Matlab课后习题解答.doc

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1、Four short words sum up what has lifted most successful individuals above the crowd: a little bit more.-author-dateMatlab课后习题解答P16P16Q2: 计算表达式在和时的函数值。function y=jie(x)y=tan(-x.2)*acos(x); jie(0.25)ans = -0.0825 jie(0.78*pi)ans = 0 + 0.4418iQ3：编写M命令文件，求的值。a=0;b=0;for i=1:50 a=a+i*i;endfor j=1:10 b=b+

2、1/j;endc=a+b; cc = 4.2928e+004P27Q2：矩阵,计算,并比较两者的区别。 A=1 2 3;4 5 6;7 8 9; B=4 6 8;5 5 6;3 2 2; A*Bans = 23 22 26 59 61 74 95 100 122 A.*Bans = 4 12 24 20 25 3621 16 18A*B表示A与B两矩阵相乘。A.*B表示A与B对应元素相乘。P34Q2：编写一个转换成绩等级的程序，其中成绩等级转换标准为：考试分数在显示为优秀；分数在的显示为良好；分数在的显示为及格；分数在的显示为不及格。if x=90 disp(优秀); elseif x=80

3、disp(良好); elseif x=60 disp(及格); else disp(不及格);end x=85x = 85良好Q3：编写函数，计算 sum=0; for i=1:50a=1;for j=1:ia=a*j;endsum=sum+a;end sumsum = 3.1035e+064P79Q1: 绘制的图像，要求用蓝色的星号符号画图；并且画出其包络线的图像，用红色的点划线画图。 x=0:pi/25:4*pi; y1=exp(x/3).*sin(3*x);y2=exp(x/3);y3=-exp(x/3); plot(x,y1,b*,x,y2,r-.,x,y3,r-.)P113Q8: 已

4、知矩阵，实现下列操作：（1）添加零元素使之成为一个的方阵。（2）在以上操作的基础上，将第三行元素替换为（1 3 5）。（3）在以上操作的基础上，提取矩阵中第2个元素以及第3行第2列的元素。(1) A=1,2;3,4; A=A;0,0A = 1 2 3 4 0 0 B=0;0;0; A=A,BA = 1 2 0 3 4 0 0 0 0(2) A(3,:)=1 3 5A = 1 2 0 3 4 0 1 3 5(3) a=A(1,2)a = 2 b=A(3,2)b = 3Q10: 已知矩阵A=，B=，求A+B，A-B，AB，BA，。 A=1 3;3 5; B=2 4;6 8; a=A+Ba = 3

5、7 9 13 b=A-Bb = -1 -1 -3 -3 c=A*Bc = 20 28 36 52 d=B*Ad = 14 26 30 58 e=det(A)e = -4 f=det(B)f = -8Q14: 求矩阵A= 的特征多项式、特征值和特征向量。 A=2 1 1;1 2 1;1 1 2; p=poly(A)p = 1.0000 -6.0000 9.0000 -4.0000特征多项式为：; V,D=eig(A)V = 0.4082 0.7071 0.5774 0.4082 -0.7071 0.5774 -0.8165 0 0.5774D = 1.0000 0 0 0 1.0000 0 0

6、0 4.0000返回A的特征值矩阵D中，主对角线的元素1、1、4为特征值；特征向量矩阵V的列向量分别是特征值1、1、4所对应的特征向量。Q17: 将下列矩阵转化为稀疏矩阵，之后再将转化后的稀疏矩阵还原为全元素矩阵。(1) A=2 0 0 1;0 -2 1 0;0 1 0 0;1 0 0 -2; S=sparse(A)S = (1,1) 2 (4,1) 1 (2,2) -2 (3,2) 1 (2,3) 1 (1,4) 1 (4,4) -2 A1=full(S)A1 = 2 0 0 1 0 -2 1 0 0 1 0 0 1 0 0 -2(2) A=1 0 0 -1 0;0 0 2 0 0;0 1

7、0 0 3; S=sparse(A)S = (1,1) 1 (3,2) 1 (2,3) 2 (1,4) -1 (3,5) 3 A1=full(S)A1 = 1 0 0 -1 0 0 0 2 0 0 0 1 0 0 3(3) A=1 0 0 0 2;0 0 0 3 0;0 0 1 0 0;0 3 0 0 0;2 0 0 0 1; S=sparse(A)S = (1,1) 1 (5,1) 2 (4,2) 3 (3,3) 1 (2,4) 3 (1,5) 2 (5,5) 1 A1=full(S)A1 = 1 0 0 0 2 0 0 0 3 0 0 0 1 0 0 0 3 0 0 0 2 0 0 0 1

8、Q20: 求解下列方程(1) A=1 0 3 10;2 1 4 18;1 -1 2 3; rref(A)ans = 1 0 0 1 0 1 0 4 0 0 1 3增广矩阵的秩为3，等于系数矩阵的秩，等于未知数的个数。所以有唯一解。（2） A=2 -1 3 0 13;1 4 -2 1 -8;5 3 2 1 10;2 3 1 -1 -6; rref(A)ans = 1 0 0 0 1 0 1 0 0 -2 0 0 1 0 3 0 0 0 1 5增广矩阵的秩为4，等于系数矩阵的秩，等于未知数的个数。所以有唯一解。Q23: 通过矩阵LU分解求解矩阵方程AX=b,其中A= ，b= A=1 0 2 0;0

9、 1 0 1;1 2 4 3;0 1 0 3; L,U=lu(A); b=1;2;-1;5; x=U(Lb)x = 8.5000 0.5000 -3.7500 1.5000Q25: 用QR方法求解下列方程组，然后用其他方法验证解的正确性。(1) A=5 4 5;7 8 9;12 3 8; b=1;2;3; Q,R=qr(A)Q = -0.3386 -0.2552 -0.9057 -0.4741 -0.7851 0.3985 -0.8127 0.5643 0.1449R = -14.7648 -7.5856 -12.4621 0 -5.6088 -3.8275 0 0 0.2174 x=R(Qb

10、)x = -0.5000 -1.0000 1.5000检验： L,U=lu(A); x=U(Lb)x = -0.5000 -1.0000 1.5000结果相同，说明结果正确。(2) A=3 4 5;7 8 9;12 3 8; b=4;2;3; Q,R=qr(A)Q = -0.2111 -0.4124 -0.8862 -0.4925 -0.7383 0.4608 -0.8443 0.5338 -0.0473R = -14.2127 -7.3174 -12.2426 0 -5.9544 -4.4363 0 0 -0.6617 x=R(Qb)x = -1.8214 -2.8571 4.1786检验：

11、 L,U=lu(A); x=U(Lb)x = -1.8214 -2.8571 4.1786结果相同，说明结果正确。Q26: 将下列矩阵进行Cholesky分解。(1) A=1 -1 2 1;-1 3 0 -3;2 0 9 -6;1 -3 -6 19; R=chol(A)R = 1.0000 -1.0000 2.0000 1.0000 0 1.4142 1.4142 -1.4142 0 0 1.7321 -3.4641 0 0 0 2.0000验证： R*Rans = 1.0000 -1.0000 2.0000 1.0000 -1.0000 3.0000 0 -3.0000 2.0000 0 9

12、.0000 -6.0000 1.0000 -3.0000 -6.0000 19.0000(2) a=1/sqrt(2),-1/sqrt(2),0,0; b=-1/sqrt(2),1/sqrt(2),0,0; c=0,0,1/sqrt(2),-1/sqrt(2); d=0,0,-1/sqrt(2),1/sqrt(2); A=a;b;c;d; R,p=chol(A)R = 0.8409 -0.8409 0 0 0 0.0000 0 0 0 0 0.8409 -0.8409 0 0 0 0.0000p = 0P=0说明A是个对称正定矩阵。P130Q3: 若多项式，求f(-3),f(7)及f（A）的值

13、，其中A=。 p=4 -3 1;x=-3 7;A=1 2;-2 3; y=polyval(p,x)y = 46 176 Y=polyval(p,A)Y = 2 1123 28Q5: 求多项式与的商及余子式。 p1=8,6,-1,4;p2=2,-1,-1; ps,pr=deconv(p1,p2)ps = 4 5pr = 0 0 8 9 ps=poly2str(ps,x)ps = 4 x + 5 pr=poly2str(pr,x)pr = 8 x + 9以上两个多项式的商为，余子式为pr=8 x + 9.Q8:在钢线碳含量对于电阻的效应的研究中，得到以下数据。分别用一次、三次、五次多项式拟合曲线来

14、拟合这组数据并画出图形。碳含量x0.100.300.400.550.700.800.95电阻y1518192122.623.826 x=0.1,0.3,0.4,0.55,0.7,0.8,0.95;y=15,18,19,21,22.6,23.8,26;p1=polyfit(x,y,1);p3=polyfit(x,y,3);p5=polyfit(x,y,5);disp(一阶拟合函数),f1=poly2str(p1,x)disp(三阶拟合函数),f3=poly2str(p3,x)disp(五阶拟合函数),f5=poly2str(p5,x)x1=0.1,0.3,0.4,0.55,0.7,0.8,0.9

15、5;y1=polyval(p1,x1);y3=polyval(p3,x1);y5=polyval(p5,x1);plot(x,y,rp,x1,y1,x1,y3,x1,y5);legend(拟合点,一次拟合,三次拟合,五次拟合)一阶拟合函数f1 = 12.5503 x + 13.9584三阶拟合函数f3 = 8.9254 x3 - 14.6277 x2 + 19.2834 x + 13.2132五阶拟合函数f5 = 146.1598 x5 - 386.879 x4 + 385.5329 x3 - 178.8558 x2 + 49.9448 x + 11.4481P165Q1:用MATLAB软件求

16、下列数列极限：(1) syms n limit(-2)n+3*n)/(-2)(n+1)+3(n+1),n,inf) ans = 0(4) syms n limit(sqrt(n+2)-2*sqrt(n+1)+sqrt(n),n,inf) ans = 0 Q2:用MATLAB软件求下列函数极限：(1) syms x limit(1+x)(1/3)-1)/x),x,0) ans = 1/3 (4) syms x limit(x3-x2+x/2).*exp(1/x)-sqrt(x6+1),x,inf) ans = 1/6 Q3:求下列函数的导数。(1) syms x f=sqrt(x+sqrt(x+

17、sqrt(x); diff(f,x) ans = 1/2/(x+(x+x(1/2)(1/2)(1/2)*(1+1/2/(x+x(1/2)(1/2)*(1+1/2/x(1/2) (2) syms x f=sqrt(x+2)*(3-x)4/(x+1)5; diff(f,x) ans = 1/2/(x+2)(1/2)*(3-x)4/(1+x)5-4*(x+2)(1/2)*(3-x)3/(1+x)5-5*(x+2)(1/2)*(3-x)4/(1+x)6 (3) syms x f=(1+sin(x)/(1+cos(x); diff(f,x) ans = cos(x)/(1+cos(x)+(1+sin(x

18、)/(1+cos(x)2*sin(x) (4) syms x f=x.*cos(2*x).*cos(3*x); diff(f,x) ans = cos(2*x)*cos(3*x)-2*x*sin(2*x)*cos(3*x)-3*x*cos(2*x)*sin(3*x) Q4:求高阶导数。（1）已知，求。（2）求的40阶导数。（3）已知，求。(1) syms x b f=x*sin(b*x); diff(f,x,3) ans = -3*sin(b*x)*b2-x*cos(b*x)*b3 (2) x=sym(x) x = x diff(x4*cos(7*x),40) ans = 5816200368

19、074729538861238724268369360*cos(7*x)-4401448927191687218597694169716603840*x*sin(7*x)-1216189835145071468296731283737482640*x2*cos(7*x)+145526988820777782531232803182262880*x3*sin(7*x)+6366805760909027985741435139224001*x4*cos(7*x) (3) x=sym(x); diff(sqrt(x*sin(sqrt(3(exp(x)-log(x),3) ans = 3/8/(x*s

20、in(3(exp(x)-log(x)(1/2)(5/2)*(sin(3(exp(x)-log(x)(1/2)+1/2*x*cos(3(exp(x)-log(x)(1/2)*(3(exp(x)-log(x)(1/2)*(exp(x)-1/x)*log(3)3-3/4/(x*sin(3(exp(x)-log(x)(1/2)(3/2)*(sin(3(exp(x)-log(x)(1/2)+1/2*x*cos(3(exp(x)-log(x)(1/2)*(3(exp(x)-log(x)(1/2)*(exp(x)-1/x)*log(3)*(cos(3(exp(x)-log(x)(1/2)*(3(exp(x)

21、-log(x)(1/2)*(exp(x)-1/x)*log(3)-1/4*x*sin(3(exp(x)-log(x)(1/2)*3(exp(x)-log(x)*(exp(x)-1/x)2*log(3)2+1/4*x*cos(3(exp(x)-log(x)(1/2)*(3(exp(x)-log(x)(1/2)*(exp(x)-1/x)2*log(3)2+1/2*x*cos(3(exp(x)-log(x)(1/2)*(3(exp(x)-log(x)(1/2)*(exp(x)+1/x2)*log(3)+1/2/(x*sin(3(exp(x)-log(x)(1/2)(1/2)*(-3/4*sin(3(

22、exp(x)-log(x)(1/2)*3(exp(x)-log(x)*(exp(x)-1/x)2*log(3)2+3/4*cos(3(exp(x)-log(x)(1/2)*(3(exp(x)-log(x)(1/2)*(exp(x)-1/x)2*log(3)2+3/2*cos(3(exp(x)-log(x)(1/2)*(3(exp(x)-log(x)(1/2)*(exp(x)+1/x2)*log(3)-1/8*x*cos(3(exp(x)-log(x)(1/2)*(3(exp(x)-log(x)(3/2)*(exp(x)-1/x)3*log(3)3-3/8*x*sin(3(exp(x)-log(

23、x)(1/2)*3(exp(x)-log(x)*(exp(x)-1/x)3*log(3)3-3/4*x*sin(3(exp(x)-log(x)(1/2)*3(exp(x)-log(x)*(exp(x)-1/x)*log(3)2*(exp(x)+1/x2)+1/8*x*cos(3(exp(x)-log(x)(1/2)*(3(exp(x)-log(x)(1/2)*(exp(x)-1/x)3*log(3)3+3/4*x*cos(3(exp(x)-log(x)(1/2)*(3(exp(x)-log(x)(1/2)*(exp(x)-1/x)*log(3)2*(exp(x)+1/x2)+1/2*x*cos

24、(3(exp(x)-log(x)(1/2)*(3(exp(x)-log(x)(1/2)*(exp(x)-2/x3)*log(3) Q8:求由方程所确定的隐函数的导数。 syms x y F=y5+2*y-x-3*x7; dF_dx=diff(F,x); dF_dy=diff(F,y); pretty(-dF_dx/dF_dy) 6 1 + 21 x - 4 5 y + 2即。代入x=0，即。Q9:求下列函数的，和：(1) syms x y z=sin(x*y)+cos(x*y).2; diff(z,x,2) ans = -sin(x*y)*y2+2*sin(x*y)2*y2-2*cos(x*y

25、)2*y2 diff(z,y,2) ans = -sin(x*y)*x2+2*sin(x*y)2*x2-2*cos(x*y)2*x2 diff(diff(z,x),y) ans = -sin(x*y)*x*y+cos(x*y)+2*sin(x*y)2*x*y-2*cos(x*y)2*x*y-2*cos(x*y)*sin(x*y) (4) syms x y z=exp(-(1/x+1/y); diff(z,x,2) ans = -2/x3*exp(-1/x-1/y)+1/x4*exp(-1/x-1/y) diff(z,y,2) ans = -2/y3*exp(-1/x-1/y)+1/y4*exp

26、(-1/x-1/y) diff(diff(z,x),y) ans = 1/x2/y2*exp(-1/x-1/y) Q12:求下列函数的极值。(1) x=-4:0.1:4; y=x.(2/3).*(x.2-8); plot(x,y)从图看出函数有极大极小值。 f=x.(2/3).*(x.2-8); xmin=fminbnd(f,-4,4)xmin = 1.278404849559600 - 0.260098701930077i x=xmin; miny=x.(2/3).*(x.2-8)miny = -7.718305008002356 + 0.237838653225849i x=-4:0.1:

27、4; f=-x.(2/3).*(x.2-8); xmax=fminbnd(f,-4,4) Exiting: Maximum number of function evaluations has been exceeded - increase MaxFunEvals option. Current function value: -3.905290 xmax = -1.506656018341058 + 0.169172989824599i x=xmax; maxy=x.(2/3).*(x.2-8)maxy = 3.905290280403644 - 6.554360315624097i(3)

28、 x=-2:0.1:4; y=x.3-4.*x.2-3.*x; plot(x,y)从图看出函数有极大极小值。 f=x.3-4.*x.2-3.*x; xmin=fminbnd(f,-4,4)xmin = 2.999998479983027 x=xmin; miny=x.3-4.*x.2-3.*xminy = -17.999999999988447 x=-2:0.1:4; f=-(x.3-4.*x.2-3.*x); xmax=fminbnd(f,-4,4)xmax = -0.333331121184471 x=xmax; maxy=x.3-4.*x.2-3.*xmaxy = 0.518518518

29、494050 (5) syms xy=(log(x).2/x;y1=diff(y) y1 = 2*log(x)/x2-log(x)2/x2 x0=solve(y1)xmin=fminbnd(log(x).2/x,0.5, 9) x=xmin;miny=(log(x).2/x x0 = 1 exp(2) xmin = 1.000000702315345miny = 4.932461516052710e-013 xmax=fminbnd(-(log(x).2/x),0.5, 9)xmax = 7.389063295324329Q14:求下列不定积分：(1) syms x f=sin(x)*cos(

30、x)/(1+sin(x)*4); int(f,x) ans = 1/4*sin(x)-1/16*log(1+4*sin(x)(3) syms x f=asin(x)/sqrt(1-x); int(f,x) ans = -2*(1-x)(1/2)*asin(x)-4*(1-x)(1/2)*(-1-x)/(2-2*x-(1-x)2)(1/2)(5) syms x f=(x6+x4-4*x2-2)/(x3*(x2+1)2); int(f,x) ans = 1/2*log(x2+1)+1/x2-1/(x2+1)Q15:求下列定积分：(2) syms x int(x2*(2-3*x2),0,1) ans

31、 = 1/15(4) syms x int(sin(5*x)*cos(4*x),0,pi/2) ans = 5/9(6) syms x int(x*(cos(x)2),0,pi*2) ans = pi2 Q17:用三种方法求下列积分的数值解：(1)精确值为： syms x int(exp(-0.5*x)*sin(x+pi/6),0,3) ans = 1.0917292622819022219011193208276 近似值为： h=0.01;x=0:h:3; y=exp(-0.5).*x).*sin(x+pi/6); format long t=length(x); z1=sum(y(1:(t

32、-1)*h z2=sum(y(2:t)*h z3=trapz(x,y) z4=quad(exp(-0.5).*x).*sin(x+pi/6),0,3)矩形法：z1 = 1.094638645293393z2 = 1.088806854576796梯形法：z3 = 1.091722749935095辛普生法：z4 = 1.091729265255455(3)精确值为： syms x int(exp(-x),1,2.5) ans = exp(-1)-exp(-5/2) z=exp(-1)-exp(-5/2)z= 0.285794442547544近似值为： h=0.01;x=1:h:2.5; y=

33、exp(-x); format long t=length(x); z1=sum(y(1:(t-1)*h z2=sum(y(2:t)*h z3=trapz(x,y) z4=quad(exp(-x),1,2.5)矩形法：z1 = 0.287225796376666z2 = 0.284367851951191梯形法：z3 = 0.285796824163929辛普生法：z4 = 0.285794449331167(5)精确值为： syms x int(exp(3*x)*sin(2*x),0,2) ans = 2/13-2/13*exp(6)*cos(4)+3/13*exp(6)*sin(4) z=

34、2/13-2/13*exp(6)*cos(4)+3/13*exp(6)*sin(4)z = -29.734649084972837近似值为： h=0.01;x=0:h:2; y= exp(3.*x).*sin(2.*x); format long t=length(x); z1=sum(y(1:(t-1)*h z2=sum(y(2:t)*h z3=trapz(x,y) z4=quad( exp(3.*x).*sin(2.*x),0,2)矩形法：z1 = -28.220113908276339z2 = -31.273273084220026梯形法：z3 = -29.746693496248191辛普生法：z4 = -29.734649479620661Q20:判别下列级数的敛散性，如果收敛，求级数的和：(1) syms n u=1/(n(3/2); limit(u,n,inf) ans = 0 级数收敛。 syms n s=symsum(1/(n(3/2),1,inf) s = zeta(3/2) (3) syms n u=1/(nn); limit(u,n,inf)