matlab數學實驗第一至第四章答案(胡良劍)

matlab數學實驗第一至第四章答案(胡良劍版)第一章%Page20,ex1(5)等于[exp(1),exp(2);exp(3),exp(4)](7)3=1*3,8=2*4(8)a為各列最小值，b為最小值所在的行號(10)1>=4,false,2>=3,false,3>=2,ture,4>=1,ture2(30>=20)4(40>=10)21(30>=20)22列元素滿足不等式(40>=10)%Page20,ex2a,b,c1，但數據類型分別為數值，字符，邏輯，注意a與c相等，但他們不等于bdouble(fun)輸出的分別是字符a,b,s,(,x,)的ASCII碼%Page20,ex3>>r=2;p=0.5;n=12;>>T=log(r)/n/log(1+0.01*p)T=11.5813%Page20,ex4>>x=-2:0.05:2;f=x.^4-2.^x;>>[fmin,min_index]=min(f)fmin=-1.3907 %最小值min_index=54 %最小值點編址>>x(min_index)ans=0.6500 %最小值點>>[f1,x1_index]=min(abs(f)) %f1=0.0328x1_index=24>>x(x1_index)ans=-0.8500>>x(x1_index)=[];f=x.^4-2.^x; %刪去絕對值最小的點以求函數絕對值次小的點>>[f2,x2_index]=min(abs(f)) %f2=0.0630x2_index=65>>x(x2_index)ans=1.2500%Page20,ex5>>z=magic(10)z=929918156774515840988071416735557644148188202254566370478587192136062697128869325296168755234172476839042492633652358289914830323966796139597293138457210129496783537444653111810077843643502759>>ans=505505505505505505505505505505>>ans=505>>z(:,2)/sqrt(3)ans=57.157746.188046.765450.229553.693613.85642.88683.46416.928210.3923>>z(8,:)=z(8,:)+z(3,:)z=9299181567745158409880714167355576441481882022545663704785871921360626971288693252961687552341724768390424926336523582899148303239668387101115119838710111511910129496783537444653111810077843643502759第二章%Page40ex1先在編輯器窗口寫下列M函數，保存為eg2_1.mfunction[xbar,s]=ex2_1(x)n=length(x);xbar=sum(x)/n;s=sqrt((sum(x.^2)-n*xbar^2)/(n-1));例如>>x=[81706551766690876177];>>[xbar,s]=ex2_1(x)xbar=72.4000s=12.1124%Page40ex2s=log(1);n=0;whiles<=100n=n+1;s=s+log(1+n);endm=n計算結果m=37%Page40ex3clear;F(1)=1;F(2)=1;k=2;x=0;e=1e-8;a=(1+sqrt(5))/2;whileabs(x-a)>ek=k+1;F(k)=F(k-1)+F(k-2);x=F(k)/F(k-1);enda,x,k計算至k=21可滿足精度%Page40ex4clear;tic;s=0;fori=1:1000000s=s+sqrt(3)/2^i;ends,toctic;s=0;i=1;whilei<=1000000s=s+sqrt(3)/2^i;i=i+1;ends,toctic;s=0;i=1:1000000;s=sqrt(3)*sum(1./2.^i);s,toc%Page40ex5t=0:24;c=[15141414141516182022232528...313231292725242220181716];plot(t,c)%Page40ex6%(1)x=-2:0.1:2;y=x.^2.*sin(x.^2-x-2);plot(x,y)y=inline('x^2*sin(x^2-x-2)');fplot(y,[-22])%(2)參數方法t=linspace(0,2*pi,100);x=2*cos(t);y=3*sin(t);plot(x,y)%(3)x=-3:0.1:3;y=x;[x,y]=meshgrid(x,y);z=x.^2+y.^2;surf(x,y,z)%(4)x=-3:0.1:3;y=-3:0.1:13;[x,y]=meshgrid(x,y);z=x.^4+3*x.^2+y.^2-2*x-2*y-2*x.^2.*y+6;surf(x,y,z)%(5)t=0:0.01:2*pi;x=sin(t);y=cos(t);z=cos(2*t);plot3(x,y,z)%(6)theta=linspace(0,2*pi,50);fai=linspace(0,pi/2,20);[theta,fai]=meshgrid(theta,fai);x=2*sin(fai).*cos(theta);y=2*sin(fai).*sin(theta);z=2*cos(fai);surf(x,y,z)%(7)x=linspace(0,pi,100);y1=sin(x);y2=sin(x).*sin(10*x);y3=-sin(x);plot(x,y1,x,y2,x,y3)%page41,ex7x=-1.5:0.05:1.5;y=1.1*(x>1.1)+x.*(x<=1.1).*(x>=-1.1)-1.1*(x<-1.1);plot(x,y)%page41,ex8分別使用whichtrapz,typetrapz,dirC:\MATLAB7\toolbox\matlab\datafun\%page41,ex9clear;close;x=-2:0.1:2;y=x;[x,y]=meshgrid(x,y);a=0.5457;b=0.7575;p=a*exp(-0.75*y.^2-3.75*x.^2-1.5*x).*(x+y>1);p=p+b*exp(-y.^2-6*x.^2).*(x+y>-1).*(x+y<=1);p=p+a*exp(-0.75*y.^2-3.75*x.^2+1.5*x).*(x+y<=-1);mesh(x,y,p)%page41,ex10lookforlyapunovhelplyap>>A=[123;456;780];C=[2-5-22;-5-24-56;-22-56-16];>>X=lyap(A,C)X=1.0000-1.0000-0.0000-1.00002.00001.0000-0.00001.00007.0000第三章%Chapter3%Exercise1>>a=[1,2,3];b=[2,4,3];a./b,a.\b,a/b,a\bans=0.5000 0.5000 1.0000ans=2 2 1ans=0.6552 %一元方程組x[2,4,3]=[1,2,3]ans=0000000.66671.33331.0000%矩陣方程[1,2,3][x11,x12,x13;x21,x22,x23;x31,x32,x33]=[2,4,3]的特解%Exercise2(1)>>A=[41-1;32-6;1-53];b=[9;-2;1];>>rank(A),rank([A,b]) %[A,b]ans=ans=

33 %可見方程組唯一解>>x=A\bx=2.38301.48942.0213%Exercise2(2)>>A=[4-33;32-6;1-53];b=[-1;-2;1];>>rank(A),rank([A,b])ans=ans=

33 可見方程組唯一解>>x=A\bx=-0.4706-0.29410%Exercise2(3)>>A=[41;32;1-5];b=[1;1;1];>>rank(A),rank([A,b])ans=ans=

23 可見方程組無解>>x=A\bx=0.3311-0.1219%最小二乘近似解%Exercise2(4)>>a=[2,1,-1,1;1,2,1,-1;1,1,2,1];b=[123]';%注意b的寫法>>rank(a),rank([a,b])ans=ans=

33 %rank(a)==rank([a,b])<4說明有無窮多解>>a\bans=1010 一個特解%Exercise3>>a=[2,1,-1,1;1,2,1,-1;1,1,2,1];b=[1,2,3]';>>x=null(a),x0=a\bx=-0.62550.6255-0.20850.4170x0=1010%通解kx+x0%Exercise4>>x0=[0.20.8]';a=[0.990.05;0.010.95];>>x1=a*x,x2=a^2*x,x10=a^10*x>>x=x0;fori=1:1000,x=a*x;end,xx=0.83330.1667>>x0=[0.80.2]';>>x=x0;fori=1:1000,x=a*x;end,xx=0.83330.1667>>[v,e]=eig(a)v=0.9806 -0.70710.1961 0.7071e=1.0000 00 0.9400>>v(:,1)./xans=1.17671.1767 成比例，說明x是最大特征值對應的特征向量%Exercise5%用到公式(3.11)(3.12)>>B=[6,2,1;2.25,1,0.2;3,0.2,1.8];x=[25520]';>>C=B/diag(x)C=0.2400 0.4000 0.05000.0900 0.2000 0.01000.1200 0.0400 0.0900>>A=eye(3,3)-CA=0.7600 -0.4000 -0.0500-0.0900 0.8000 -0.0100-0.1200 -0.0400 0.9100>>D=[171717]';x=A\Dx=37.569625.786224.7690%Exercise6(1)>>a=[41-1;32-6;1-53];det(a),inv(a),[v,d]=eig(a)ans=-94ans=0.2553-0.02130.04260.1596-0.1383-0.22340.1809-0.2234-0.0532v=0.0185-0.9009-0.3066-0.7693-0.1240-0.7248-0.6386-0.41580.6170d=-3.05270003.67600008.3766%Exercise6(2)>>a=[11-1;02-1;-120];det(a),inv(a),[v,d]=eig(a)ans=1ans=2.0000-2.00002.0000-2.00001.00001.0000-1.00001.00002.0000-3.00002.0000=-0.57730.5774+0.0000i0.5774-0.0000i-0.57730.57740.5774-0.57740.5773-0.0000i0.5773+0.0000i=1.00000001.0000+0.0000i0d0 0 1.0000-0.0000i%Exercise6(3)>>A=[5765;71087;68109;57910]A=5 7 6 57 10 8 76 8 10 95 7 9 10>>det(A),inv(A),ans=1ans=68.0000-41.0000-17.000010.0000-41.000025.000010.0000-6.0000-17.000010.00005.0000-3.000010.0000-6.0000-3.00002.0000v=0.83040.09330.39630.3803-0.5016-0.30170.61490.5286-0.20860.7603-0.27160.55200.1237-0.5676-0.62540.5209d=0.010200000.843100003.8581000030.2887%Exercise6(4)(以n=5為例)%關鍵是矩陣的定義%方法一（三個for）n=5;fori=1:n,a(i,i)=5;endfori=1:(n-1),a(i,i+1)=6;endfori=1:(n-1),a(i+1,i)=1;enda%方法二（一個for）n=5;a=zeros(n,n);a(1,1:2)=[56];fori=2:(n-1),a(i,[i-1,i,i+1])=[156];enda(n,[n-1n])=[15];a%方法三（不用for）n=5;a=diag(5*ones(n,1));b=diag(6*ones(n-1,1));c=diag(ones(n-1,1));a=a+[zeros(n-1,1),b;zeros(1,n)]+[zeros(1,n);c,zeros(n-1,1)]%下列計算>>det(a)ans=665>>inv(a)ans=0.3173-0.58651.0286-1.62411.9489-0.09770.4887-0.85711.3534-1.62410.0286-0.14290.5429-0.85711.0286-0.00750.0376-0.14290.4887-0.58650.0015-0.00750.0286-0.09770.3173>>[v,d]=eig(a)v=-0.7843 -0.7843 -0.9237 0.9860 -0.92370.5546-0.5546-0.3771-0.00000.3771-0.2614-0.26140.0000-0.16430.00000.0924-0.09240.0628-0.0000-0.0628-0.0218-0.02180.02570.02740.0257d=0.7574000009.2426000007.4495000005.0000000002.5505%Exercise7(1)>>a=[41-1;32-6;1-53];[v,d]=eig(a)v=0.0185 -0.9009 -0.3066-0.7693 -0.1240 -0.7248-0.6386 -0.4158 0.6170d=-3.0527 0 003.67600008.3766>>det(v)ans=-0.9255 %v,特征向量線性相關，可對角化>>inv(v)*a*v %ans=-3.05270.0000-0.00000.00003.6760-0.0000-0.0000-0.00008.3766>>[v2,d2]=jordan(a) %也可用v2=0.07980.00760.91270.1886-0.31410.1256-0.1605-0.26070.4213%特征向量不同d2=8.3766000-3.0527-0.0000i00 0 3.6760+0.0000i>>v2\a*v2ans=8.376600.00000.0000-3.05270.00000.00000.00003.6760>>v(:,1)./v2(:,2) %對應相同特征值的特征向量成比例ans=2.44912.44912.4491%Exercise7(2)>>a=[11-1;02-1;-120];[v,d]=eig(a)-0.57730.5774+0.0000i-0.57730.5774+0.0000i0.5774-0.0000i-0.57730.57740.5774-0.57740.5773-0.0000i0.5773+0.0000i=1.00000001.0000+0.0000i0d0 0 1.0000-0.0000i>>det(v)ans=-5.0566e-028-5.1918e-017i %v0,>>[v,d]=jordan(a)v=1011001-10d=110011001%jordan標準形不是對角的，所以不可對角化%Exercise7(3)>>A=[5765;71087;68109;57910]A=5 7 6 57 10 8 76 8 10 95 7 9 10>>[v,d]=eig(A)v=0.83040.09330.39630.3803-0.5016-0.30170.61490.5286-0.20860.7603-0.27160.55200.1237-0.5676-0.62540.5209d=0.010200000.843100003.8581000030.2887>>inv(v)*A*vans=0.01020.0000-0.00000.00000.00000.8431-0.0000-0.0000-0.00000.00003.8581-0.0000-0.0000-0.0000030.2887%本題用jordan不行,原因未知%Exercise7(4)參考6(4)和7(1),略%Exercise8只有(3)對稱,且特征值全部大于零,所以是正定矩陣.%Exercise9(1)>>a=[4-313;2-135;1-1-1-1;3-234;7-6-70]>>ans=3>>rank(a(1:3,:))ans=2>>rank(a([124],:))%1,2,4ans=3>>b=a([124],:)';c=a([35],:)';>>b\cans=%線性表示的系數0.50005.0000-0.50001.00000-5.0000%Exercise10>>a=[1-22;-2-24;24-2]>>[v,d]=eig(a)v=0.33330.9339-0.12930.6667-0.3304-0.6681-0.66670.1365-0.7327d=-7.00000002.00000002.0000>>v'*vans=1.00000.00000.00000.00001.000000.000001.0000%v確實是正交矩陣%Exercise11%設經過6個電阻的電流分別為i1,...,i6.列方程組如下%20-2i1=a;5-3i2=c;a-3i3=c;a-4i4=b;c-5i5=b;b-3i6=0;%i1=i3+i4;i5=i2+i3;i6=i4+i5;%計算如下>>A=[100200000;001030000;10-100-3000;1-10000-400;0-110000-50;01000000-3;00010-1-100;0000-1-1010;000000-1-11];>>b=[2050000000]';A\bans=13.34536.44018.54203.3274-1.18071.60111.72630.42042.1467%Exercise12>>A=[123;456;780];>>left=sum(eig(A)),right=sum(trace(A))left=6.0000right=6>>left=prod(eig(A)),right=det(A)%原題有錯,(-1)^n應刪去left=27.0000right=27>>fA=(A-p(1)*eye(3,3))*(A-p(2)*eye(3,3))*(A-p(3)*eye(3,3))fA=1.0e-012*0.08530.14210.02840.14210.14210-0.0568-0.11370.1705>>norm(fA) %f(A)0ans=2.9536e-013第四章%Exercise1(1)roots([111])%Exercise1(2)roots([30-402-1])%Exercise1(3)p=zeros(1,24);p([1171822])=[5-68-5];roots(p)%Exercise1(4)p1=[23];p2=conv(p1,p1);p3=conv(p1,p2);p3(end)=p3(end)-4;%原p3最后一個分量-4roots(p3)%Exercise2fun=inline('x*log(sqrt(x^2-1)+x)-sqrt(x^2-1)-0.5*x');fzero(fun,2)%Exercise3fun=inline('x^4-2^x');fplot(fun,[-22]);gridon;fzero(fun,-1),fzero(fun,1),fminbnd(fun,0.5,1.5)%Exercise4fun=inline('x*sin(1/x)','x');fplot(fun,[-0.10.1]);x=zeros(1,10);fori=1:10,x(i)=fzero(fun,(i-0.5)*0.01);end;x=[x,-x]%Exercise5fun=inline('[9*x(1)^2+36*x(2)^2+4*x(3)^2-36;x(1)^2-2*x(2)^2-20*x(3);16*x(1)-x(1)^3-2*x(2)^2-16*x(3)^2]','x');[a,b,c]=fsolve(fun,[000])%Exercise6fun=@(x)[x(1)-0.7*sin(x(1))-0.2*cos(x(2)),x(2)-0.7*cos(x(1))+0.2*sin(x(2))];[a,b,c]=fsolve(fun,[0.50.5])%Exercise7clear;close; t=0:pi/100:2*pi;x1=2+sqrt(5)*cos(t);y1=3-2*x1+sqrt(5)*sin(t);x2=3+sqrt(2)*cos(t);y2=6*sin(t);plot(x1,y1,x2,y2);gridon;%作圖發(fā)現4個解的大致位置，然后分別求解y1=fsolve('[(x(1)-2)^2+(x(2)-3+2*x(1))^2-5,2*(x(1)-3)^2+(x(2)/3)^2-4]',[1.5,2])y2=fsolve('[(x(1)-2)^2+(x(2)-3+2*x(1))^2-5,2*(x(1)-3)^2+(x(2)/3)^2-4]',[1.8,-2])y3=fsolve('[(x(1)-2)^2+(x(2)-3+2*x(1))^2-5,2*(x(1)-3)^2+(x(2)/3)^2-4]',[3.5,-5])y4=fsolve('[(x(1)-2)^2+(x(2)-3+2*x(1))^2-5,2*(x(1)-3)^2+(x(2)/3)^2-4]',[4,-4])%Exercise8(1)clear;fun=inline('x.^2.*sin(x.^2-x-2)');fplot(fun,[-22]);gridon;%作圖觀察x(1)=-2;x(3)=fminbnd(fun,-1,-0.5);x(5)=fminbnd(fun,1,2);fun2=inline('-x.^2.*sin(x.^2-x-2)');x(2)=fminbnd(fun2,-2,-1);x(4)=fminbnd(fun2,-0.5,0.5);x(6)=2feval(fun,x)%答案:x(1)(3)(5)x(1)小，x(2)最大。%Exercise8(2)clear;fun=inline('3*x.^5-20*x.^3+10');fplot(fun,[-33]);gridx(1)=-3;x(3)=fminsearch(fun,2.5);fun2=inline('-(3*x.^5-20*x.^3+10)');x(2)=fminsearch(fun2,-2.5);x(4)=3;feval(fun,x)%Exercise8(3)fun=inline('abs(x^3-x^2-x-2)');fplot(fun,[03]);gridon;%作圖觀察fminbnd(fun,1.5,2.5)fun2=inline('-abs(x^3-x^2-x-2)');fminbnd(fun2,0.5,1.5)%Exercise9close;x=-2:0.1:1;y=-7:0.1:1;[x,y]=meshgrid(x,y);z=y.^3/9+3*x.^2.*y+9*x.^2+y.^2+x.*y+9;mesh(x,y,z);gridon;%作圖觀察fun=inline('x(2)^3/9+3*x(1)^2*x(2)+9*x(1)^2+x(2)^2+x(1)*x(2)+9');x=fminsearch(fun,[00])%求極小值fun2=inline('-(x(2)^3/9+3*x(1)^2*x(2)+9*x(1)^2+x(2)^2+x(1)*x(2)+9)');x=fminsearch(fun2,[0-5])%求極大值%Exercise10clear;t=0:24;c=[15141414141516182022232528...313231292725242220181716];p2=polyfit(t,c,2)p3=polyfit(t,c,3)fun=inline('a(1)*exp(a(2)*(t-14).^2)','a','t');a=lsqcurvefit(fun,[0f=feval(fun,a,t)norm(f-c)%擬合效果plot(t,c,t,f)%作圖檢驗fun2=inline('b(1)*sin(pi/12*t+b(2))+20','b','t');%原題修改f(x)+20b=lsqcurvefit(fun2,[00],t,c)figuref2=feval(fun2,b,t)norm(f2-c)%擬合效果plot(t,c,t,f2)%作圖檢驗%Exercise11fun=inline('(1-x)*sqrt(10.52+x)-3.06*x*sqrt(1+x)*sqrt(5)');x=fzero(fun,0,1)%Exercise12r=5.04/12/100;N=20*12;x=7500*180%房屋總價格y=x*0.3%首付款額x0=x-y%貸款總額a=(1+r)^N*r*x0/((1+r)^N-1)%月付還款額r1=4.05/12/100;x1=10*10000;%公積金貸款a1=(1+r1)^N*r1*x1/((1+r1)^N-1)x2=x0-x1%商業(yè)貸款a2=(1+r)^N*r*x2/((1+r)^N-1)a=a1+a2%Exercise13%列方程th*R^2+(pi-2*th)*r^2-R*r*sin(th)=pi*r^2/2%化簡得sin(2*th)-2*th*cos(2*th)=pi/2%以下Matlab計算clear;fun=inline('sin(2*th)-2*th*cos(2*th)-pi/2','th')th=fsolve(fun,pi/4)R=20*cos(th)%Exercise14%先在Editor窗口寫M函數保存functionx=secant(fname,x0,x1,e)whileabs(x0-x1)>e,x=x1-(x1-x0)*feval(fname,