matlab光盤-ch2符號(hào)計(jì)算2007a所謂指解算數(shù)學(xué)表達(dá)式、方程不是在離散化數(shù)值點(diǎn)上進(jìn)行

2章符號(hào)計(jì)本書之所以把符號(hào)計(jì)算內(nèi)容放在第2章，是出于以下考慮：一，相對(duì)于的這樣的安排，將使讀者在閱讀完本章后，就有可能運(yùn)用的符號(hào)計(jì)算能力去解決相符號(hào)對(duì)象和符號(hào)表符號(hào)對(duì)象的創(chuàng)建和衍一 生成符號(hào)對(duì)象的基本規(guī)則 符號(hào)數(shù)asa=Ca=Csa=ans= 符號(hào)參 符號(hào)變2.1-2】用符號(hào)計(jì)算研究方程uz2vzw0symsuvw result_1ans=wresult_2=symsabxXYans=X,Y,a,b,c,delta,theta,x,ans=xans=ans=symsabtuvxyA[a+b*x,sin(t)+u][x*exp(-t),log(y)+v]ans=x符號(hào)計(jì)算中的算符號(hào)計(jì)算中的函數(shù)指符號(hào)對(duì)象的識(shí)Mn McMs=[a,[c,=22=19=22CMn=CMc=CMs=ans1ans1ans1whosMnMc Bytes 1832312符號(hào)數(shù)字及表達(dá)式的操數(shù)值數(shù)字與符號(hào)數(shù)字之間的轉(zhuǎn)一 數(shù)值數(shù)字向符號(hào)數(shù)字的轉(zhuǎn)換 符號(hào)數(shù)字向雙精度數(shù)字轉(zhuǎn)符號(hào)數(shù)字的任意精度計(jì)2.2-1】digits,vpa指令的使用。%%e32r=vpa(abs(p0-pr))Digits=p0=pr=pd e32re16=e32d符號(hào)表達(dá)式的基本操3 163 1612xsymsg1=g2=表達(dá)式中的置換操一 子表達(dá)式置換操作cc

ddclearsymsabcdW[V,D]=eig([ab;cd])V D= RVD=[-(1/2*d-1/2*a-1/2*W)/c,-(1/2*d-1/2*a+1/2*W)/c] W= 通用置換指【例2.2-4】用簡單算例演示subs的置換規(guī)則symsaf= f1=ans= f2=ans= f3=ans= f4 ans= f5 ans=符號(hào)微積極限和導(dǎo)數(shù)的符號(hào)計(jì)【例2.3-1】試求lim1

1xxsymsxLim_f= t3

d2

d2

lnxdx

dt

symsatx;f=[a,t^3;t*cos(x),log(x)];df= [-t*sin(x), dfdt2= 0, dfdxdt [- 1 x 1

f12【例2.3-3】求f(x,x) 的Jacobian矩陣f2

f2

x

f

f23x2symsx1x2;f=[x1*exp(x2);x2;cos(x1)*sin(x2)];v=[x1x2];fjac=jacobian(f,v)fjac [-sin(x1)*sin(x2),2.3-4f(xsinxfx(0fx(xsymsxsymsd %df_p0=limit((subs(f_p,x,d)-subs(f_p,x,0))/d,d,0)%df_p=df_p0= % df_ndf_n0 %dfdx=holdon %legend(char(f),char(df_n),char(df_p),'Location','NorthEast')%<16>gridonholdoff2.3-12.3-5】設(shè)cosxsiny)sinydysymsx g=dgdx=-sin(x+sin(y(x)))*(1+cos(y(x))*diff(y(x),x))= % 2.3-6fx)xexx08Maclaurinsyms %忽略9階及9階以上小量的展r=2.3-7】求sinx2yx0y08TaylorTL1ans=序列/級(jí)數(shù)的符號(hào)求tk1k1

(1)k【例2.3-8】求[t,t

](2k1)2 symskt;f1=[tk^3];f2=[1/(2*k-1)^2,(-1)^k/k];s1[1/2*t*(t-1), s2=[1/8*pi^2,-符號(hào)積1111 symsxf=s

dxs2.3-10

bx2sinxdxsymsabx;f=[a*x,b*x^2;1/x,sin(x)];disp('Theintegraloffis');pretty(int(f))Theintegraloffis22.3-1112symsxy

2x[[1/2a[2x[[1/2a[21/3bx][x2 -cos(x)

z)dzdydxF2 VF2 【例2.3-12】求阿基米德（Archimedes）螺線ra (a0)在0到間的曲線度函數(shù)，并求出a1,2symsarthetaphipositiveLL_2pi=L_2pi_vpa= gridonholdontitle('')holdoff2.3-2阿基米德螺線（粗紅）和螺線長度函數(shù)（細(xì)藍(lán)微分方程的符號(hào)解符號(hào)解法和數(shù)值解法的互補(bǔ)作求微分方程符號(hào)解的一般指微分方程符號(hào)解示2.4-1dx

dyxS=x:[1x1sym]y:[1x1 [-C1*cos(t)+C2*sin(t),2.4-2yxyy)2 clf,hold%%%fork=-2:0.5:2;ezplot(subs(y(2),'C1',k),[-6,6,-4,8],1);endholdoff,title('\fontname{隸書}\fontsize{16}通解和奇解')y

2.4-12.4-3xy3yx2y(1)0y(5)0yholdontitle(['x*D2y-3*Dy=x^2',',y(1)=0,y(5)=0'])holdoff2.4-2符號(hào)變換和符號(hào)卷Fourier變換及其反變2.5-1f(t)

tFouriertsymstwUT=Ut=0【例2.5-2】根據(jù)Fourier變換定義，用積分指令 脈沖yA0

/2t/的FouriersymsAtsymstaoYw=Yt=yt3Yw3=100t12332100w246圖2.5-1時(shí) 及其Fourier變e(t

t2.5-3f(t)

FourierxttsymstxF1F2=F3=Laplace變換及其反變(t u(t2.5-4eatsin

t2

Laplacesymstsymsabpositive MS [ 【例2.5-5】驗(yàn)證Laplace時(shí)移性質(zhì)：L{f(tt0)U(tt0)} symstsymst0ftFS=FS_t=Z變換及其反變 【例2.5-6】求序列f(n)

nn0Zsyms

n [1,symsdisp('FZ=')fn= FZ=

24z+2FZ_n 符號(hào)卷2.5-7】已知系統(tǒng)沖激響應(yīng)h(t)symsTttao

2z-3z+et/TU(t，求u(t)etU(tTyt=2.5-8Laplace變換和反變換求上例的輸出響應(yīng)。symssyt2.5-9】求函數(shù)u(t)U(t)U(t1)和h(t)tetU(tsymstaoytyt=符號(hào)矩陣分析和代數(shù)方程符號(hào)矩陣分2.6-1A

a12 asymsa11a12a21a22A=[a11,[a21,a22]DA=IA=

22[a22/(a11*a22-a12*a21),-a12/(a11*a22-a12*a21)][-a21/(a11*a22-a12*a21),a11/(a11*a22-a12*a21)]EA=

sint對(duì)矩陣A3 132132syms

sin

cost

1 A[ 1/2,sqrt(3/4)]GA=[1/2*cos(t)*3^(1/2)-1/2*sin(t),1/2*cos(t)-1/2*sin(t)*3^(1/2)][1/2*sin(t)*3^(1/2)+1/2*cos(t),axis([-1,1,-1,1]),axissquareholdontitle('GivensRotation')legend('v1','v2','u1','u2','Location','South')holdoffgridGivensGivens10-- - 2.6-1Givens

22qndqp10qd4pqpn8d1A=sym([11/21/2-1;11-11;1-1/4-11;-8-111]);b=01X1=889disp(['d','n','p','q'])dnpq[1,8,8,9]線性方程組的符號(hào)一般代數(shù)方程組的2.6-4】求方程組uy^2vzw0yzw0yz的解。Sy:[2x1sym]z:[2x12.6-5】solve指令求dnpqndqp10qdnp symsdnp Warning:3equationsin4>InsolveatInsym.solveat 8,4*d+4,2.6-6】求(x2x2的解。clearall,symsx;s 利用MAPLE的符號(hào)計(jì)算資經(jīng)典特殊函數(shù)的調(diào)2.7-1】求x1dtx0,1。注意：被積函數(shù)在t0無定義，在t1處為0lnsymstxsx=x sx_n MAPLE庫函數(shù)的檢索mhelpIndexofhelpdescriptionsCallingSequence help(index,expressionoperatorsforformingexpressions listofMaplefunctions miscellaneousfacilities topicsrelatedtomodules descriptionsoflibrary topicsrelatedtoproceduresandprogramming listofMaplestatementsToaccessthesehelppages,youmustprefixthecategorywiththusmhelpIndexofdescriptionsforstandardlibraryfunctions-ThefollowingarethenamesofMaple'sstandardlibraryfunctions.Forinformation,see?fwherefisanyofthese _Seelibname,MAPLE的計(jì)算潛2.7-2f(n)3f(n12f(n2 gs1 gs2

xyzHessianFH1matrix([[0,z,y],[z,0,x],[y,x,FH2=matrix([[0,z,y],[z,0,x],[y,x,FH=[0,z,[z,0,[y,x,符號(hào)計(jì)算結(jié)果的可直接可視化符號(hào)表達(dá)式一 單獨(dú)立變量符號(hào)函數(shù)的可視2.8-1ysymst

3e2cos3

t和它的積分s(t)

ty(t)dt在[0,4]間的圖形（t%gridongridontitle('s=0-02468ts=02468t2.8-1ezplot 雙獨(dú)立變量符號(hào)函數(shù)的可視【例2.8-2】借助可視化，加深Taylor展開的鄰域近似概念。圖形研究函f(xysinx2yx0y08Taylor展開（2.3-7討論過）TL1=Fxy_TL1=[Fxy,'-(',TL1,')']Fxy=Fxy_TL1=shadinginterp2.8-2shadinginterp2.8-3Taylorshadinginterp2.8-4shadinginterp2.8-5Taylorshadinginterp2.8-62.8-3】使用球坐標(biāo)參量畫部分球殼（2.8-7）shadinginterp圖2.8- ezsurf在參變量格式下繪制的圖符號(hào)計(jì)算結(jié)果的數(shù)值化繪【例2.8-4】在(01區(qū)間，圖示1和x1dt（在例2.7-1中已計(jì)算過積分）symstxsx=

0lngridonholdonChar2='{\int_0^x}1/ln(t)dt'; 2.8-6習(xí)題a=b=sym(@c=sym(@,'d'dsym(@ %在此，@分別代表具體數(shù)值7/3,pi/3,pi*3^(1/3) ；而異同通過vpa(abs(a-d)),vpa(abs(b-d)),vpa(abs(c-d))等來觀察。ab=c d=v1=v2=v3ab=c d=v1=v2=v3abc d=v1v2v33/7+0.1,sym(3/7+0.1),c1c2c3 sym('sin(w*t)'),sym('a*exp(-X)'),sym('z*exp(j*theta)')ans=ans=ans=

a1323A23

a

a33A[a11,a12,[a21,a22,[a31,a32,DAIAs[(a22*a33-a23*a32)/d,-(a12*a33-a13*a32)/d,-(-a12*a23+a13*a22)/d][-(a21*a33-a23*a31)/d,(a11*a33-a13*a31)/d,-(a11*a23-a13*a21)/d][(-a21*a32+a22*a31)/d,-(a11*a32-a12*a31)/d,(a11*a22-a12*a21)/d]d

f

kk

a為正實(shí)數(shù)時(shí)，求f(k)zk

Z變換問題。Z1 x12k對(duì)于x0，求2k