Matlab7.0符號(hào)計(jì)算 上機(jī)實(shí)驗(yàn)

Matlab7.0符號(hào)計(jì)算實(shí)驗(yàn)報(bào)告⒈目的本實(shí)驗(yàn)旨在向?qū)W生介紹一種解決專(zhuān)業(yè)問(wèn)題的快速有效且具有強(qiáng)大功能的科學(xué)與工程計(jì)算軟件。通過(guò)本實(shí)驗(yàn)，應(yīng)使學(xué)生掌握的內(nèi)容是：MATLAB7.0符號(hào)對(duì)象的創(chuàng)建和使用，MATLAB7.0任意精度的計(jì)算，MATLAB7.0符號(hào)表達(dá)式的化簡(jiǎn)和替換，MATLAB7.0符號(hào)矩陣計(jì)算，MATLAB7.0符號(hào)微積分，MATLAB7.0積分變換，MATLAB7.0符號(hào)代數(shù)方程和符號(hào)微分方程求解。該實(shí)驗(yàn)主要為上機(jī)實(shí)驗(yàn)，要求學(xué)生按要求上機(jī)實(shí)現(xiàn)相關(guān)的程序的設(shè)計(jì)，自己動(dòng)手編寫(xiě)程序并驗(yàn)證程序的正確性。⒉實(shí)驗(yàn)任務(wù)分解通過(guò)一些實(shí)例初步掌握MATLAB7.0的基本符號(hào)計(jì)算功能和應(yīng)用。實(shí)驗(yàn)任務(wù)可分解為：MATLAB7.0符號(hào)計(jì)算的基礎(chǔ)，MATLAB7.0符號(hào)計(jì)算在高等數(shù)學(xué)中的應(yīng)用，MATLAB7.0符號(hào)方程（組）求解和符號(hào)矩陣計(jì)算。⒊實(shí)驗(yàn)環(huán)境介紹實(shí)驗(yàn)一MATLAB7.0符號(hào)計(jì)算的基礎(chǔ)實(shí)驗(yàn)?zāi)康模赫莆誐ATLAB7.0符號(hào)對(duì)象的創(chuàng)建和使用；掌握MATLAB7.0任意精度的計(jì)算；掌握MATLAB7.0符號(hào)表達(dá)式的化簡(jiǎn)和替換。實(shí)驗(yàn)要求:給出程序和實(shí)驗(yàn)結(jié)果。實(shí)驗(yàn)題目與結(jié)果：一、計(jì)算符號(hào)表達(dá)式在處的值，并將結(jié)果設(shè)置為以下五種精度，即分別為小數(shù)點(diǎn)之后1位、2位、10位、20位、50位有效數(shù)字；>>symsx;x=pi-1f=x+cos(x)-sin(x);vpa(f,1)vpa(f,2)vpa(f,10)vpa(f,20)vpa(f,50)x=2.1416ans=.8ans=.76ans=.7598193629ans=.75981936291375673509ans=.75981936291375673508952104384661652147769927978516二、設(shè)為符號(hào)變量，，，試進(jìn)行如下運(yùn)算。（1），（2），（3）對(duì)進(jìn)行因式分解，（4）求的反函數(shù)。>>symsx;f1=x^4+2*x^2+1;f2=x^3+6*x^2+3*x+5;f1+f2f1*f2factor(f1)finverse(f2)ans=x^4+8*x^2+6+x^3+3*xans=(x^4+2*x^2+1)*(x^3+6*x^2+3*x+5)ans=(x^2+1)^2Warning:finverse(x^3+6*x^2+3*x+5)isnotunique.>Insym.finverseat43ans=1/2*(-60+4*x+4*(117-30*x+x^2)^(1/2))^(1/3)+6/(-60+4*x+4*(117-30*x+x^2)^(1/2))^(1/3)-2三、指出下面各條語(yǔ)句的輸出結(jié)果>>f=sym('2*u');>>subs(f,'u',2)>>f2=subs(f,'u','u+2')>>a=3;>>subs(f2,'u',a+2)>>subs(f2,'u','a+2')>>symsxy>>f3=subs(f,'u',x+y)>>subs(f3,[x,y],[1,2])>>subs(f3,[x,y],[x+y,x+y])>>f=sym('2*u');subs(f,'u',2)f2=subs(f,'u','u+2')a=3;subs(f2,'u',a+2)subs(f2,'u','a+2')symsxyf3=subs(f,'u',x+y)subs(f3,[x,y],[1,2])subs(f3,[x,y],[x+y,x+y])ans=4f2=2*(u+2)ans=14ans=2*((a+2)+2)f3=2*x+2*yans=6ans=6*x+4*y四、簡(jiǎn)化>>symsx;f1=[x^(-3)+6*x^(-2)+12*x^(-1)+8]^(1/3);simple(f1)simplify:((2*x+1)^3/x^3)^(1/3)radsimp:(2*x+1)/xcombine(trig):((1+6*x+12*x^2+8*x^3)/x^3)^(1/3)factor:((2*x+1)^3/x^3)^(1/3)expand:(1/x^3+6/x^2+12/x+8)^(1/3)combine:(1/x^3+6/x^2+12/x+8)^(1/3)convert(exp):(1/x^3+6/x^2+12/x+8)^(1/3)convert(sincos):(1/x^3+6/x^2+12/x+8)^(1/3)convert(tan):(1/x^3+6/x^2+12/x+8)^(1/3)collect(x):(1/x^3+6/x^2+12/x+8)^(1/3)mwcos2sin:(1/x^3+6/x^2+12/x+8)^(1/3)ans=(2*x+1)/x五、設(shè)a、b定義如下，試上機(jī)輸出factor(a)和factor(b)的結(jié)果，并指出哪個(gè)結(jié)果才是12345678901234567890的因式分解，為什么？a=sym('12345678901234567890')b=sym(12345678901234567890)y1=factor(a)y2=factor(b)y1=(2)*(3)^2*(5)*(101)*(3803)*(3607)*(27961)*(3541)y2=^11*(7)*(7324703)*(117570121)第一種正確>>a=sym('12345678901234567890');b=sym(12345678901234567890);y1=factor(a)y2=factor(b)y1=(2)*(3)^2*(5)*(101)*(3803)*(3607)*(27961)*(3541)y2=(2)^11*(7)*(7324703)*(117570121)六、替換函數(shù)subs的使用。(1)試指出下面程序中的f1、f2、f3、f4、f5的值。symsax;f=a*sin(x)+5;f1=subs(f,'sin(x)',sym('y'))f2=subs(f,{a,x},{2,sym(pi/3)})f3=subs(f,{a,x},{2,pi/3})f4=subs(subs(f,a,2),x,0:pi/6:pi)f5=subs(f,{a,x},{0:6,0:pi/6:pi})>>symsax;f=a*sin(x)+5;f1=subs(f,'sin(x)',sym('y'))f2=subs(f,{a,x},{2,sym(pi/3)})f3=subs(f,{a,x},{2,pi/3})f4=subs(subs(f,a,2),x,0:pi/6:pi)f5=subs(f,{a,x},{0:6,0:pi/6:pi})f1=a*y+5f2=3^(1/2)+5f3=6.7321f4=Columns1through45.00006.00006.73217.0000Columns5through76.73216.00005.0000f5=Columns1through45.00005.50006.73218.0000Columns5through78.46417.50005.0000(2)試指出下面替換的結(jié)果，并說(shuō)明理由。>>symsxy;f=2*x+y;>>subs(f,{x,y},{x+y,x-y})>>symsxy;f=2*x+y;subs(f,{x,y},{x+y,x-y})ans=5*x-3*y因?yàn)镽=subs(S,Old,New)表示用新符號(hào)變量New代替原來(lái)的符號(hào)表達(dá)式S中的變量Old。七、創(chuàng)建符號(hào)變量有幾種方法？答：兩種Sym()Syms八、下面三種表示方法有什么不同的含義？（1）f=3*x^2+5*x+2（2）f='3*x^2+5*x+2'（3）x=sym('x')f=3*x^2+5*x+2答：定義了x后將3*x^2+5*x+2的數(shù)值運(yùn)算結(jié)果賦值給f表示將字符串‘3*x^2+5*x+2’賦值給f表示x是符號(hào)變量，f=3*x^2+5*x+2是符號(hào)函數(shù)，所以f也是符號(hào)變量九、用符號(hào)計(jì)算驗(yàn)證三角等式：sin(1)cos(2)-cos(1)sin(2)=sin(1-2)>>symsphi1phi2;y=simple(sin(phi1)*cos(phi2)-cos(phi1)*sin(phi2))y=-sin(-phi1+phi2)十、因式分解：>>symsx;f=x^4-5*x^3+5*x^2+5*x-6;factor(f)ans=(x-1)*(x-2)*(x-3)*(x+1)實(shí)驗(yàn)二MATLAB7.0符號(hào)計(jì)算在高等數(shù)學(xué)中的應(yīng)用實(shí)驗(yàn)?zāi)康模赫莆誐ATLAB7.0符號(hào)計(jì)算在極限中的應(yīng)用；掌握MATLAB7.0符號(hào)計(jì)算在微分、積分中的應(yīng)用；掌握MATLAB7.0符號(hào)計(jì)算在級(jí)數(shù)求和以及泰勒級(jí)數(shù)中的應(yīng)用。實(shí)驗(yàn)要求:給出程序和實(shí)驗(yàn)結(jié)果。實(shí)驗(yàn)內(nèi)容與結(jié)果：一、求在上的定積分； >>symsx;f=sin(x)+x;R=int(f,x,0,8)R=-cos(8)+33二、求符號(hào)表達(dá)式的5次微分；>>symsx;f=sin(x)+x^5;diff(f,'x',5)ans=cos(x)+120三、計(jì)算和lnx表示成log(x)

lgx表示成log10(x)>>symsxhn;L=[log(x+h)-log(x)]/h;M=(1-x/n)^n;limit(L,h,0)limit(M,n,inf)ans=1/xans=exp(-x)四、1計(jì)算級(jí)數(shù)及其前100項(xiàng)的部分和；>>symsn;S=1/n^2;s1=symsum(S,1,inf)s2=symsum(S,1,100)s1=1/6*pi^2s2=1589508694133037873112297928517553859702383498543709859889432834803818131090369901/9721861444343810305896579766726231441619755839957462417827203547055179861652480002計(jì)算函數(shù)級(jí)數(shù)>>symsnx;S=x/n^2;s1=symsum(S,n,1,inf)s1=1/6*x*pi^2五、(1)設(shè)求復(fù)合函數(shù)f(g(z));(2)設(shè)求復(fù)合函數(shù)f(g(z))。（1）>>symsuytxz;f1=x^2/1+u;f2=cos(y+2*t);compose(f1,f2,x,y,z)ans=cos(z+2*t)^2+u（2）>>symsuytxz;f1=x^2/1+u;f2=cos(y+2*t);compose(f1,f2,u,t,z)ans=x^2+cos(y+2*z)六、計(jì)算sin(t)在t=0處的taylor級(jí)數(shù)、Fourier變換和Laplace變換。>>symstws;f=sin(t);s1=taylor(f)fw=fourier(f,t,w)fs=laplace(f,t,s)s1=t-1/6*t^3+1/120*t^5fw=i*pi*(dirac(w+1)-dirac(w-1))fs=1/(s^2+1)七、計(jì)算的和。>>symsxy;z=y*x+3*y^2*x+2*y^3;a=diff(z,y)b=diff(a,x)a=x+6*y*x+6*y^2b=1+6*y八、因式分解：>>symsx;f=x^4-5*x^3+5*x^2+5*x-6;R=factor(f)R=(x-1)*(x-2)*(x-3)*(x+1)九符號(hào)函數(shù)繪圖法繪制函數(shù)x=sin(3t)cos(t)，y=sin(3t)sin(t)的圖形，t的變化范圍為[0,2]。>>symst;x=sin(3*t)*cos(t);y=sin(3*t)*sin(t);ezplot(x,y,[0,2*pi])十、，用符號(hào)微分求df/dx。>>symsax;f=[a,x^2,1/x;exp(a*x),log(x),sin(x)];diff(f,x)ans=[0,2*x,-1/x^2][a*exp(a*x),1/x,cos(x)]實(shí)驗(yàn)三MATLAB7.0符號(hào)方程（組）求解和符號(hào)矩陣計(jì)算實(shí)驗(yàn)?zāi)康模赫莆誐ATLAB7.0符號(hào)代數(shù)方程和符號(hào)代數(shù)方程組求解；掌握MATLAB7.0符號(hào)符號(hào)微分方程和符號(hào)微分方程組求解；掌握MATLAB7.0符號(hào)矩陣計(jì)算。實(shí)驗(yàn)要求:給出程序和實(shí)驗(yàn)結(jié)果。實(shí)驗(yàn)題目與結(jié)果：一、使用sym函數(shù)生成如下符號(hào)矩陣，，，試對(duì)這兩個(gè)符號(hào)矩陣分別進(jìn)行如下操作。（1）（2）（3）求的行列式（4）求的逆（5）求的秩（6）求>>symsx;a=sym('[1/x,1/(x+1);1/(x+2),1/(x+3)]');b=sym('[x,1;x+2,0]');s1=a-bs2=a*bs3=det(a)s4=inv(b)s5=rank(a)s6=a^3s1=[1/x-x,1/(x+1)-1][1/(x+2)-x-2,1/(x+3)]s2=[1+1/(x+1)*(x+2),1/x][1/(x+2)*x+1/(x+3)*(x+2),1/(x+2)]s3=2/x/(x+3)/(x+1)/(x+2)s4=[0,1/(x+2)][1,-1/(x+2)*x]s5=2s6=[1/x*(1/x^2+1/(x+1)/(x+2))+1/(x+1)*(1/(x+2)/x+1/(x+3)/(x+2)),1/x*(1/x/(x+1)+1/(x+1)/(x+3))+1/(x+1)*(1/(x+1)/(x+2)+1/(x+3)^2)][1/(x+2)*(1/x^2+1/(x+1)/(x+2))+1/(x+3)*(1/(x+2)/x+1/(x+3)/(x+2)),1/(x+2)*(1/x/(x+1)+1/(x+1)/(x+3))+1/(x+3)*(1/(x+1)/(x+2)+1/(x+3)^2)]二、求解非線性方程組>>S=solve('x-0.7*sin(x)-0.2*cos(y)=0','y-0.7*cos(x)-0.2*sin(y)=0','x','y')S.xS.yS=x:[1x1sym]y:[1x1sym]ans=.44415725748356642278916008786253ans=.77152736448567154642848671505449三、已知，求該方程的解。>>a=dsolve('D2y+2*Dy+2*y=0','y(0)=1','Dy(0)=1');>>sy=simple(a)sy=exp(-t)*(2*sin(t)+cos(t))>>expand(sy)ans=2/exp(t)*sin(t)+1/exp(t)*cos(t)四、用符號(hào)函數(shù)法求解方程at2+b*t+c=0。>>r=solve('a*t^2+b*t+c=0','t')r=1/2/a*(-b+(b^2-4*a*c)^(1/2))1/2/a*(-b-(b^2-4*a*c)^(1/2))五、用符號(hào)計(jì)算驗(yàn)證三角等式：sin(1)cos(2)-cos(1)sin(2)=sin(1-2)symsphi1phi2;y=simple(sin(p1)*cos(p2)-cos(p1)*sin(p2))y=-sin(-p1+p2)>>z=sin(p1-p2)z=-sin(-p1+p2)六、求矩陣的行列式值、逆和特征根。symsa11a12aA=[a11,a12;a21,a22]A=[a11,a12][a21,a22]>>AD=det(A)AD=a11*a22-a12*a21AI=inv(A)AI=[a22/(a11*a22-a12*a21),-a12/(a11*a22-a12*a21)][-a21/(a11*a22-a12*a21),a11/(a11*a22-a12*a21)]AE=eig(A)AE=1/2*a11+1/2*a22+1/2*(a11^2-2*a11*a22+a22^2+4*a12*a21)^(1/2)1/2*a11+1/2*a22-1/2*(a11^2-2*a11*a22+a22^2+4*a12*a21)^(1/2)七、，用符號(hào)微分求df/dx。symsax;f=[a,x^2,1/x;exp(a*x),log(x),sin(x)];df=diff(f)df=[0,