中南大学matlab最新题库.docx

p.31 ex 2.2 数值驻足和字符串转换 a=1:5; b=num2str(a); aa = 1 2 3 4 5 bb =1 2 3 4 5 b*2ans =98 64 64 100 64 64 102 64 64 104 64 64 106p.44 ex 2.9 比较用左除和右除分别求解恰定方程（线性方程组如果方程数等于未知数个数，叫做恰定方程组，如果方程多于未知数，叫做超定方程组，反之称为欠定。换个角度说，系数矩阵如果是方阵，就是恰定方程组）的解见课本p.48 ex 2.14 计算矩阵的指数 并比较不同函数的结果 b=magic(3); expm(b)ans = 1.0e+006 * 1.0898 1.0896 1.0897 1.0896 1.0897 1.0897 1.0896 1.0897 1.0897 expmdemo2(b)ans = 1.0e+006 * 1.0898 1.0896 1.0897 1.0896 1.0897 1.0897 1.0896 1.0897 1.0897 expm1(b)ans = 1.0e+003 * 2.9800 0.0017 0.4024 0.0191 0.1474 1.0956 0.0536 8.1021 0.0064 expmdemo3(b)ans = 1.0e+006 * 1.0898 1.0896 1.0897 1.0896 1.0897 1.08971.0896 1.0897 1.0897p50 ex 2.18 计算矩阵的特征值条件数 a=rand(3)a = 0.9649 0.9572 0.1419 0.1576 0.4854 0.4218 0.9706 0.8003 0.9157 v,d,s=condeig(a)v = 0.4913 0.6696 0.6696 0.3158 -0.4476 + 0.2831i -0.4476 - 0.2831i 0.8117 -0.2332 - 0.4655i -0.2332 + 0.4655id = 1.8146 0 0 0 0.2757 + 0.3061i 0 0 0 0.2757 - 0.3061is = 1.1792 1.2366 1.2366p62 ex 2.29矩阵的抽取、三角抽取 a=pascal(4)a = 1 1 1 1 1 2 3 4 1 3 6 10 1 4 10 20 diag(a,-2)ans = 1 4 v=diag(diag(a)v = 1 0 0 0 0 2 0 0 0 0 6 0 0 0 0 20%diag简单来说就是抽取矩阵各对角线上的元素，如上是抽取主对角线以下第二条对角线之元素，其另一功能是建立对角矩阵 tril(a)ans = 1 0 0 0 1 2 0 0 1 3 6 0 1 4 10 20 triu(a)ans = 1 1 1 1 0 2 3 4 0 0 6 10 0 0 0 20%triu&tril用法与diag非常类似，用途是提取下、上三角矩阵p62 ex2.30 建立多项式之伴随矩阵这道题有点凌乱了.求解释p64 ex 2.31 数组的幂运算 a=2 1 -3 -1;3 1 0 7;-1 2 4 -2;1 0 -1 5; a3ans = 32 -28 -101 34 99 -12 -151 239 -1 49 93 8 51 -17 -98 139 a.3ans = 8 1 -27 -1 27 1 0 343 -1 8 64 -8 1 0 -1 125p66 ex 2.32 数组之逻辑运算 a=1:3;4:6;7:9; b=0 1 0;1 0 1;0 0 1; x=5;y=ones(3)*5; x ab=a&bab = 0 1 0 1 0 1 0 0 1%此处为与运算，就是同真才为真（同为非零数） bans = 1 0 1 0 1 0 1 1 0%逻辑非运算，即全都非，真变假假变真；还有逻辑或运算，看下面即懂： a|bans = 1 1 1 1 1 1 1 1 1%总结多项式运算的函数：poly: polynomial with specified roots特征多项式的生成p=poly(a)a为n阶特征矩阵，所得一般为n阶特征多项式；poly2sym数值2符号；polyval 求多项式的值；roots 求多项式的根；conv 多项式的乘法(向量之卷积)conv(p,d);polyder 多项式微分；polyfit 多项式拟合。p71 ex 2.41 多项式拟合，用5阶多项式对正弦函数进行最小二乘拟合 x=0:pi/20:pi/2; y=sin(x); a=polyfit(x,y,5); x1=0:pi/30:pi*2; y1=sin(x1); y2=a(1)*x1.5+a(2)*x1.4+a(3)*x1.2+a(4)*x1.2+a(5)*x1+a(6); plot(x1,y1,b-,x1,y1,r*) legend(原曲线,拟合曲线) axis(0,7,-1.2,4) axis(0,7,-1.2,1.5) %调整坐标轴显示范围 2.符号运算命名和基本运算法则p 79 符号矩阵的运算 a=sym(a,b;c,d) a = a, b c, d syms a b c d b=a,b;c,d b = a, b c, d b-a ans = 0, 0 0, 0 a/b ans = 1, 0 0, 1 a2 ans = a2 + b*c, a*b + b*d a*c + c*d, d2 + b*c a.2 ans = a2, b2 c2, d2 det(a) ans = a*d - b*c %行列式 inv(a) ans = d/(a*d - b*c), -b/(a*d - b*c) -c/(a*d - b*c), a/(a*d - b*c) %逆 rank(a) ans = 2%秩%summary 矩阵的分解：奇异值分解 u,s,v=svd(a)；特征值分解v,d=eig(a)；正交分解q,r=qr(a)；三角分解l,u=lu(a)；p88 ex 3.7 利用函数gradient绘制一个矢量图 x,y=meshgrid(-2:.2:2,-2:.2:2); z=x.*exp(-x.2-y.2); px,py=gradient(z,.2,.2); contour(z),%等高线绘制函数 hold on quiver(px,py)%矢量图绘制函数 %绘图summary 1.二维plot 不解释；fplot：绘制y=f(x)图形 fplot(fname,lims,s)；ezplot：绘制隐函数图形 help吧；极坐标polar,对数坐标semilogx,semilogy,loglog;bar 条形图，pie饼状图；hist 直方图 有些疑惑！；grid on/off：控制是否画网格线。box on/off： 控制是否加边框线。hold on/off 控制是否刷新当前轴及图形%2.三维：plot3; meshgrid函数：产生平面区域内的网格坐标矩阵;mesh 画格子；surf 面；figure(n)开窗户；subplot:割图。3.二维图形函数运用p98 基本绘图命令 y=rand(100,1); plot(y) x=rand(100,1); z=x+y.*i; plot(z) p101 ex4.1 绘制带有显示属性的二维属性 x=1:0.1*pi:2*pi; y=sin(x); z=cos(x); plot(x,y,-k,x,z,-.rd) p104 ex4.5 条状图和矢量图 x=1:10; y=rand(10,1); bar(x,y) x=:0.1*pi:2*pi; x=0:0.1*pi:2*pi; y=x.*sin(x); feather(x,y) p104 ex4.6 函数图形绘制 lim=0,2*pi,-1,1; fplot(sin(x),cos(x),lim) p105 ex 4.7 绘制饼状图 x=2,4,6,8; pie(x,math,english,chinese,music) 4.三维图形函数应用p107 ex4.9 绘制三维螺旋线 x=0:pi/50:10*pi; y=sin(x);z=cos(x); plot3(x,y,z) p107 ex 4.10 绘制参数为矩阵的三维图 x,y=meshgrid(-2:0.1:2,-2:0.1:2); z=x.*exp(-x.2-y.2); plot3(x,y,z) p109 ex 4.11 使用mesh函数绘制三维面图 x=-8:0.5:8;y=x; a=ones(size(y)*x; b=y*ones(size(x); c=sqrt(a.2+b.2)+eps; z=sin(c)./c; mesh(z) p110 ex4.13 meshc函数绘制的三维面图 x,y=meshgrid(-4:0.5:4); z=sqrt(x.2+y.2); meshc(z) p111 ex 4.16 绘制三维饼状图 clear x=2,4,6,8; pie(x) pie(x,0,0,1,0) pie3(x,0,0,1,0) pie3(x,0,0,1,1) p113 ex 4.19 绘制如图柱面图 x=0:pi/20:pi*3; r=5+cos(x); a,b,c=cylinder(r,30); mesh(a,b,c) p113 ex 4.20 绘制地球表面的气温分布示意图 a,b,c=sphere(40); t=abs(c); surf(a,b,c,t); axis(equal) axis(square) colormap(hot) 5.图形控制命令p118 ex 4.24 坐标标注函数应用 x=1:0.1*pi:2*pi; y=sin(x);plot(x,y) xlabel(x(0-2pi) ylabel(y=sin(x) title(正弦函数,fontsize)error using title (line 29)incorrect number of input argumentserror in title (line 23) h = title(gca,varargin:); title(正弦函数,fontsize,12) title(正弦函数,fontsize,12,fontweight,bold) %课本上不知由于版本问题还是什么，与2011b不同，最后尝试用bold成功解决 add 已解决，用单引号引起bold即可undefined function or variable bold. title(正弦函数,fontsize,12,fontweight,bold)undefined function or variable bold. title(正弦函数,fontsize,12,fontweight,bold) title(正弦函数,fontsize,12,fontweight,bold,fontname,隶书) p123 ex 4.30 在同一张途中绘制几个三角函数图 x=0:0.1*pi:2*pi; y=sin(x); z=cos(x); plot(x,y,-*) hold on plot (x,z,-o) plot(x,y+z,-+) legend(sin(x),cos(x),sin(x)+cos(x),0) p124 ex 4.31 在4个子图中绘制不同的三角函数 x=0:0.1*pi:2*pi; subplot(2,2,1); plot(x,sin(x),-*) title(sin(x) subplot(2,2,2) plot(x,cos(x),-o) title(cos(x) subplot(2,2,3) plot(x,sin(x).*cos(x),-x) title(sin(x)*cos(x) subplot(2,2,4) plot(x,sin(x)+cos(x),-h) title(sin(x)+cos() title(sin(x)+cos(x) % summary interp1:1-d data interpolation (table lookup) % yi = interp1(x,y,xi,method,extrap) method:nearest/linear/echip(hermite)/spline,etcp222 ex 7.3 正弦曲线的差值实例 x=0:0.05:10; y=sin(x); xi=0:.125:10; yi=interp1(x,y,xi);%一维插值 plot(x,y,*,xi,yi)p227 例7.7二次拟合 x=0.5:0.5:3.0; y=1.75 2.45 3.81 4.80 8.00 8.60; a=polyfit(x,y,2)%用最小二乘法拟合a = 0.4900 1.2501 0.8560 x1=0.5:0.05:3.0； y1=a(1)*x1.2+a(2)*x1+a(3);%拟合出的函数 plot(x1,y1,-or,x,y,-+b)p228 例7.8拟合，用求解矩阵的方法解，求解超定方程 xi=19:6:44xi = 19 25 31 37 43 yi=19.0 32.3 49.0 73.3 98.8; format short a=polyfit(xi,yi,2)a = 0.0635 -0.5932 7.2811 x1=19:.1:44; y1=a(1)*x1.2+a(2)*x1+a(2);%此处发生了一点错误，将a（3）误输为a(2),后面会有改正 plot(x1,y1,-x) plot(x1,y1,-) x2=xi.2;%采用求解矩阵的方法来求解此拟合问题 x2=ones(5,1),x2x2 = 1 361 1 625 1 961 1 1369 1 1849 ab=x2yiab = -1.3522 0.0540 y3=ab(1)+ab(2)*x1.2; hold on;plot(x1,y3,-g) y1=a(1)*x1.2+a(2)*x1+a(3); plot(x1,y1,-)%从图中可以看出，两种拟合方法较为吻合p233 例7.10 分别用矩形和梯形公式求积分 y=inline(exp(-0.5*t).*sin(t+pi/6);%内联函数 d=pi/1000; t=0:d:3*pi; nt=length(t)nt = 3001 y1=y(t); sc=cumsum(y1)*d; scf=sc(nt)scf = 0.9016 format long scfscf = 0.901618619310013%矩形方法 z=trapz(y1)*dz = 0.900840276606885%梯形方法p237 ex7.11 采用自适应simpson公式求积分 f=inline(x./(x.2+4); quad(f,0,1)ans = 0.111571765994935p237 ex 7.12 用quadl求积分 f=inline(exp(-x/2); f1=quadl(f,1,3)f1 = 0.7668 f2=quad(f,1,3,1e-10)f2 =0.7668 format long e f1f1 = 7.668009991284686e-001 f2f2 =7.668009991284349e-001p237 7.12 quadl gauss-labatto 求积分 quadl(exp(-x/2),1,3)ans = 0.7668%求线性方程组的方法一般有两种：左除求法和linsolve求法p246 ex 7.17 求线性方程组 a=rand(4,4);%生成随机矩阵 b=rand(4,1); x=abx = 1.728190838792039e+001 8.395388076704506e-001 -1.590669694986821e+0011.088276299038511e+000%此处与书上不同，为生成的随机矩阵%summary矩阵左除 ab =a-1* b 等效于a*x=b求x inv(a) 注：.数组左除 a.b bij/aij ; /:矩阵右除 a/b =a*b-1 等效于x*b=a求x format short x1=linsolve(a,b)%summary用linsolve求解线性方程；用solve、fzero求解非线性方程；dsolve ordinary differential equation and system solver,用dsolve求解微分方程；fsolve solve system of nonlinear equations, x,fval = fsolve(myfun,x0,options) % call solverx1 = 17.2819 0.8395 -15.9067 1.0883p246 ex 7.18 对矩阵进行lu分解 a=ones(4,4)a = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 l,u=lu(a)l = 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 1u = 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 l*uans = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1p265 ex 7.39 求方程组之符号解 x1,x2=solve(x1-0.7*sin(x1)-0.2*cos(x2)=0,x2-0.7*cos(x1)+0.2*sin(x2)=0) x1 = 0.52652262191818418730769280519209 x2 = 0.50791971903684924497183722688768%方法一：用solve求符号解，注意使用x1,x2方式调用才可求得具体数值 另有不动点迭代法和newton法，见课本p263editfc.mfunction f=fc(x)f(1)=x(1)-0.7*sin(x(1)-0.2*cos(x(2);f(2)=x(2)-0.7*cos(x(1)+0.2*sin(x(2);f=f(1) f(2);%编写m文件 x0=0.5 0.5; fsolve(fc,x0)equation solved.fsolve completed because the vector of function values is near zeroas measured by the default value of the function tolerance, andthe problem appears regular as measured by the gradient.ans = 0.5265 0.5079%用fsolve求解，显然麻烦很多p273 ex 7.42 微分方程数值解f=inline(-y+x+1); x,y=ode23(f,0,1,1)x = 0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.0000y = 1.0000 1.0048 1.0187 1.0408 1.0703 1.1065 1.1488 1.1966 1.2493 1.3066 1.3679 plot(x,y)%ode、euler等方法都是数值法求解微分方程 x1,y1=ode45(f,0,1,1); plot(x,y,-*r,x1,y1,-o)p273 ex 7.45 r-k法求解微分方程 f=inline(-2*y+2*x.2+2*x)f = inline function: f(x,y) = -2*y+2*x.2+2*x x,y=ode23(f,0,.5,1);%此处定义域可再取大一些 plot(x,y)p274 ex 7.46 求解刚性方程fgang.mfunction f=fgang(x,y)f=-2 0;998 -999*y+2*sin(x);999*(cos(x)-sin(x); ode23(fgang,0 10,2,3); %此题略神奇，记下吧p275 ex 7.47 求常微分方程符号解 a=dsolve(dy=y+sin(t),y(pi/2)=0) a = exp(t)/(2*exp(pi/2) - sin(t)/2 - cos(t)/2%subs 代入数值至函数p307 ex 9.21 检验三台机器生产的铝合金报班有无显著性差异（方差分析） x=.236 .238 .248 .245 .243;.257 .253 .255 .254 .261;.258 .264 .259 .267 .262; a=anova1(x)a = 1.3431e-005 %anova1为单因素试验的方差分析，此处a的大小，是原假设均值相等的几率p308 9.22 用三种火箭推进器、四种燃料做射程实验，每种推进器和燃料的组合各发射火箭两次，考察推进器和燃料对射程是否有显著影响%见课本，课本中直接输入为8x3的矩阵，使用anova2来求解，题目中的两次发射表现在anova(a,2)中的数字2，意为：行列对所拥有的观察点数，即题中所说的发射几次火箭。返回值有三个，老师讲是：第一个：第一种影响因子的原假设均值相等的几率；第二种同理；第三种是二者之乘积p309 ex 9.23 回归分析，研究温度对产品得率之影响，做y=a+bx型之回归%其实就是拟合 x=100:10:190; y=45 51 54 61 66 70 74 78 85 89; a,b=polyfit(x,y,1)a = 0.4830 -2.7394b = r: 2x2 double df: 8 normr: 2.6878 f=a(1)*x+a(2); plot(x,y,or,x,f,-g)操作题 43 积分的符号法和数值法 syms x; int(log(x+1),0,1) ans = log(4) - 1 vpa(ans) ans = 0.38629436111989061883446424291635%符号法，使用int语句积分，功能强大 quad(log(1+x),0,1)ans = 0.3863%用quad语句对于简单的积分十分方便 x=0:.01:1; f=log(x+1); ff=trapz(x,f)ff = 0.3863%用trapz梯形积分，用quad进行自适应辛卜生法，用quad8进行neton-cotes法，gauss-labatto:quadl,二重积分 i=dblquad(fname,xmin,xmax,ymin,ymax,tol ,trace)，三重积分：i = triplequad(fun,xmin,xmax,ymin,ymax,zmin,zmax,tol)操作题 44多阶常微分 用ode23、ode45、ode113求y.mfunction y=y(x,y)dy=zeros(3,1)dy(1)=y(2)dy(2)=y(3)dy(3)=2*y(3)/x3+3*y(2)/x3+3*exp(2*x)/x3x23,y23=ode23(y,1,10,1 10 30);x45,y45=ode45(y,1,10,1 10 30);x113,y113=ode113(y,1,10,1 10 30);plot(x23,y23,-xb,x45,y45,-+g,x113,y113,-or)操作题45 求方程的根 fzero(x.2+x-1,0.5)ans = 0.6180%fzero: find root of continuous function of one variable. 格式：x = fzero(fun,x0)；x0可以是某个数值，也可以是定义域；roots是求多项式的，也可以用来求本题 c=1 1 -1; roots(c)ans = -1.6180 0.6180操作题 46 求梯度和法向量%数值法求梯度，使用fx=gradient(f)只显示fx/fx,fy=gradient(f) f=1 1.2 1.4 2.35;.06 3 4 2;-1 77.2 9 1.4; fx,fy=gradient(f)%二元多元函数求导一般用gradient和nx,ny,nz=surfnorm,gradient: numerical gradient；surfnorm: compute and display 3-d surface normal(法向量)fx = 0.2000 0.2000 0.5750 0.9500 2.9400 1.9700 -0.5000 -2.0000 78.2000 5.0000 -37.9000 -7.6000fy = -0.9400 1.8000 2.6000 -0.3500 -1.0000 38.0000 3.8000 -0.4750 -1.0600 74.2000 5.0000 -0.6000 fx=gradient(f)fx = 0.2000 0.2000 0.5750 0.9500 2.9400 1.9700 -0.5000 -2.0000 78.2000 5.0000 -37.9000 -7.6000 n=surfnorm(f)n = -0.1485 -0.0058 -0.3170 -0.7910 -0.9404 -0.0518 0.1262 0.9534 -1.0000 -0.0452 0.9865 -0.9985%求法向量用surfnorm操作题 47 add path,pathcan be used:1.addpath:add folders to search path addpath(f:download) addmypath(1)f = -3ans = -32.pathtool: open set path dialog box to view and change search pat3.genpath generate path string p = genpath(fullfile(matlabroot,toolbox,images)p =f:matric laboratarytoolboximages;f:matric laboratarytoolboximagescolorspaces;f:matric laboratarytoolboximagescolorspacesja;f:matric laboratarytoolboximagesicons;f:matric laboratarytoolboximagesimages;f:matric laboratarytoolboximagesimageseml;f:matric laboratarytoolboximagesimagesja;f:matric laboratarytoolboximagesimdemos;f:matric laboratarytoolboximagesimdemosdemosearch;f:matric laboratarytoolboximagesimdemoshtml;f:matric laboratarytoolboximagesimdemoshtmlja;f:matric laboratarytoolboximagesimdemosja;f:matric laboratarytoolboximagesimdemosjademosearch;f:matric laboratarytoolboximagesimuitools;f:matric laboratarytoolboximagesimuitoolsja;f:matric laboratarytoolboximagesiptformats;f:matric laboratarytoolboximagesiptformatsja;f:matric laboratarytoolboximagesiptutils;f:matric laboratarytoolboximagesiptutilsja;操作题 48 求多项式的根并生成原多项式 p=1 21 20 0; roots(p)ans = 0 -20 -1%roots: polynomial roots, r = roots(c) returns a column vector whose elements are the roots of the polynomial c. f=poly2sym(p) f = x3 + 21*x2 + 20*x操作题 49 求极限1. 含参数 syms x m n; a=limit(m./(1-x.m)-n./(1-x.n),x,1) a = (- n2/2 + n/2)/n - (- m2/2 + m/2)/m2.x=sym(x); a1=limit(x.3+x.2+x+1).3-sqrt(x.2+x+1).*(log(exp(x)+x)./x),inf) a1 = inf3. syms x y; a2=limit(3.x+9.x).(1/x),x,1./y) a2 = 3*(3(1/y) + 1)y4. a3=limit(sqrt(1./x+sqrt(1./x+sqrt(1./x)-sqrt(1./x-sqrt(1./x+sqrt(1./x),x,0,right) a3 = 1%求极限使用limit即可，调用格式：limit(1/x, x, 0, right)%求积分summary 符号法求积分：int :int(expr,var,a,b,name,value),求定积分、不定积分皆可 quad/quadl/quad8数值法求积分：numerically evaluate integral, adaptive simpson quadrature 调用q = quad(fun,a,b,tol,trace)；trapz:梯形积分：trapezoidal numerical integration 调用：z = trapz(x,y)操作题 50 求复杂函数的导数 diff(5*x3+3*x2-2*x+7)/(-4*x3+8*x+3),1) ans = (15*x2 + 6*x - 2)/(- 4*x3 + 8*x + 3) + (12*x2 - 8)*(5*x3 + 3*x2 - 2*x + 7)/(- 4*x3 + 8*x + 3)2%此式中不可有.等操作题 51 积分1. syms x f=(x+sin(x)/(1+cos(x) f = x + sin(x)/(cos(x) + 1) int(f) ans = x2/2 - log(cos(x) + 1)2. quad(sqrt(log(1./x),0,1)ans = 0.88623. syms x f=cos(x) x.2;2.x log(2+x); int(f) ans = sin(x), x3/3 2x/log(2), (log(x + 2) - 1)*(x + 2)操作题52 taylor 幂级数展开%taylor幂级数展开：taylor series expansion：taylor(f,n,a) taylor(sin(x),3,1) ans = sin(1) - (sin(1)*(x - 1)2)/2 + cos(1)*(x - 1) vpa(ans) ans = 0.54030230586813971740093660744298*x - 0.42073549240394825332625116081515*(x - 1.0)2 + 0.30116867893975678925156571418732操作题 53 非线性方程、求偏导 syms u v x y; u,v=solve(x*u+y*v=0,y*u