扬州大学MATLAB实验7.1.3

1、1.实验目的（1） 学习MATLAB语言的基本符号运算；（2） 学习MATLAB语言的矩阵符号运算；（3） MAPLE函数调用。2.实验内容在下面的实验操作中，认真理解和查询每项操作的作用和目的。（1） 基本符号运算1） 符号微分，积分。 syms t f1=sin(2*t); df1=diff(f1) df1 = 2*cos(2*t) if1=int(f1) if1 = -1/2*cos(2*t)2）泰勒级数展开。 tf1=taylor(f1,8) tf1 = 2*t-4/3*t3+4/15*t5-8/315*t7 tf1=taylor(f1,8) tf1 = 2*t-4/3*t3+4/15

2、*t5-8/315*t7 3)符号代数方程求解。 syms a b c x f=a*x2+b*x+c; ef=sole(f) ef=solve(f) ef = 1/2/a*(-b+(b2-4*a*c)(1/2) 1/2/a*(-b-(b2-4*a*c)(1/2)4） 符号微分方程求解 f=D2x+2*Dx+10*x=0;g=Dx(0)=1,x(0)=0; dfg=dsolve(f,g) dfg = 1/3*exp(-t)*sin(3*t)5） 积分变换 syms t f1=exp(-2*t)*sin(5*t); F1=laplace(f1) F1 = 5/(s+2)2+25) syms s F

3、2=1/(s+2)2; f2=ilaplace(F2) f2 = t*exp(-2*t) syms t f1=sin(t); Fz1=ztrans(f1) Fz1 = z*sin(1)/(z2-2*z*cos(1)+1) syms z Fz2=z/(z-2)+z/(z-3); fz2=iztrans(Fz2) fz2 = 2n+3n6)计算精度。 g=sym(pi/2 2;3 exp(1) g = pi/2, 2 3, exp(1) g1=vpa(g) g1 = 1.5707963267948966192313216916398, 2. 3., 2.7182818284590452353602

4、874713527 digits Digits = 32 g1=vpa(g,5) g1 = 1.5708, 2. 3., 2.7183（2）符号矩阵运算1）创建与修改符号矩阵。 G1=sym(1/(s+1),s/(s+1)/(s+2);1/(s+1)/(s+2),s/(s+2) G1 = 1/(s+1), s/(s+1)/(s+2) 1/(s+1)/(s+2), s/(s+2) G2=subs(G1,G1(2,2),0) G2 = 1/(s+1), s/(s+1)/(s+2) 1/(s+1)/(s+2), 0 G3=G1(1,1) G3 = 1/(s+1)2） 符号线性代数inv 符号矩阵求逆

5、 det 符号矩阵行列式 eig 符号矩阵特征值 transpose 符号矩阵转置charpoly 符号特征多项式3） 常规符号运算。 syms s d1=1/(s+1);d2=1/(s+2);d=d1*d2 d = 1/(s+1)/(s+2) ad=sym(s+1 s;0 s+2);G=d*ad G = 1/(s+2), s/(s+1)/(s+2) 0, 1/(s+1) n1=1 2 3 4 5;n2=1 2 3; p1=poly2sym(n1);p2=poly2sym(n2); p=p1+p2 p = x4+2*x3+4*x2+6*x+8 pn=sym2poly(p)pn = 1 2 4

6、6 8 a=0 1;-2 -3; gcha=poly2sym(poly(eig(a) gcha = x2+3*x+2 sym s ans = s mg=s*eye(2)-a mg = s, -1 2, s+3 geig=transpose(eig(mg) geig = s+2, s+1 gdet=det(mg) gdet = s2+3*s+2 groot=transpose(solve(gdet) groot = -2, -1 G=inv(mg) G = (s+3)/(s2+3*s+2), 1/(s2+3*s+2) -2/(s2+3*s+2), s/(s2+3*s+2) g21=ilaplac

7、e(G(1,1); g22=ilaplace(G(2,2); g12=ilaplace(G(1,2); g21=ilaplace(G(2,1); g=g11 g12;g21 g22 g = -exp(-2*t)+2*exp(-t), exp(-t)-exp(-2*t) -2*exp(-t)+2*exp(-2*t), 2*exp(-2*t)-exp(-t)（3）MAPLE函数调用 maple 调用maple内核函数 mhelp maple库函数帮助命令 1）MAPLE下的泰勒级数展开。 maple(readlib(mtaylor);); tmap=maple(mtaylor(sin(2*t),t

8、)tmap =2*t-4/3*t3+4/15*t5 syms t;f=sin(2*t); tmat=taylor(f) tmat = 2*t-4/3*t3+4/15*t5 mhelp mtaylor mtaylor - multivariate Taylor series expansionCalling Sequence: mtaylor(f, v) mtaylor(f, v, n) mtaylor(f, v, n, w)Parameters: f - an algebraic expression v - a list or set of names or equations n - (o

9、ptional) non-negative integer w - (optional) list of positive integersDescription:- The mtaylor function computes a multivariate Taylor series expansion of the input expression f, with respect to the variables v, to order n, using the variable weights w. - The result type has changed in Maple V to b

10、e simply a polynomial in the variables with no order term. - The variables v can be a list or set of names or equations. If vi is an equation, then the left-hand side of vi is the variable, and the right-hand side is the point of expansion. If vi is a name, then vi = 0 is assumed as the point of exp

11、ansion. - If the third argument n is present then it specifies the truncation order of the series. The concept of truncation order used is total degree in the variables. If n is not present, the truncation order used is the value of the global variable Order, which is 6 by default. - If the fourth a

12、rgument w is present it specifies the variable weights to be used (by default all 1). A weight of 2 will halve the order in the corresponding variable to which the series is computed. - Note: mtaylor restricts its domain to pure Taylor series, those series with non-negative powers in the variables.

13、- This function must be loaded using readlib(mtaylor).Examples: readlib(mtaylor): mtaylor(sin(x2+y2), x,y); 2 2 x + y mtaylor(sin(x2+y2), x,y, 8); 2 2 6 2 4 4 2 6 x + y - 1/6 x - 1/2 y x - 1/2 y x - 1/6 y mtaylor(sin(x2+y2), x,y, 8, 2,1); 2 2 6 y + x - 1/6 y mtaylor(sin(x2+y2), x=1,y, 3); 2 2 sin(1)

14、 + 2 cos(1) (x - 1) + (-2 sin(1) + cos(1) (x - 1) + cos(1) y mtaylor(f(x,y), x,y, 3); 2f(0, 0) + D1(f)(0, 0) x + D2(f)(0, 0) y + 1/2 D1, 1(f)(0, 0) x 2 + x D1, 2(f)(0, 0) y + 1/2 D2, 2(f)(0, 0) ySee Also: taylor, series, coeftayl 2）MAPLE库函数与MATLAB函数。 FZ1=maple(ztrans(exp(-2*k),k,z);)FZ1 =z/exp(-2)/(

15、z/exp(-2)-1) syms k;f=exp(-2*k); FZ2=ztrans(f) FZ2 = z/exp(-2)/(z/exp(-2)-1) help ztrans - help for sym/ztrans.m - ZTRANS Z-transform. F = ZTRANS(f) is the Z-transform of the scalar sym f with default independent variable n. The default return is a function of z: f = f(n) = F = F(z). The Z-transform

16、 of f is defined as: F(z) = symsum(f(n)/zn, n, 0, inf), where n is fs symbolic variable as determined by FINDSYM. If f = f(z), then ZTRANS(f) returns a function of w: F = F(w). F = ZTRANS(f,w) makes F a function of the sym w instead of the default z: ZTRANS(f,w) F(w) = symsum(f(n)/wn, n, 0, inf). F

17、= ZTRANS(f,k,w) takes f to be a function of the sym variable k: ZTRANS(f,k,w) F(w) = symsum(f(k)/wk, k, 0, inf). Examples: syms k n w z ztrans(2n) returns z/(z-2) ztrans(sin(k*n),w) returns sin(k)*w/(1-2*w*cos(k)+w2) ztrans(cos(n*k),k,z) returns z*(-cos(n)+z)/(-2*z*cos(n)+z2+1) ztrans(cos(n*k),n,w)

18、returns w*(-cos(k)+w)/(-2*w*cos(k)+w2+1) ztrans(sym(f(n+1) returns z*ztrans(f(n),n,z)-f(0)*z See also IZTRANS, LAPLACE, FOURIER. mhelp ztrans ztrans - Z transformCalling Sequence: ztrans(f, n, z)Parameters: f - expression n - name z - name Description:- The function ztrans finds the Z transformation

19、 of f(n) with respect to z. Formally, ztrans(f(n),n,z) = sum(f(n)/zn,n=0.infinity). - ztrans recognizes and specially handles a large class of expressions, and only resorts to using the definition to calculate the transformation if the given expression has an unknown form. If the Z transform of the

20、given expression cannot be found in a closed form, then the left-hand side of the formal definition is returned, rather than the right-hand side. - ztrans recognizes the delta function as charfcn(.) and the step function as Heaviside(.). - This function has to be defined using readlib(ztrans) before

21、 it can be used. Examples: ztrans(f(n+1),n,z); z ztrans(f(n), n, z) - f(0) z ztrans(sin(Pi/2*t),t,z); z - 2 z + 1 ztrans(3n/n!,n,w); exp(3/w) ztrans(beta*5n*(n2+3*n+1),n,z); / z (1/5 z + 1) z z beta |1/5 - + 3/5 - + 1/5 -| | 3 2 1/5 z - 1| (1/5 z - 1) (1/5 z - 1) / ztrans(charfcn5(t)*Psi(t),t,w); 25

22、 - - gamma 12 - 5 w ztrans(n*Heaviside(n-3),n,z); 3 z - 2 - 2 2 z (z - 1) ztrans(invztrans(f(z),z,n),n,z); f(z)See Also: invztrans , inttranslaplace , rsolve 3）MAPLE库函数装入命令readlib。求符号函数F（s）=2/(s+3)2+4),s,t);) F=maple(invlaplace(2/(s+3)2+4),s,t);)F =-1/8*(-16)(1/2)*(exp(-3+1/2*(-16)(1/2)*t)-exp(-3-1/2*(-16)(1/2)*t)4)MAPLE工具包装入命令with。寻找二次多项式的完全平方。 maple(with(student);); a=maple(completesquare(x2+2*x+2)a =(x+1)2+15）MATLAB函数与NAPLE函数。 maple(mtaylo