數(shù)學(xué)實驗練習(xí)題參考答案

1、專業(yè) 姓名 學(xué)號 成績第一次練習(xí)教學(xué)要求：熟練掌握Matlab軟件的基本命令和操作，會作二維、三維幾何圖形，能夠用Matlab軟件解決微積分、線性代數(shù)與解析幾何中的計算問題。補(bǔ)充命令vpa(x,n)顯示x的n位有效數(shù)字，教材102頁fplot(f(x),a,b)函數(shù)作圖命令，畫出f(x)在區(qū)間a,b上的圖形在下面的題目中為你的學(xué)號的后3位（1-9班）或4位（10班以上）1.1 計算與程序：syms xlimit(1001*x-sin(1001*x)/x3,x,0)結(jié)果：1003003001/6程序：syms xlimit(1001*x-sin(1001*x)/x3,x,inf)結(jié)果：01.2

2、，求 程序：syms xdiff(exp(x)*cos(1001*x/1000),2)結(jié)果：-2001/1000000*exp(x)*cos(1001/1000*x)-1001/500*exp(x)*sin(1001/1000*x)1.3 計算程序：dblquad(x,y) exp(x.2+y.2),0,1,0,1)結(jié)果：2.139350195142281.4 計算程序：syms xint(x4/(10002+4*x2)結(jié)果：1/12*x3-1002001/16*x+1003003001/32*atan(2/1001*x)1.5 程序：syms xdiff(exp(x)*cos(1000*x)

3、,10)結(jié)果：-1009999759158992000960720160000*exp(x)*cos(1001*x)-10090239998990319040000160032*exp(x)*sin(1001*x)1.6 給出在的泰勒展式(最高次冪為4). 程序：syms xtaylor(sqrt(1001/1000+x),5)結(jié)果：1/100*10010(1/2)+5/1001*10010(1/2)*x-1250/1002001*10010(1/2)*x2+625000/1003003001*10010(1/2)*x3-390625000/1004006004001*10010(1/2)*x

4、41.7 Fibonacci數(shù)列的定義是，用循環(huán)語句編程給出該數(shù)列的前20項（要求將結(jié)果用向量的形式給出）。程序：x=1,1;for n=3:20 x(n)=x(n-1)+x(n-2);endx結(jié)果：Columns 1 through 10 1 1 2 3 5 8 13 21 34 55 Columns 11 through 20 89 144 233 377 610 987 1597 2584 4181 67651.8 對矩陣，求該矩陣的逆矩陣，特征值，特征向量，行列式，計算，并求矩陣（是對角矩陣），使得。程序與結(jié)果：a=-2,1,1;0,2,0;-4,1,1001/1000;inv(a)

5、0.50100100100100 -0.00025025025025 -0.50050050050050 0 0.50000000000000 0 2.00200200200200 -0.50050050050050 -1.00100100100100eig(a)-0.49950000000000 + 1.32230849275046i -0.49950000000000 - 1.32230849275046i 2.00000000000000p,d=eig(a)p = 0.3355 - 0.2957i 0.3355 + 0.2957i 0.2425 0 0 0.9701 0.8944 0.8

6、944 0.0000 注：p的列向量為特征向量d = -0.4995 + 1.3223i 0 0 0 -0.4995 - 1.3223i 0 0 0 2.0000 a6 11.9680 13.0080 -4.9910 0 64.0000 0 19.9640 -4.9910 -3.0100 1.9 作出如下函數(shù)的圖形（注：先用M文件定義函數(shù)，再用fplot進(jìn)行函數(shù)作圖）：函數(shù)文件f.m: function y=f(x)if 0=x&x=1/2 y=2.0*x;else 1/2x&x f=inline(x+1000/x)/2);x0=3;for i=1:20;x0=f(x0);fprintf(%g

7、,%gn,i,x0);end運(yùn)行結(jié)果：1,168.167 11,31.62282,87.0566 12,31.62283,49.2717 13,31.62284,34.7837 14,31.62285,31.7664 15,31.62286,31.6231 16,31.62287,31.6228 17,31.62288,31.6228 18,31.62289,31.6228 19,31.622810,31.6228 20,31.6228由運(yùn)行結(jié)果可以看出，數(shù)列收斂，其值為31.6228。2.2 求出分式線性函數(shù)的不動點，再編程判斷它們的迭代序列是否收斂。解：取m=1000.（1）程序如下：f=

8、inline(x-1)/(x+1000);x0=2;for i=1:20;x0=f(x0);fprintf(%g,%gn,i,x0);end運(yùn)行結(jié)果：1,0.000998004 11,-0.0010012,-0.000999001 12,-0.0010013,-0.001001 13,-0.0010014,-0.001001 14,-0.0010015,-0.001001 15,-0.0010016,-0.001001 16,-0.0010017,-0.001001 17,-0.0010018,-0.001001 18,-0.0010019,-0.001001 19,-0.00100110,-

9、0.001001 20,-0.001001由運(yùn)行結(jié)果可以看出，分式線性函數(shù)收斂，其值為-0.001001。易見函數(shù)的不動點為-0.001001（吸引點）。（2）程序如下：f=inline(x+1000000)/(x+1000);x0=2;for i=1:20;x0=f(x0);fprintf(%g,%gn,i,x0);end運(yùn)行結(jié)果：1,998.006 11,618.3322,500.999 12,618.3023,666.557 13,618.3144,600.439 14,618.3095,625.204 15,618.3116,615.692 16,618.317,619.311 17,

10、618.3118,617.929 18,618.319,618.456 19,618.3110,618.255 20,618.31由運(yùn)行結(jié)果可以看出，分式線性函數(shù)收斂，其值為618.31。易見函數(shù)的不動點為618.31（吸引點）。2.3 下面函數(shù)的迭代是否會產(chǎn)生混沌？（56頁練習(xí)7（1）解：程序如下：f=inline(1-2*abs(x-1/2);x=;y=;x(1)=rand();y(1)=0;x(2)=x(1);y(2)=f(x(1);for i=1:100;x(1+2*i)=y(2*i);x(2+2*i)=x(1+2*i);y(2+2*i)=f(x(2+2*i);endplot(x,y,

11、r);hold on;syms x;ezplot(x,0,1/2);ezplot(f(x),0,1);axis(0,1/2,0,1); hold off運(yùn)行結(jié)果：2.4 函數(shù)稱為Logistic映射，試從“蜘蛛網(wǎng)”圖觀察它取初值為產(chǎn)生的迭代序列的收斂性，將觀察記錄填人下表，若出現(xiàn)循環(huán)，請指出它的周期（56頁練習(xí)8）3.33.53.563.5683.63.84序列收斂情況T=2T=4T=8T=9混沌混沌解：當(dāng)=3.3時，程序代碼如下：f=inline(3.3*x*(1-x);x=;y=;x(1)=0.5;y(1)=0;x(2)=x(1);y(2)=f(x(1);for i=1:1000;x(1+

12、2*i)=y(2*i);x(2+2*i)=x(1+2*i);y(1+2*i)=x(1+2*i);y(2+2*i)=f(x(2+2*i);endplot (x,y,r);hold on;syms x;ezplot(x,0,1);ezplot(f(x),0,1);axis(0,1,0,1);hold off運(yùn)行結(jié)果：當(dāng)=3.5時，上述程序稍加修改，得：當(dāng)=3.56時，得：當(dāng)=3.568時，得：當(dāng)=3.6時，得：當(dāng)=3.84時，得：2.5 對于Martin迭代，取參數(shù)為其它的值會得到什么圖形？參考下表（取自63頁練習(xí)13）mmm-m-mm-mm/1000-mm/1000m/10000.5m/1000

13、m-mm/100m/10-10-m/10174解：取m=10000；迭代次數(shù)N=20000；在M-文件里面輸入代碼：function Martin(a,b,c,N)f=(x,y)(y-sign(x)*sqrt(abs(b*x-c);g=(x)(a-x);m=0;0;for n=1:N m(:,n+1)=f(m(1,n),m(2,n),g(m(1,n); end plot(m(1,:),m(2,:),kx); axis equal在命令窗口中執(zhí)行Martin（10000，10000，10000，20000），得：執(zhí)行Martin（-10000，-10000，10000，20000），得：執(zhí)行Ma

14、rtin（-10000，10，-10000，20000），得：執(zhí)行Martin（10，10，0.5，20000），得：執(zhí)行Martin（10，10000，-10000，20000），得：執(zhí)行Martin（100，1000，-10，20000），得：執(zhí)行Martin（-1000，17，4，20000），得：2.6 能否找到分式函數(shù)（其中是整數(shù)）,使它產(chǎn)生的迭代序列（迭代的初始值也是整數(shù)）收斂到(對于為整數(shù)的學(xué)號，請改為求)。如果迭代收斂,那么迭代的初值與收斂的速度有什么關(guān)系.寫出你做此題的體會.提示：教材54頁練習(xí)4的一些分析。若分式線性函數(shù)的迭代收斂到指定的數(shù)，則為的不動點，因此化簡得：。若為

15、整數(shù)，易見。取滿足這種條件的不同的以及迭代初值進(jìn)行編。解：取m=10000；根據(jù)上述提示，?。篴=e=1,b=10000,c=1,d=0.程序如下（初值為1200）：f=inline(x+10000)/(x2+1);x0=1200;for i=1:100;x0=f(x0);fprintf(%g,%gn,i,x0);end運(yùn)行結(jié)果如下：1,0.007777772,9999.43,0.0002000184,100005,0.00026,100007,0.00028,100009,0.000210,1000011,0.000212,1000013,0.000214,1000015,0.000216,

16、1000017,0.000218,1000019,0.000220,1000021,0.000222,1000023,0.000224,1000025,0.000226,1000027,0.000228,1000029,0.000230,1000031,0.000232,1000033,0.000234,1000035,0.000236,1000037,0.000238,1000039,0.000240,1000041,0.000242,1000043,0.000244,1000045,0.000246,1000047,0.000248,1000049,0.000250,1000051,0.0

17、00252,1000053,0.000254,1000055,0.000256,1000057,0.000258,1000059,0.000260,1000061,0.000262,1000063,0.000264,1000065,0.000266,1000067,0.000268,1000069,0.000270,1000071,0.000272,1000073,0.000274,1000075,0.000276,1000077,0.000278,1000079,0.000280,1000081,0.000282,1000083,0.000284,1000085,0.000286,10000

18、87,0.000288,1000089,0.000290,1000091,0.000292,1000093,0.000294,1000095,0.000296,1000097,0.000298,1000099,0.0002100,10000若初值取為1000，運(yùn)行結(jié)果：1,0.0112,9998.83,0.0002000364,100005,0.00026,100007,0.00028,100009,0.000210,1000011,0.000212,1000013,0.000214,1000015,0.000216,1000017,0.000218,1000019,0.000220,1000

19、021,0.000222,1000023,0.000224,1000025,0.000226,1000027,0.000228,1000029,0.000230,1000031,0.000232,1000033,0.000234,1000035,0.000236,1000037,0.000238,1000039,0.000240,1000041,0.000242,1000043,0.000244,1000045,0.000246,1000047,0.000248,1000049,0.000250,1000051,0.000252,1000053,0.000254,1000055,0.00025

20、6,1000057,0.000258,1000059,0.000260,1000061,0.000262,1000063,0.000264,1000065,0.000266,1000067,0.000268,1000069,0.000270,1000071,0.000272,1000073,0.000274,1000075,0.000276,1000077,0.000278,1000079,0.000280,1000081,0.000282,1000083,0.000284,1000085,0.000286,1000087,0.000288,1000089,0.000290,1000091,0

21、.000292,1000093,0.000294,1000095,0.000296,1000097,0.000298,1000099,0.0002100,10000若初值取為-1，運(yùn)行結(jié)果：1,4999.52,0.00060013,100004,0.00025,100006,0.00027,100008,0.00029,1000010,0.000211,1000012,0.000213,1000014,0.000215,1000016,0.000217,1000018,0.000219,1000020,0.000221,1000022,0.000223,1000024,0.000225,100

22、0026,0.000227,1000028,0.000229,1000030,0.000231,1000032,0.000233,1000034,0.000235,1000036,0.000237,1000038,0.000239,1000040,0.000241,1000042,0.000243,1000044,0.000245,1000046,0.000247,1000048,0.000249,1000050,0.000251,1000052,0.000253,1000054,0.000255,1000056,0.000257,1000058,0.000259,1000060,0.0002

23、61,1000062,0.000263,1000064,0.000265,1000066,0.000267,1000068,0.000269,1000070,0.000271,1000072,0.000273,1000074,0.000275,1000076,0.000277,1000078,0.000279,1000080,0.000281,1000082,0.000283,1000084,0.000285,1000086,0.000287,1000088,0.000289,1000090,0.000291,1000092,0.000293,1000094,0.000295,1000096,

24、0.000297,1000098,0.000299,10000100,0.0002 第三次練習(xí)教學(xué)要求：理解線性映射的思想，會用線性映射和特征值的思想方法解決諸如天氣等實際問題。3.1 對，求出的通項. 程序：A=sym(4,2;1,3);P,D=eig(A)Q=inv(P)syms n; xn=P*(D.n)*Q*1;2 結(jié)果：P = 2, -1 1, 1D = 5, 0 0, 2Q = 1/3, 1/3 -1/3, 2/3xn =2*5n-2n 5n+2n3.2 對于練習(xí)1中的，求出的通項. 程序：A=sym(2/5,1/5;1/10,3/10); %沒有sym下面的矩陣就會顯示為小數(shù)P,

25、D=eig(A)Q=inv(P)xn=P*(D.n)*Q*1;2 結(jié)果：P = 2, -1 1, 1D = 1/2, 0 0, 1/5Q = 1/3, 1/3 -1/3, 2/3xn = 2*(1/2)n-(1/5)n (1/2)n+(1/5)n3.3 對隨機(jī)給出的，觀察數(shù)列.該數(shù)列有極限嗎? A=4,2;1,3;a=;x=2*rand(2,1)-1;for i=1:20 a(i,1:2)=x; x=A*x; end for i=1:20 if a(i,1)=0 else t=a(i,2)/a(i,1); fprintf(%g,%gn,i,t); endend 結(jié)論：在迭代18次后，發(fā)現(xiàn)數(shù)列存

26、在極限為0.53.4 對120頁中的例子，繼續(xù)計算.觀察及的極限是否存在. （120頁練習(xí)9） A=2.1,3.4,-1.2,2.3;0.8,-0.3,4.1,2.8;2.3,7.9,-1.5,1.4;3.5,7.2,1.7,-9.0;x0=1;2;3;4;x=A*x0;for i=1:1:100a=max(x);b=min(x);m=a*(abs(a)abs(b)+b*(abs(a) A=2.1,3.4,-1.2,2.3;0.8,-0.3,4.1,2.8;2.3,7.9,-1.5,1.4;3.5,7.2,1.7,-9.0;P,D=eig(A)P = -0.3779 -0.8848 -0.08

27、32 -0.3908 -0.5367 0.3575 -0.2786 0.4777 -0.6473 0.2988 0.1092 -0.7442 -0.3874 -0.0015 0.9505 0.2555D = 7.2300 0 0 0 0 1.1352 0 0 0 0 -11.2213 0 0 0 0 -5.8439結(jié)論：A的絕對值最大特征值等于上面的的極限相等，為什么呢？還有，P的第三列也就是-11.2213對應(yīng)的特征向量和上題求解到的y也有系數(shù)關(guān)系，兩者都是-11.2213的特征向量。3.6 設(shè)，對問題2求出若干天之后的天氣狀態(tài)，并找出其特點（取4位有效數(shù)字）. （122頁練習(xí)12） A2=

28、3/4,1/2,1/4;1/8,1/4,1/2;1/8,1/4,1/4;P=0.5;0.25;0.25;for i=1:1:20 P(:,i+1)=A2*P(:,i);endPP = Columns 1 through 14 0.5000 0.5625 0.5938 0.6035 0.6069 0.6081 0.6085 0.6086 0.6087 0.6087 0.6087 0.6087 0.6087 0.6087 0.2500 0.2500 0.2266 0.2207 0.2185 0.2178 0.2175 0.2174 0.2174 0.2174 0.2174 0.2174 0.217

29、4 0.2174 0.2500 0.1875 0.1797 0.1758 0.1746 0.1741 0.1740 0.1739 0.1739 0.1739 0.1739 0.1739 0.1739 0.1739 Columns 15 through 21 0.6087 0.6087 0.6087 0.6087 0.6087 0.6087 0.6087 0.2174 0.2174 0.2174 0.2174 0.2174 0.2174 0.21740.1739 0.1739 0.1739 0.1739 0.1739 0.1739 0.1739結(jié)論：9天后，天氣狀態(tài)趨于穩(wěn)定P*=（0.6087,

30、0.2174,0.1739）T3.7 對于問題2，求出矩陣的特征值與特征向量，并將特征向量與上一題中的結(jié)論作對比. （122頁練習(xí)14） P,D=eig(A2)P = -0.9094 -0.8069 0.3437 -0.3248 0.5116 -0.8133 -0.2598 0.2953 0.4695D = 1.0000 0 0 0 0.3415 0 0 0 -0.0915分析：事實上，q=k(-0.9094, -0.3248, -0.2598)T均為特征向量，而上題中P*的3個分量之和為1，可令k(-0.9094, -0.3248, -0.2598)T=1，得k=-0.6696.有q=(0.

31、6087, 0.2174, 0.1739),與P*一致。3.8 對問題1，設(shè)為的兩個線性無關(guān)的特征向量，若，具體求出上述的，將表示成的線性組合，求的具體表達(dá)式，并求時的極限，與已知結(jié)論作比較. （123頁練習(xí)16） A=3/4,7/18;1/4,11/18;P,D=eig(A);syms k pk;a=solve(u*P(1,1)+v*P(1,2)-1/2,u*P(2,1)+v*P(2,2)-1/2,u,v);pk=a.u*D(1,1).k*P(:,1)+a.v*D(2,2).k*P(:,2) pk = -5/46*(13/36)k+14/23 5/46*(13/36)k+9/23或者：p0=

32、1/2;1/2;P,D=eig(sym(A);B=inv(sym(P)*p0 B = 5/46 9/23syms kpk=B(1,1)*D(1,1).k*P(:,1)+B(2,1)*D(2,2).k*P(:,2) pk = -5/46*(13/36)k+14/23 5/46*(13/36)k+9/23 vpa(limit(pk,k,100),10) ans = .6086956522 .3913043478結(jié)論：和用練習(xí)12中用迭代的方法求得的結(jié)果是一樣的。第四次練習(xí)教學(xué)要求：會利用軟件求勾股數(shù)，并且能夠分析勾股數(shù)之間的關(guān)系。會解簡單的近似計算問題。4.1 求滿足，的所有勾股數(shù)，能否類似于（1

33、1.8），把它們用一個公式表示出來？程序：for b=1:998 a=sqrt(b+2)2-b2); if(a=floor(a) fprintf(a=%i,b=%i,c=%in,a,b,b+2) endend運(yùn)行結(jié)果：a=4,b=3,c=5a=6,b=8,c=10a=8,b=15,c=17a=10,b=24,c=26a=12,b=35,c=37a=14,b=48,c=50a=16,b=63,c=65a=18,b=80,c=82a=20,b=99,c=101a=22,b=120,c=122a=24,b=143,c=145a=26,b=168,c=170a=28,b=195,c=197a=30,b

34、=224,c=226a=32,b=255,c=257a=34,b=288,c=290a=36,b=323,c=325a=38,b=360,c=362a=40,b=399,c=401a=42,b=440,c=442a=44,b=483,c=485a=46,b=528,c=530a=48,b=575,c=577a=50,b=624,c=626a=52,b=675,c=677a=54,b=728,c=730a=56,b=783,c=785a=58,b=840,c=842a=60,b=899,c=901a=62,b=960,c=962勾股數(shù)，的解是： 以下是推導(dǎo)過程：由，有顯然，從而是2的倍數(shù).設(shè)，代

35、入上式得到：因為，從而.4.2 將上一題中改為,分別找出所有的勾股數(shù).將它們與時的結(jié)果進(jìn)行比較，然后用公式表達(dá)其結(jié)果。（1）時通項：a=8,b=6,c=10a=12,b=16,c=20a=16,b=30,c=34a=20,b=48,c=52a=24,b=70,c=74a=28,b=96,c=100a=32,b=126,c=130a=36,b=160,c=164a=40,b=198,c=202a=44,b=240,c=244a=48,b=286,c=290a=52,b=336,c=340a=56,b=390,c=394a=60,b=448,c=452a=64,b=510,c=514a=68,b=

36、576,c=580a=72,b=646,c=650a=76,b=720,c=724a=80,b=798,c=802a=84,b=880,c=884a=88,b=966,c=970（2）5時通項： a=15,b=20,c=25a=25,b=60,c=65a=35,b=120,c=125a=45,b=200,c=205a=55,b=300,c=305a=65,b=420,c=425a=75,b=560,c=565a=85,b=720,c=725a=95,b=900,c=905（3）6時通項a=12,b=9,c=15a=18,b=24,c=30a=24,b=45,c=51a=30,b=72,c=78

37、a=36,b=105,c=111a=42,b=144,c=150a=48,b=189,c=195a=54,b=240,c=246a=60,b=297,c=303a=66,b=360,c=366a=72,b=429,c=435a=78,b=504,c=510a=84,b=585,c=591a=90,b=672,c=678a=96,b=765,c=771a=102,b=864,c=870a=108,b=969,c=975（4）7時通項a=21,b=28,c=35a=35,b=84,c=91a=49,b=168,c=175a=63,b=280,c=287a=77,b=420,c=427a=91,b=588,c=595a=105,b=784,c=791綜上：當(dāng)c-b=k為奇數(shù)時，通項當(dāng)c-b=k為偶數(shù)時，通項4.3 對，（），對哪些存在本原勾股數(shù)？（140頁練習(xí)12）程序：for k=1:200 for b=1:999 a=sqrt(b+k)2-b2); if(a=floor(a)&gcd(gcd(a,b),(b+k)=1) fprintf(%i,k); break; end endend運(yùn)行結(jié)果：1,2,8,9,18,25,32,49,50,72,81,98,121,128,162,169,200,4.4 設(shè)方程(11.15)的解構(gòu)成