数学软件实验报告实验三.docx

1、数学软件实验报告实验三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11月8日实验三 MATLAB的符号矩阵运算与符号微积分一实验目的MATLAB 不仅具有数值运算功能，还开发了在matlab环境下实现符号计算的工具包Symbolic Math Toolbox。本次实验的目的对所学的符号矩阵的创建与修改、各种符号运算进行巩固，学会使用数学软件来求极限、微分、积分，解方程和解微分方程等。二实验要求理解符号变量、符号表达式、符号矩阵等概念，掌握符号矩阵和符号表达式的创建，了解符号运算与数值运算的不同点，会修改已有的符号矩阵，并会符号矩阵与

2、数值矩阵的相互转换，掌握符号矩阵矩阵的运算。熟练掌握符号求极限、符号求微分（导数）、符号求积分（不定积分和定积分），掌握符号代数方程（组）求解、符号微分方程（组）求解，了解符号积分变换。三实验内容实验四第三节 矩阵特征值与特征向量求矩阵的特征值与特征向量 A=1 4 2;0 -3 4;0 4 3; v,d=eig(A)v = 1.0000 0.4082 -0.6667 0 -0.8165 -0.3333 0 0.4082 -0.6667d = 1 0 0 0 -5 0 0 0 5 B=-9 -3 -16;13 7 16;3 3 10; c=eig(B)c = -6.0000 10.0000 4

3、.0000 v1,d1=eig(A,B)v1 = 0.8469 0.8279 + 0.1721i 0.8279 - 0.1721i -1.0000 0.1751 - 0.0982i 0.1751 + 0.0982i -0.1798 -0.2898 + 0.0785i -0.2898 - 0.0785id1 = 2.0114 0 0 0 -0.1724 + 0.1486i 0 0 0 -0.1724 - 0.1486i矩阵的对角化 D=1 2 2;2 1 2;2 2 1; trigle(D)ans = 1 1,k=eig(D)? Error: An array for multiple LHS

4、assignment cannot contain numeric value. inv(1)*D*1ans = 1 2 2 2 1 2 2 2 1 F=0 1 1 -1;1 0 -1 1;1 -1 0 1;-1 1 1 0; Fd,Fv=eig(F)Fd = -0.5000 0.2887 0.7887 0.2113 0.5000 -0.2887 0.2113 0.7887 0.5000 -0.2887 0.5774 -0.5774 -0.5000 -0.8660 0 0Fv = -3.0000 0 0 0 0 1.0000 0 0 0 0 1.0000 0 0 0 0 1.0000 Fd*F

5、dans = 1.0000 0.0000 0 0 0.0000 1.0000 -0.0000 0.0000 0 -0.0000 1.0000 0 0 0.0000 0 1.0000 Fd*F*Fdans = -3.0000 0 0 0 -0.0000 1.0000 0 0 0 -0.0000 1.0000 0.0000 0 0.0000 0.0000 1.0000矩阵相似与Jordan标准形第3章 符合运算功能1、符号表达式的生产（1）创建符号函数 f=lpg(x)f =lpg(x)（2）创建符号方程 eqation=a*x2+b*x+c=0eqation =a*x2+b*x+c=0（3）创建

6、符号微分方程 diffeq=Dy-y=xdiffeq =Dy-y=x f=sym(sin(x)f = sin(x) f=sym(sin(x)2=0)f = 2 sin(x) = 0 syms x f=sin(x)+cos(x)f = sin(x) + cos(x)2、符号和数值之间的转换(1). digits函数、vpa函数、subs函数的常用调用格式 s=solve(3*x2-exp(x)=0)s = 1/2 -2 lambertw(- 1/6 3 ) 1/2 -2 lambertw(-1, - 1/6 3 ) 1/2 -2 lambertw(1/6 3 ) vpa(s)ans = 0.91

7、00075730 3.733079028 -0.4589622676 vpa(s,6)ans = 0.910012 3.73306 -0.458964 x=sym(x)x = x f=x-cos(x)f = x - cos(x) f1=subs(f,pi,x)f1 = x - cos(x) digits(25) vpa(f1)ans = x - 1. cos(x)3、符号函数的运算(1). 复合函数运算：功能函数compose syms x y z t u f=1/(1+x2)f = 1 - 2 x + 1 g=sin(y)g = sin(y) h=xth = t x p=exp(-y/u)

8、p = exp(- y/u) compose(f,g)ans = 1 - 2 sin(y) + 1 compose(f,g,t)ans = 1 - 2 sin(t) + 1 compose(h,g,x,z)ans = t sin(z) compose(h,g,t,z)ans = sin(z) x compose(h,p,x,y,z)ans = t exp(- z/u) compose(h,p,t,u,z)ans = exp(- y/z) x(2). 反函数运算：由函数finverse实现 syms s y syms x y f=x2+yf = 2 x + y finverse(f,y)ans

9、= 2 -x + y finverse(f)Warning, finverse(x2+y) is not uniqueans = 1/2 (-y + x)4、符号矩阵的创立(1). 使用sym函数直接创建符号函数 a=sym(1/s+x,sin(x) cos(x)2/(b+x);9,exp(x2+y2),log(tanh(y)Evaluating:Matrix(, , scan=columns );Warning: deprecated syntax used in sym constructor In sym.sym at 163a = 2 cos(x) 1/s + x sin(x) - b

10、 + x 2 2 2 y - 1 9 exp(x ) + exp(y ) - 2 y + 1 (2). 用创建子阵的方法创建符号矩阵 exp(-i),3,x3+y9ans =exp(-i),3,x3+y9 b=a;exp(-i),3,x3+y9b = 2 cos(x) 1/s + x , sin(x) , - b + x 2 2 2 y - 1 9 , exp(x ) + exp(y ) , - 2 y + 1 9 3 9 3 exp(- i), 3, y + x , exp(- i), 3, y + x , 9 3 exp(- i), 3, y + x (3). 将数值矩阵转化为符号矩阵 a

11、=2/3,sqrt(2),0.222;1/4,1/0.23,log(3)a = 0.6667 1.4142 0.2220 0.2500 4.3478 1.0986 b=sym(a) b = 0.666666666666666630 , 1.41421356237309515 , 0.222000000000000003 0.250000000000000000 , 4.34782608695652151 , 1.09861228866810956(4). 符号矩阵的索引和修改 b(2,3)ans = 1.09861228866810956 b(2,3)=99 b = 0.66666666666

12、6666630 , 1.41421356237309515 , 0.222000000000000003 0.250000000000000000 , 4.34782608695652151 , 99.5、符号矩阵的运算（1）基本运算a、符号矩阵的四则运算加减法 a=sym(1/x,1/(x+1);1/(x+2),1/(x+3)a = 1 1/x - x + 1 1 1 - - x + 2 x + 3 b=sym(x,1;x+2,0)b = x 1 x + 2 0 b-a ans = 1 x - 1/x 1 - - x + 1 1 1 x + 2 - - - - x + 2 x + 3 乘除法

13、 baans = 1 1 - - 2 (x + 2) (x + 3) (x + 2) x + 1 2 x + 3 4 - 2 - 2 (x + 3) (x + 1) (x + 2) (x + 2) x abans = 2 2 -(2 x + 7 x + 6) x 1/2 (x + 3 x + 2) x 2 2 (x + 1) (x + 3) - 1/2 (x + 3) x (x + 1)矩阵的转置 aans = 1 1 - - conj(x) 2 + conj(x) 1 1 - - 1 + conj(x) 3 + conj(x)符号矩阵行列式的运算 det(a)ans = 2 - (x + 3

14、) (x + 1) (x + 2) x矩阵的逆 inv(b)ans = 1 0 - x + 2 x 1 - - x + 2符号矩阵的秩 rank(a)ans = 2符号矩阵的幂运算 a2ans = 1 1 1 1 - + - - + - 2 (x + 1) (x + 2) x (x + 1) (x + 1) (x + 3) x 1 1 1 1 - + - - + - (x + 2) x (x + 2) (x + 3) (x + 1) (x + 2) 2 (x + 3) 符号矩阵的指数运算 exp(b)ans = exp(x) exp(1) exp(x + 2) 1 （2）矩阵分解 x,y=ei

15、g(b)x = 1 1 - , - 2 1/2 2 1/2 - 1/2 x + 1/2 (x + 4 x + 8) - 1/2 x - 1/2 (x + 4 x + 8) 1 , 1y = 2 1/2 1/2 x + 1/2 (x + 4 x + 8) 0 2 1/2 0 1/2 x - 1/2 (x + 4 x + 8) syms t real A=0 1;-1 0A = 0 1 -1 0 E=expm(t*A)E = cos(t) sin(t) -sin(t) cos(t) sigma=svd(E)sigma = 2 2 1/2 (cos(t) + sin(t) ) 2 2 1/2 (co

16、s(t) + sin(t) ) simplify(sigma)ans = 2 2 1/2 (cos(t) + sin(t) ) 2 2 1/2 (cos(t) + sin(t) ) a=sym(1 1 2;0 1 3;0 0 2)a = 1 1 2 0 1 3 0 0 2 x,y=jordan(a)x = 5 -5 -5 3 0 -5 1 0 0y = 2 0 0 0 1 1 0 0 1 z=sym(x*y xa sin(y);ta log(y) b;y exp(t) x)Evaluating:Matrix(, , , scan=columns );Warning: deprecated sy

17、ntax used in sym constructor In sym.sym at 163z = a x . y x sin(y) a t log(y) b y exp(t) x triu(z)ans = a x . y x sin(y) 0 log(y) b 0 0 x diag(z)ans = x . y log(y) x tril(t,-1)ans = 0 colspace(a)ans = 1 0 0 0 1 0 0 0 1 a 1 2 3? a 1 2 3 |Error: Unbalanced or unexpected parenthesis or bracket. a= 1 2

18、3 1 2 3 1 2 3a = 1 2 3 1 2 3 1 2 3 null(a)ans = 0.9636 0 -0.1482 -0.8321 -0.2224 0.5547 null(a,r)ans = -2 -3 1 0 0 1 sym xans = x syms x factor(x9-1)ans = 2 6 3 (x - 1) (x + x + 1) (x + x + 1) factor(sym(12345678901234567890)ans = 1, 2, 3, 3, 5, 101, 3803, 3607, 3541, 27961 syms x y expand(x+1)3)ans

19、 = 3 2 x + 3 x + 3 x + 1 expand(sin(x+y)ans = sin(x) cos(y) + cos(x) sin(y) syms x y collect(x2*y+y*x-x2-2+x)ans = 2 -2 + (y - 1) x + (y + 1) x符号矩阵的简化 syms x simplify(sin(x)2+cos(x)2)ans = 2 2 sin(x) + cos(x) syms c alpha beta simplify(exp(c*log(sqrt(alpha+beta)ans = exp(1/2 log(alpha + beta) c) n,d

20、=numden(x/y+y/x)n = 2 2 x + yd = x y horner(x3-6*x2+11*x-6)ans = -6 + (11 + (-6 + x) x) x syms x y expand(x+1)3)ans = 3 2 x + 3 x + 3 x + 1 expand(sin(x+y)ans = sin(x) cos(y) + cos(x) sin(y)6、符号微积分符号极限 syms x a t h limit(sin(x)/x)ans = 1 limit(1+2*t/x)(3*x),x,inf)ans = exp(6 t) limit(1/x,x,0,right)a

21、ns = infinity符号积分 syms x x1 alpha u t; A=cos(x*t),sin(x*t);-sin(x*t),cos(x*t)A = cos(t x) sin(t x) -sin(t x) cos(t x) int(A,t)ans = sin(t x) cos(t x) - - - x x cos(t x) sin(t x) - - x x int(x1*log(1+x1),0,1)ans = 1/4 syms k a simple(symsum(k)ans = 2 1/2 k - 1/2 k simple(symsum(k2,0,n)ans = 6 4 2 2 4

22、 6 4 2 2 4 2 2 1/3 x + x y + x y + 1/3 y + 1/2 x + x y + 1/2 y + 1/6 x + 1/6 y symsum(k2,0,10)ans = 385符号微分和差分 x=sym(x)x = x t=sym(t)t = t diff(sin(x2)ans = 2 2 cos(x ) x diff(t6,6)ans = 720 x=sym(x); y=sym(y); z=sym(z); jacobian(x2+y2;x2-y2,x y)ans = 2 x 2 y 2 x -2 y7、符号代数方程求解线性方程组的符号解法x,y,z=solve(

23、10*x-y=9,-x+10*y-2*z=7,-x+10*y-2*z=7,-2*y+10*z=6)Warning, solving 4 equations for 3 variablesx = 473 - 475y = 91 - 95 z = 376 - 475 vpa(x,y,z)ans = 0.9957894737, 0.9578947368, 0.7915789474非线性方程组符号解法 x0=0.5 0.5; fsolve(fc,x0)ans = 0.5265 0.50798、符号微分方程求解 dsolve(Dx=-a*x)ans = _C1 exp(-a t) y=dsolve(Dy=1/sqrt(y),y(0)=1)y = 2/3 1/4 (12 t + 8)四、实验总结 在这次试验中，我做了矩阵的特征值与特征向量的内容。在做矩阵相似于Jordon标准形的时候，出现了一些错误，到最后还是没有把它做出来。 之后做了第3章,符合运算功