1、数学软件实验报告实验二数学软件实验报告学院名称：理学院 专业年级： 姓 名： 学 号：课 程：数学软件实验 报告日期：2014年10月25日实验二 MATLAB的基本数值运算一实验目的MATLAB具有非常强大的数值计算能力，对各种常量（包括一般常量和固定常量）、各种变量（包括数值变量、字符变量、单元形变量和结构变量）熟悉其用法，向量及其运算，矩阵及其运算，数组及其运算，多项式及其各种运算，线性方程组的求解，数值统计的基本函数及其应用，简单插值函数，简单优化函数，微分方程数值解的基本函数等。二实验要求理解常量、变量、向量、矩阵、多项式等概念，掌握向量在MATLAB中的表示，熟练掌握矩阵及其运算，
2、数组及其运算，多项式运算，线性方程组求解，了解数值统计的基本函数，简单插值函数，简单优化函数，微分方程数值解的基本函数等。三实验内容实验一：第二节 基本数学运算一：简单矩阵的建立与矩阵元素（1）直接输入矩阵 a=1 2 3;4 5 6;7 8 9a = 1 2 3 4 5 6 7 8 9（2）通过语句或者函数产生矩阵 b=sin(a)b = 0.8415 0.9093 0.1411 -0.7568 -0.9589 -0.2794 0.6570 0.9894 0.4121 c=a+0.1*(1+b/2)c = 1.1421 2.1455 3.1071 4.0622 5.0521 6.0860 7
3、.1328 8.1495 9.1206 d=ones(3)+eye(3)d = 2 1 1 1 2 1 1 1 2（3）利用文件创建矩阵 load filel.txt filelfilel = 1 2 3 4 5 6 7 8 9（4）从外部数据文件中装入已有矩阵load filel.txt x=-1.3 sqrt(3) (1+2+3)*4/5x = -1.3000 1.7321 4.8000 定义或修改某一元素load filel.txt x(5)=abs(x(1)x = -1.3000 1.7321 4.8000 0 1.3000小矩阵生成大矩阵load filel.txt e=a,be =
4、 1.0000 2.0000 3.0000 0.8415 0.9093 0.1411 4.0000 5.0000 6.0000 -0.7568 -0.9589 -0.2794 7.0000 8.0000 9.0000 0.6570 0.9894 0.4121 e=a,de = 1 2 3 2 1 1 4 5 6 1 2 1 7 8 9 1 1 2大矩阵抽取元素变为小矩阵f=e(:,2,5)f = 2 1 5 2 8 12：常量，变量与表达式t=how about this character stringt =how about this character string v=I cant f
5、ind the litterv =I cant find the litter A=HelloA =Hello B=2*AB = 144 202 216 216 2223：命令窗口常用的命令与标点符号的使用help cd CD Change current working directory. CD directory-spec sets the current directory to the one specified. CD . moves to the directory above the current one. CD, by itself, prints out the curr
6、ent directory. WD = CD returns the current directory as a string. Use the functional form of CD, such as CD(directory-spec), when the directory specification is stored in a string. See also pwd. Overloaded methods: ftp/cd Reference page in Help browser doc cd二：算数运算1：加减法运算a=1 2;3 4;b=ones(2);c=a+bc =
7、 2 3 4 5 d=1 2 3;e=a+d? Error using = plusMatrix dimensions must agree. c1=c-1c1 = 1 2 3 42：乘除法运算f=a*bf = 3 3 7 7 g=b*ag = 4 6 4 6 h=pi*ah = 3.1416 6.2832 9.4248 12.5664 F=a.*cF = 2 6 12 20 G=c.*aG = 2 6 12 20 a=1 2 3;4 2 6;7 4 9 b=4 1 2 a*x=b? a=1 2 3;4 2 6;7 4 9 b=4 1 2 a*x=b |Error: Unexpected MA
8、TLAB expression.四a=rand(3),b=rand(3),a = 0.8147 0.9134 0.2785 0.9058 0.6324 0.5469 0.1270 0.0975 0.9575b = 0.9649 0.9572 0.1419 0.1576 0.4854 0.4218 0.9706 0.8003 0.9157 A1=a/bA1 = 0.7571 0.3356 0.0323 0.2462 -0.4341 0.7590 -0.9446 0.4093 1.0035 A2=abA2 = -2.5775 -1.3591 -0.0618 3.0365 2.0130 -0.086
9、3 1.0462 0.8110 0.9734 A3=baA3 = -1.8233 -1.1435 -0.2172 2.7367 2.1961 0.3685 -0.3205 -0.6006 0.9537 A4=b/aA4 = 0.8306 0.3601 -0.2991 1.0730 -0.8795 0.6307 0.3442 0.6978 0.4577 A5=(a/b)A5 = -1.8233 -1.1435 -0.2172 2.7367 2.1961 0.3685 -0.3205 -0.6006 0.9537 A6=a.bA6 = 1.1843 1.0479 0.5095 0.1740 0.7
10、676 0.7712 7.6433 8.2046 0.9564 A7=a.bA7 = 1.1843 1.0479 0.5095 0.1740 0.7676 0.7712 7.6433 8.2046 0.9564 A8=1./A7A8 = 0.8444 0.9542 1.9628 5.7469 1.3028 1.2967 0.1308 0.1219 1.04563.乘幂运算g=1 2 3 4;5 6 7 8;9 10 11 12g = 1 2 3 4 5 6 7 8 9 10 11 12 g=1 2 3 4;5 6 7 8;9 10 11 12; g.2 %对g中的元素求平方ans = 1 4
11、9 16 25 36 49 64 81 100 121 144 h=1 1 1 1;2 2 2 2;3 3 3 3; g.(h-1) %求以g元素为底，以h中相应元素减一为幂指数产生的矩阵ans = 1 1 1 1 5 6 7 8 81 100 121 144 2.g %以2为底，以中相应元素为幂指数产生的矩阵ans = 2 4 8 16 32 64 128 256 512 1024 2048 40964：转置运算x=1 2 3;4 5 6;7 8 9x = 1 2 3 4 5 6 7 8 9 y=xy = 1 4 7 2 5 8 3 6 9 a=1+2i 2-3i;4+5i 5-6ia =
12、1.0000 + 2.0000i 2.0000 - 3.0000i 4.0000 + 5.0000i 5.0000 - 6.0000i b=ab = 1.0000 - 2.0000i 4.0000 - 5.0000i 2.0000 + 3.0000i 5.0000 + 6.0000i b=a.b = 1.0000 + 2.0000i 4.0000 + 5.0000i 2.0000 - 3.0000i 5.0000 - 6.0000i conj(a)ans = 1.0000 + 2.0000i 4.0000 + 5.0000i 2.0000 - 3.0000i 5.0000 - 6.0000i三
13、：关系运算与逻辑运算1：关系运算a=-1 2 4;5 4 8;b=0 1 5;5 1 2; c=abc = 0 1 0 0 1 12.逻辑运算（1）逻辑运算与（&）a=-1 2 4;5 4 8;b=0 1 5;5 1 2; c=a&bc = 0 1 1 1 1 1（2）逻辑运算与（|）c=a|bc = 1 1 1 1 1 1（3）逻辑非 c=ac = 0 0 0 0 0 0四：建立特殊数据组1.用特殊函数建立数组2.用小数组建大数组a=1 2;3 4a = 1 2 3 4 b=a,eye(2,3);ones(3,2),rand(3)b = 1.0000 2.0000 1.0000 0 0 3.
14、0000 4.0000 0 1.0000 0 1.0000 1.0000 0.7922 0.0357 0.6787 1.0000 1.0000 0.9595 0.8491 0.7577 1.0000 1.0000 0.6557 0.9340 0.74313.利用冒号建立数组x=1:5x = 1 2 3 4 5 y=0:pi/4:piy = 0 0.7854 1.5708 2.3562 3.1416 z=6:-1:1z = 6 5 4 3 2 1 a=0:0.2:1;b=exp(-a).*sin(a);a,bans = 0 0 0.2000 0.1627 0.4000 0.2610 0.6000
15、 0.3099 0.8000 0.3223 1.0000 0.30964.空数组x=x = y=1:-3y = Empty matrix: 1-by-0 a=1 2 3;4 5 6a = 1 2 3 4 5 6 a(:,3)=a = 1 2 4 5实验四第一节 1、由文件生成和保存矩阵 clear %清除当前工作空间中的变量 myfile %执行M文件A = 3 4 -1 1 -9 10 6 5 0 7 4 -16 1 -4 7 -1 6 -8 2 -4 5 -6 12 -8 -3 6 -7 8 -1 1 8 -4 9 1 3 0 who %查看工作空间中的变量Your variables a
16、re:A load txefile.txt %装入txtfile.txt文件 whoYour variables are:A save matfile %保存工作空间变量到matfile.mat文件中 clear who load matfile whoYour variables are:A txtfile %显示变量txtfile的内容2、由函数生成矩阵 eye(5,6)ans = 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 eye(5)ans = 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0
17、 0 1 0 0 0 0 0 1 ones(8)ans = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 zeros(4)ans = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 rand(3)ans = 0.8147 0.9134 0.2785 0.9058 0.6324 0.5469 0.1270 0.0975 0.9575 diag(4)ans = 43、符号矩阵
18、的创建 exam=sym(1,x/a,sin(x);y/x,1+1/y,tan(x/y);1=0,3+3,4*r)? Error using = sym.sym at 198Error using = sym.sym at 165Error using = maplemex; unexpected syms x y z a b cf=a*x2+b*x+c;g=x*y*z;h=(f+g)*b/a;e1=sym( a*x2+b*x+c=0); e2=sym(x*y*z=0); e3=sym(h=0);M=1 2 3 xf g h ye1 e2 e3 zM = 1 , 2 , 3 , x 2 2 b
19、 (a x + x y z + b x + c) a x + b x + c , z y x , - , y a 2 a x + b x + c = 0 , z y x = 0 , h = 0 , z syms x y c ra=sin(c+(r-1)*3);b=exp(r+(c-1)*4);c=(c+(r-1)*3)*x+(r+(c-1)*4)*y;A=symmat(3,3,a)forC = 1 2 34、矩阵的修改 A=rand(5)A = 0.8147 0.0975 0.1576 0.1419 0.6557 0.9058 0.2785 0.9706 0.4218 0.0357 0.127
20、0 0.5469 0.9572 0.9157 0.8491 0.9134 0.9575 0.4854 0.7922 0.9340 0.6324 0.9649 0.8003 0.9595 0.6787 A(1 3,:)=A = 0.9058 0.2785 0.9706 0.4218 0.0357 0.9134 0.9575 0.4854 0.7922 0.9340 0.6324 0.9649 0.8003 0.9595 0.6787 A(2:3,2:3)=eye(2)A = 0.9058 0.2785 0.9706 0.4218 0.0357 0.9134 1.0000 0 0.7922 0.9
21、340 0.6324 0 1.0000 0.9595 0.6787二：矩阵与向量的基本运算（1）矩阵的运算A+txtfile %矩阵加法ans = 4 6 3 7 -12 12 13 14 16 2 12 -23 9 7 27 0 11 -2 12 11 33 7 11 1 9 25 29 33 -8 24 10 0 15 -2 3 5 A-txtfile %矩阵减法ans = 2 2 -5 -5 -6 8 -1 -4 -16 12 -4 -9 -7 -15 -13 -2 1 -14 -8 -19 -23 -19 13 -17 -15 -13 -43 -17 6 -22 6 -8 3 4 3
22、-5 A*txtfile %矩阵乘法ans = -55 -85 -180 -245 80 -176 127 174 348 250 -13 52 75 110 220 194 -41 161 82 129 260 283 -91 244 53 76 138 21 21 -36 98 151 284 165 -33 176 3*A+7*txtfile %数乘矩阵ans = 16 26 25 45 -48 44 67 78 112 -14 68 -97 59 65 161 4 53 18 76 93 211 73 29 39 75 151 231 199 -52 164 38 16 69 -18
23、9 35 A %矩阵转置ans = 3 6 1 2 -3 8 4 5 -4 -4 6 -4 -1 0 7 5 -7 9 1 7 -1 -6 8 1 -9 4 6 12 -1 3 10 -16 -8 -8 1 0 txtfile/m %矩阵右除ans = 0.9337 -0.1687 1.5921 -0.2383 0.7938 0.0596 5.9273 -0.0162 5.5138 3.0371 1.8754 -2.0808 7.2759 -0.7621 6.9845 3.3142 3.4372 -1.5695 9.1056 -1.0471 10.1798 2.7908 4.9550 -1.4
24、914 11.3503 -1.9092 13.2537 2.7827 7.2407 -0.9615 2.8057 -0.3272 2.3154 1.3410 0.9586 -0.8219 Atxtfile %矩阵左除ans = -2.1783 -3.1414 -6.0051 -3.2105 0.1122 -3.2168 5.0944 7.5348 14.2952 7.7724 -0.7515 5.6699 3.6894 5.4000 10.1410 4.5299 0.0113 4.5038 0.2344 0.4858 0.8027 0.2483 -0.2744 1.5049 2.1216 3.
25、3950 6.3834 4.2520 -1.2436 3.7915 0.9706 1.6754 3.1624 2.7091 -1.1237 2.6093det(A)ans = 245295det(txtfile)ans = 0inv(A) %求逆矩阵ans = -0.0737 0.0604 -0.2297 0.0067 -0.0804 0.1042 0.3142 0.0036 0.2408 0.1605 0.1259 -0.1436 0.2099 -0.0395 0.3155 0.0364 0.0834 -0.0663 -0.0827 -0.0123 0.0088 -0.0777 0.0779
26、 0.0878 0.0134 -0.0335 -0.0159 0.1129 0.1061 0.0337 0.0377 -0.0525 -0.0110 0.0469 0.0698 0.0411 pinv(txtfile) %求广义逆矩阵ans = -0.0187 -0.0022 0.0340 -0.0084 0.0040 -0.0398 0.0363 0.0274 -0.0564 0.0272 -0.0092 0.0849 0.0214 0.0265 -0.0275 0.0221 -0.0037 0.0387 -0.0247 -0.0217 0.0469 -0.0110 0.0244 -0.16
27、94 -0.1294 -0.0547 0.2146 -0.0805 0.0231 -0.2439 -0.0621 -0.0659 0.0926 -0.0516 0.0248 -0.0000矩阵的迹,翻书，条件数与秩：trace(A)ans = 8 trace(txtfile)ans = 41 norm(A)ans = 28.5398 norm(A,1)ans = 43cond(A)ans = 18.6569 cond(A,1)ans = 35.3343 rank(A)ans = 6 rank(txtfile)ans = 4第二节;解线性方程组（1）其次线性方程组的求解A为奇次线性方程组对应的系
28、数矩阵A=1 -2 3 -4;0 1 -1 1;-1 0 -1 2;1 -3 4 -5; a=null(A)a = -0.1402 0.8044 0.4723 -0.3321 0.8044 0.1402 0.3321 0.4723若求方程中含有最多零元素个数的解：a=null(A,r)a = -1 2 1 -1 1 0 0 1（2）恰定方程组求解 A=rand(100)*1.e2;x=ones(100,1);b=A*x;ticy=inv(A)*b;tocElapsed time is 26.456033 seconds.err=norm(y-x)err = 1.1380e-012res=norm(A*y-b)res = 4.6191e-010ticy=Ab;tocElapsed time is 54.438244 seconds.err=norm(y-x)err =
