205312 -1.73205113 -1.732051Please input initial value Xo:0.8k Xk0 0.8000001 -0.9481482 5.6253703 3.8726254 2.7661975 2.

18、1213676 1.8182927 1.7378228 1.7320799 1.73205110 1.732051Please input initial value Xo:0.85k Xk0 0.8500001 -1.4753752 -1.8194443 -1.7379694 -1.7320815 -1.7320516 -1.732051Please input initial value Xo:0.9k Xk0 0.9000001 -2.5578952 -2.0129153 -1.7816624 -1.7340495 -1.7320546 -1.7320517 -1.732051Pleas

19、e input initial value Xo:0.99k Xk0 0.9900001 -32.5058292 -21.6910813 -14.4915214 -9.7072385 -6.5409066 -4.4649667 -3.1338408 -2.3260759 -1.90230310 -1.75247811 -1.73240312 -1.73205113 -1.732051计算结果显示，迭代序列局部收敛于-1.732051，即根 ，局部收敛于 1.730251，即根 。*1x*3x区间 上取 100，60，20 ，10，7 ，6，4，3 ，1.5，计算结果如下 :(1,)Please

20、 input initial value Xo:100k Xk0 100.0000001 66.6733342 44.4588913 29.6542634 19.7920165 13.2284476 8.8696517 5.9892318 4.1073249 2.91075510 2.20018911 1.84868712 1.74223513 1.73213914 1.73205115 1.732051Please input initial value Xo:60k Xk0 60.0000001 40.0111142 26.6907493 17.8188454 11.9167625 8.0

21、008486 5.4185467 3.7397368 2.6851519 2.07836010 1.80296711 1.73602712 1.73206413 1.73205114 1.732051Please input initial value Xo:20k Xk0 20.0000001 13.3667502 8.9613233 6.0495474 4.1463286 2.2136057 1.8541268 1.7431369 1.73215610 1.73205111 1.732051Please input initial value Xo:10k Xk0 10.0000001 6

22、.7340072 4.5905703 3.2128404 2.3716535 1.9229816 1.7571757 1.7325808 1.7320519 1.732051Please input initial value Xo:7k Xk0 7.0000001 4.7638892 3.3223183 2.4355334 1.9529155 1.7646306 1.7329317 1.7320518 1.732051Please input initial value Xo:6k Xk0 6.0000001 4.1142862 2.9150683 2.2025784 1.8496505 1

23、.7423926 1.7321427 1.7320518 1.732051Please input initial value Xo:4k Xk0 4.0000001 2.8444442 2.1637243 1.8342814 1.7400075 1.7321056 1.7320517 1.732051Please input initial value Xo:3k Xk0 3.0000001 2.2500002 1.8692313 1.7458104 1.7322125 1.7320516 1.732051Please input initial value Xo:1.5k Xk0 1.5000001 1.8000002 1.7357143 1.7320624 1.7320515 1.732051结果显示，以上初值迭代序列均收敛于 1.732051，即根 。*3x综上所述：(- ,-1)区间收敛于-1.73205,(-1,) 区间局部收敛于 1.73205,局部收敛于-1.73205,(-, )区间收敛于 0,(,1)区间类似于(-1,) 区间， (1,)收敛于 1.73205。通过本上机题，明白了对于多根方程，Newton 法求方程根时，迭代序列收敛于某一个根有一定的区间限制，在一个区间上，可能会局部收敛于不同的根。