数学实验4-1微积分问题的计算机求解

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1、单击此处编辑母版标题样式,,单击此处编辑母版文本样式,,第二级,,第三级,,第四级,,第五级,,,*,第四讲 微积分问题的计算机求解（上）,微积分问题的解析解,,函数的级数展开与级数求和问题求解,,数值微分,,数值积分问题,,曲线积分与曲面积分的计算,,,3.1 微积分问题的解析解,3.1.1 极限问题的解析解,单变量函数的极限,,,,格式1：,L= limit( fun, x, x0),,,,,格式2：,L= limit( fun, x, x0, ‘left’ 或 ‘right’),,例: 试求解极限问题,,>> syms x a b;,,>> f=x*(1+a/x)^x*sin(b/x)

2、;,,>> L=limit(f,x,inf),,L =,,exp(a)*b,,例:求解单边极限问题,,>> syms x;,,>> limit((exp(x^3)-1)/(1-cos(sqrt(x-sin(x)))),x,0,'right'),,ans =,,12,,在(-0.1,0.1)区间绘制出函数曲线：,,>> x=-0.1:0.001:0.1;,,>> y=(exp(x.^3)-1)./(1-cos(sqrt(x-sin(x))));,,Warning: Divide by zero.,,(Type "warning off,,MATLAB:,,divideByZero" to,,su

3、ppress this warning.),,>> plot(x,y,'-',[0],,,[12],'o'),,多变量函数的极限：,,,,,格式：,L,1,=limit(limit(f,x,x,0,),y,y,0,),,或,L,1,=limit(limit(f,y,y,0,), x,x,0,),,,如果x,0,或y,0,不是确定的值，而是另一个变量的函数，如x->g(y)，则上述的极限求取顺序不能交换。,,例：求出二元函数极限值,,,,,>> syms x y a;,,>> f=exp(-1/(y^2+x^2)) … *sin(x)^2/x^2*(1+1/y^2)^(x+a^2*y^2);,,

4、>> L=limit(limit(f,x,1/sqrt(y)),y,inf),,L =,,exp(a^2),,3.1.2 函数导数的解析解,函数的导数和高阶导数,,,,格式：,y=diff(fun,x),%求导数(默认为1阶),,,y= diff(fun,x,n),%求n阶导数,,,例：,,一阶导数：,,>> syms x; f=sin(x)/(x^2+4*x+3);,,>> f1=diff(f); pretty(f1),,cos(x) sin(x) (2 x + 4),,--------------- - -------------------,,2

5、 2 2,,x + 4 x + 3 (x + 4 x + 3),,,原函数及一阶导数图：,,>> x1=0:.01:5;,,>> y=subs(f, x, x1);,,>> y1=subs(f1, x, x1);,,>> plot(x1,y,x1,y1,‘:’),,,更高阶导数：,,>> tic, diff(f,x,100); toc,,elapsed_time =,,4.6860,,原函数4阶导数,,>> f4=diff(f,x,4); pretty(f4),,2,,sin(x) cos(x)

6、 (2 x + 4) sin(x) (2 x + 4),,------------ + 4 ------------------- - 12 -----------------,,2 2 2 2 3,,x + 4 x + 3 (x + 4 x + 3) (x + 4 x + 3),,,3,,sin(x) cos(x) (2 x + 4) cos(x) (2 x + 4),,+ 12 -

7、-------------- - 24 ----------------- + 48 ----------------,,2 2 2 4 2 3,,(x + 4 x + 3) (x + 4 x + 3) (x + 4 x + 3),,,4 2,,sin(x) (2 x + 4) sin(x) (2 x + 4) sin(x),,+ 2

8、4 ----------------- - 72 ----------------- + 24 ---------------,,2 5 2 4 2 3,,(x + 4 x + 3) (x + 4 x + 3) (x + 4 x + 3),,多元函数的偏导：,,,,格式：,f=diff(diff(f,x,m),y,n),,或,f=diff(diff(f,y,n),x,m),,,例：

9、 求其偏导数并用图表示。,,,>> syms x y z=(x^2-2*x)*exp(-x^2-y^2-x*y);,,>> zx=simple(diff(z,x)),,zx =,,-exp(-x^2-y^2-x*y)*(-2*x+2+2*x^3+x^2*y-4*x^2-2*x*y),,,>> zy=diff(z,y),,zy =,,(x^2-2*x)*(-2*y-x)*exp(-x^2-y^2-x*y),,直接绘制三维曲面,,>> [x,y]=meshgrid(-3:.2:3,-2:.2:2);,,>> z=(x.^2-2*x).*exp(-x.^2-y.^2-x.*y);

10、,,>> surf(x,y,z), axis([-3 3 -2 2 -0.7 1.5]),,>> contour(x,y,z,30), hold on % 绘制等值线,,>> zx=-exp(-x.^2-y.^2-x.*y).*(-2*x+2+2*x.^3+x.^2.*y-4*x.^2-2*x.*y);,,>> zy=-x.*(x-2).*(2*y+x).*exp(-x.^2-y.^2-x.*y); % 偏导的数值解,,>> quiver(x,y,zx,zy) % 绘制引力线,,例,,,,,>> syms x y z; f=sin(x^2*y)*exp(-x^2*y-z^2);,,

11、>> df=diff(diff(diff(f,x,2),y),z); df=simple(df);,,>> pretty(df),,,2 2 2 2 2,,-4 z exp(-x y - z ) (cos(x y) - 10 cos(x y) y x + 4,,2 4 2 2 4 2 2,,sin(x y) x y+ 4 cos(x y) x y - sin(x y)),,多元函数的Jacobi矩阵:,,,,,,,,,格式

12、：,J=jacobian(Y,X),,其中，X是自变量构成的向量，Y是由各个函数构成的向量。,,例：,,试推导其 Jacobi 矩阵,,,>> syms r theta phi;,,>> x=r*sin(theta)*cos(phi);,,>> y=r*sin(theta)*sin(phi);,,>> z=r*cos(theta);,,>> J=jacobian([x; y; z],[r theta phi]),,,J =,,[ sin(theta)*cos(phi), r*cos(theta)*cos(phi), -r*sin(theta)*sin(phi)],,[ sin(t

13、heta)*sin(phi), r*cos(theta)*sin(phi), r*sin(theta)*cos(phi)],,[ cos(theta), -r*sin(theta), 0 ],,,,隐函数的偏导数：,,,,,,,,,格式：,F=-diff(f,x,j,)/diff(f,x,i,),,,例：,,,,>> syms x y; f=(x^2-2*x)*exp(-x^2-y^2-x*y);,,>> pretty(-simple(diff(f,x)/diff(

14、f,y))),,,,3 2 2,,-2 x + 2 + 2 x + x y - 4 x - 2 x y,,- -----------------------------------------,,x (x - 2) (2 y + x),,3.1.3 积分问题的解析解,不定积分的推导：,,,格式：,F=int(fun,x),,,例：,,用diff() 函数求其一阶导数，再积分，,检验是否可以得出一致的结果。,,>> syms x; y=sin(x)/(x^2+4*x+3); y1=diff(y);,,>> y0=int(y1); pretty(y0) %

15、 对导数积分,,sin(x) sin(x),,- 1/2 ------ + 1/2 ------,,x + 3 x + 1,,对原函数求4 阶导数，再对结果进行4次积分,,>> y4=diff(y,4);,,>> y0=int(int(int(int(y4))));,,>> pretty(simple(y0)),,,sin(x),,------------,,2,,x + 4 x + 3,,,,例:,证明,,,,>> syms a x; f=simple(int(x^3*cos(a*x)^2,x)),,f = 1/16*(4*a^3*x^3*sin(2*a*x)+2

16、*a^4 *x^4+6*a^2*x^2*cos(2*a*x)-6*a*x*sin(2*a*x)-3*cos(2*a*x)-3)/a^4,,>> f1=x^4/8+(x^3/(4*a)-3*x/(8*a^3))*sin(2*a*x)+...,,(3*x^2/(8*a^2)-3/(16*a^4))*cos(2*a*x);,,>> simple(f-f1) % 求两个结果的差,,ans =,,-3/16/a^4,,定积分与无穷积分计算:,,,,,格式：,I=int(f,x,a,b),,,,,,格式：,I=int(f,x,a,inf),,,,例：,,,,>> syms x; I1=int(exp(-

17、x^2/2),x,0,1.5) ％无解,,I1 =,,1/2*erf(3/4*2^(1/2))*2^(1/2)*pi^(1/2),,>> vpa(I1,70),,ans =,,,>> I2=int(exp(-x^2/2),x,0,inf),,I2 =,,1/2*2^(1/2)*pi^(1/2),,,,多重积分问题的MATLAB求解,,例：,,,,,,,>>,syms x y z; f0=-4*z*exp(-x^2*y-z^2)*(cos(x^2*y)-10*cos(x^2*y)*y*x^2+...,,4*sin(x^2*y)*x^4*y^2+4*cos(x^2*y)*x^4*y^2-sin(

18、x^2*y));,,>> f1=int(f0,z);f1=int(f1,y);f1=int(f1,x);,,>> f1=simple(int(f1,x)),,f1 =,,exp(-x^2*y-z^2)*sin(x^2*y),,,,>> f2=int(f0,z); f2=int(f2,x); f2=int(f2,x);,,>> f2=simple(int(f2,y)),,f2 =,,2*exp(-x^2*y-z^2)*tan(1/2*x^2*y)/(1+tan(1/2*x^2*y)^2),,>> simple(f1-f2),,ans =,,0,,顺序的改变使化简结果不同于原函数，但其误差为0，表

19、明二者实际完全一致。这是由于积分顺序不同，得不出实际的最简形式。,,例：,,,>> syms x y z,,>> int(int(int(4*x*z*exp(-x^2*y-z^2),x,0,1),y,0,pi),z,0,pi),,ans =,,(Ei(1,4*pi)+log(pi)+eulergamma+2*log(2))*pi^2*hypergeom([1],[2],-pi^2),,Ei(n,z)为指数积分，无解析解，但可求其数值解：,,>> vpa(ans,60),,ans =,,,3.2 函数的级数展开与 级数求和问题求解,3.2.1 Taylor 幂级数展开,,,3.2.2 F

20、ourier 级数展开,,,3.2.3 级数求和的计算,,,3.2.1 Taylor 幂级数展开,3.2.1.1 单变量函数的 Taylor 幂级数展开,,,例:,,,,,>> syms x; f=sin(x)/(x^2+4*x+3);,,,>> y1=taylor(f,x,9); pretty(y1),,,,2 23 3 34 4 4087 5 3067 6 515273 7 386459 8,,,1/3 x - 4/9 x + -- x - ---- x + ------x - ------ x +---------- x

21、- --------- x,,54 81 9720 7290 1224720 918540,,>> taylor(f,x,9,2),,ans =,,,,,>> syms a; taylor(f,x,5,a) % 结果较冗长，显示从略,,ans =,,sin(a)/(a^2+3+4*a) +(cos(a)-sin(a)/(a^2+3+4*a)*(4+2*a))/(a^2+3+4*a)*(x-a) +(-sin(a)/(a^2+3+4*a)-1/2*sin(a)-(cos(a)*a^2+3*cos(a)+4*cos(a)*a-4

22、*sin(a)-2*sin(a)*a)/(a^2+3+4*a)^2*(4+2*a))/(a^2+3+4*a)*(x-a)^2+…,,例:对y=sinx进行Taylor幂级数展开,并观察不同阶次的近似效果。,,>> x0=-2*pi:0.01:2*pi; y0=sin(x0); syms x; y=sin(x);,,>> plot(x0,y0,'r-.'), axis([-2*pi,2*pi,-1.5,1.5]); hold on,,>> for n=[8:2:16],,p=taylor(y,x,n), y1=subs(p,x,x0); line(x0,y1) end,,p =,,x-1/

23、6*x^3+1/120*x^5-1/5040*x^7,,p =,,x-1/6*x^3+1/120*x^5-1/5040*x^7+1/362880*x^9,,p =,,x-1/6*x^3+1/120*x^5-1/5040*x^7+1/362880*x^9-1/39916800*x^11,,p =,,x-1/6*x^3+1/120*x^5-1/5040*x^7+1/362880*x^9-1/39916800*x^11+1/6227020800*x^13,,,p =,,,3.2.1.2 多变量函数的Taylor 幂级数展开,多变量函数

24、 在,,的Taylor幂级数的展开,,例：,？？？,,,,>> syms x y; f=(x^2-2*x)*exp(-x^2-y^2-x*y);,,>> F=maple('mtaylor',f,'[x,y]',8),,F =,,mtaylor((x^2-2*x)*exp(-x^2-y^2-x*y),[x, y],8),,>> maple(‘readlib(mtaylor)’);％读库，把函数调入内存,,>> F=maple('mtaylor',f,'[x,y]',8),,F =,,-2*x+x^2+2*x^3-x^4-x^5+1/2*x^6+1/3*x^7+2*y*x^2+2*y^2*x-y*

25、x^3-y^2*x^2-2*y*x^4-3*y^2*x^3-2*y^3*x^2-y^4*x+y*x^5+3/2*y^2*x^4+y^3*x^3+1/2*y^4*x^2+y*x^6+2*y^2*x^5+7/3*y^3*x^4+2*y^4*x^3+y^5*x^2+1/3*y^6*x,,>> syms a; F=maple('mtaylor',f,'[x=1,y=a]',3);,,>> F=maple('mtaylor',f,'[x=a]',3),,F =,,(a^2-2*a)*exp(-a^2-y^2-a*y)+((a^2-2*a)*exp(-a^2-y^2-a*y)*(-2*a-y)+(2*a-

26、2)*exp(-a^2-y^2-a*y))*(x-a)+((a^2-2*a)*exp(-a^2-y^2-a*y)*(-1+2*a^2+2*a*y+1/2*y^2)+exp(-a^2-y^2-a*y)+(2*a-2)*exp(-a^2-y^2-a*y)*(-2*a-y))*(x-a)^2,,3.2.2 Fourier 级数展开,,,function [A,B,F]=fseries(f,x,n,a,b),,if nargin==3, a=-pi; b=pi; end,,L=(b-a)/2;,,if a+b, f=subs(f,x,x+L+a); end％变量区域互换,,A=int(f,x,-L,L

27、)/L; B=[]; F=A/2;,,for i=1:n,,an=int(f*cos(i*pi*x/L),x,-L,L)/L;,,bn=int(f*sin(i*pi*x/L),x,-L,L)/L; A=[A, an]; B=[B,bn];,,F=F+an*cos(i*pi*x/L)+bn*sin(i*pi*x/L);,,end,,if a+b, F=subs(F,x,x-L-a); end ％换回变量区域,,例:,,,>> syms x; f=x*(x-pi)*(x-2*pi);,,>> [A,B,F]=fseries(f,x,6,0,2*pi),,A =,,[ 0, 0, 0, 0, 0,

28、0, 0],,B =,,[ -12, 3/2, -4/9, 3/16, -12/125, 1/18],,F =,,12*sin(x)+3/2*sin(2*x)+4/9*sin(3*x)+3/16*sin(4*x)+12/125*sin(5*x)+1/18*sin(6*x),,例:,,,,>> syms x; f=abs(x)/x; % 定义方波信号,,>> xx=[-pi:pi/200:pi]; xx=xx(xx~=0); xx=sort([xx,-eps,eps]); % 剔除零点,,>> yy=subs(f,x,xx); plot(xx,yy,'r-.'

29、), hold on % 绘制出理论值并保持坐标系,,>> for n=2:20,,[a,b,f1]=fseries(f,x,n), y1=subs(f1,x,xx); plot(xx,y1),,end,,a =,,[ 0, 0, 0],,b =,,[ 4/pi, 0],,f1 =,,4/pi*sin(x),,a =,,[ 0, 0, 0, 0],,b =,,[ 4/pi, 0, 4/3/pi],,f1 =,,4/pi*sin(x)+4/3/pi*sin(3*x),,……,,3.2.3 级数求和的计算,是在符号工具箱中提供的,,例:计算,,,,>> format long

30、; sum(2.^[0:63]) %数值计算,,ans =,,1.844674407370955e+019,,>> sum(sym(2).^[0:200]) % 或 syms k; symsum(2^k,0,200),,％把2定义为符号量可使计算更精确,,ans =,,,>> syms k; symsum(2^k,0,200),,ans =,,,例,:试求解无穷级数的和,,,,>> syms n; s=symsum(1/((3*n-2)*(3*n+1)),n,1,inf),,%采用符号运算工具箱,,s =,,1/3,,>> m=1:10000000; s1=sum(1./((3*m-2).

31、*(3*m+1)));%数值计算方法,双精度有效位16,“大数吃小数”,无法精确,,>> format long; s1 % 以长型方式显示得出的结果,,s1 =,,0.33333332222165,,例:求解,,,>> syms n x,,>> s1=symsum(2/((2*n+1)*(2*x+1)^(2*n+1)),n, 0,inf);,,>> simple(s1) % 对结果进行化简，MATLAB 6.5 及以前版本因本身 bug 化简很麻烦,,ans =,,log((((2*x+1)^2)^(1/2)+1)/(((2*x+1)^2)^(1/2)-1)),,%实际应为log((x+1

32、)/x),,例:求,,,>> syms m n; limit(symsum(1/m,m,1,n)-log(n),n,inf),,ans =,,eulergamma,,,,>> vpa(ans, 70) % 显示 70 位有效数字,,ans =,,,,,3.3 数值微分,3.3.1 数值微分算法,,向前差商公式：,,,向后差商公式,,,,,,,两种中心公式：,,,,,,,,,,,3.3.2 中心差分方法及其 MATLAB 实现,function [dy,dx]=diff_ctr(y, Dt, n),,,yx1=[y 0 0 0 0 0]; yx2=[0 y 0 0 0 0]; yx3=[0 0

33、 y 0 0 0];,,yx4=[0 0 0 y 0 0]; yx5=[0 0 0 0 y 0]; yx6=[0 0 0 0 0 y];,,switch n,,case 1,,dy = (-diff(yx1)+7*diff(yx2)+7*diff(yx3)- … diff(yx4))/(12*Dt); L0=3;,,case 2,,dy=(-diff(yx1)+15*diff(yx2)- 15*diff(yx3)… +diff(yx4))/(12*Dt^2);L0=3;,,％数值计算diff(X)表示数组X相邻两数的差,,case 3,,dy=(-diff(yx1)+7*diff(yx2

34、)-6*diff(yx3)-6*diff(yx4)+...,,7*diff(yx5)-diff(yx6))/(8*Dt^3); L0=5;,,case 4,,dy = (-diff(yx1)+11*diff(yx2)-28*diff(yx3)+28*… diff(yx4)-11*diff(yx5)+diff(yx6))/(6*Dt^4); L0=5;,,end,,dy=dy(L0+1:end-L0); dx=([1:length(dy)]+L0-2-(n>2))*Dt;,,,调用格式：,,,y为 等距实测数据， dy为得出的导数向量， dx为相应的自变量向量，dy、dx的数据比y短 。,,例：

35、,,求导数的解析解,再用数值微分求取原函数的1～4 阶导数，并和解析解比较精度。,,,>> h=0.05; x=0:h:pi;,,>> syms x1; y=sin(x1)/(x1^2+4*x1+3);,,% 求各阶导数的解析解与对照数据,,>> yy1=diff(y); f1=subs(yy1,x1,x);,,>> yy2=diff(yy1); f2=subs(yy2,x1,x);,,>> yy3=diff(yy2); f3=subs(yy3,x1,x);,,>> yy4=diff(yy3); f4=subs(yy4,x1,x);,,>> y=sin(x)./(x.^2+4*x+3);

36、% 生成已知数据点,,>> [y1,dx1]=diff_ctr(y,h,1); subplot(221),plot(x,f1,dx1,y1,':');,,>> [y2,dx2]=diff_ctr(y,h,2); subplot(222),plot(x,f2,dx2,y2,':'),,>> [y3,dx3]=diff_ctr(y,h,3); subplot(223),plot(x,f3,dx3,y3,':');,,>> [y4,dx4]=diff_ctr(y,h,4); subplot(224),plot(x,f4,dx4,y4,':'),,,求最大相对误差：,,>> norm((y4-…,,f

37、4(4:60))./f4(4:60)),,ans =,,3.5025e-004,,3.3.3 用插值、拟合多项式的求导数,基本思想：当已知函数在一些离散点上的函数值时，该函数可用插值或拟合多项式来近似，然后对多项式进行微分求得导数。,,选取x=0附近的少量点,,,进行多项式拟合或插值,,,g(x)在x=0处的k阶导数为,,,通过坐标变换用上述方法计算任意x点处的导数值,,令,,将g(x)写成z的表达式,,,导数为,,,,可直接用 拟合节点 得到系数,,,d=polyfit(x-a,y,length(xd)-1),,,例：数据集合如下：,,xd:

38、0 0.2000 0.4000 0.6000 0.8000 1.000,,yd: 0.3927 0.5672 0.6982 0.7941 0.8614 0.9053,,计算x=a=0.3处的各阶导数。,,>> xd=[ 0 0.2000 0.4000 0.6000 0.8000 1.000];,,>> yd=[0.3927 0.5672 0.6982 0.7941 0.8614 0.9053];,,>> a=0.3;L=length(xd);,,>> d=polyfit(xd-a,yd,L-1);fact=[1];,,>> for

39、 k=1:L-1;fact=[factorial(k),fact];end,,>> deriv=d.*fact,,deriv =,,1.8750 -1.3750 1.0406 -0.9710 0.6533 0.6376,,建立用拟合（插值）多项式计算各阶导数的poly_drv.m,,function der=poly_drv(xd,yd,a),,m=length(xd)-1;,,d=polyfit(xd-a,yd,m);,,c=d(m:-1:1); ％去掉常数项,,fact(1)=1;for i=2:m; fact(i)=i*fact(i-1);end,,der

40、=c.*fact;,,例：,,>> xd=[ 0 0.2000 0.4000 0.6000 0.8000 1.000];,,>> yd=[0.3927 0.5672 0.6982 0.7941 0.8614 0.9053];,,>> a=0.3; der=poly_drv(xd,yd,a),,der =,,0.6533 -0.9710 1.0406 -1.3750 1.8750,,3.3.4 二元函数的梯度计算,,,格式：,,,若z矩阵是建立在等间距的形式生成的网格基础上，则实际梯度为,,,例：,,计算梯度，绘制引力线图：,,>> [x,y]=me

41、shgrid(-3:.2:3,-2:.2:2); z=(x.^2-2*x).*exp(-x.^2-y.^2-x.*y);,,>> [fx,fy]=gradient(z);,,>> fx=fx/0.2; fy=fy/0.2;,,>> contour(x,y,z,30);,,>> hold on;,,>> quiver(x,y,fx,fy),,%绘制等高线与,,引力线图,,绘制误差曲面:,,>> zx=-exp(-x.^2-y.^2-x.*y).*(-2*x+2+2*x.^3+x.^2.*y-4*x.^2-2*x.*y);,,>> zy=-x.*(x-2).*(2*y+x).*exp(-x.^2-

42、y.^2-x.*y);,,>> surf(x,y,abs(fx-zx)); axis([-3 3 -2 2 0,0.08]),,>> figure; surf(x,y,abs(fy-zy)); axis([-3 3 -2 2 0,0.11]),,％建立一个新图形窗口,,为减少误差，对网格加密一倍：,,>> [x,y]=meshgrid(-3:.1:3,-2:.1:2); z=(x.^2-2*x).*exp(-x.^2-y.^2-x.*y);,,>> [fx,fy]=gradient(z); fx=fx/0.1; fy=fy/0.1;,,>> zx=-exp(-x.^2-y.^2-x.*y).

43、*(-2*x+2+2*x.^3+x.^2.*y-4*x.^2-2*x.*y);,,>> zy=-x.*(x-2).*(2*y+x).*exp(-x.^2-y.^2-x.*y);,,>> surf(x,y,abs(fx-zx)); axis([-3 3 -2 2 0,0.02]),,>> figure; surf(x,y,abs(fy-zy)); axis([-3 3 -2 2 0,0.06]),,3.4 数值积分问题,4.3.1 由给定数据进行梯形求积,Sum((2*y(1:end-1,:)+diff(y)).*diff(x))/2,,,格式：,S=trapz(x,y),,例：,,,,>> x

44、1=[0:pi/30:pi]'; y=[sin(x1) cos(x1) sin(x1/2)];,,>> x=[x1 x1 x1]; S=sum((2*y(1:end-1,:)+diff(y)).*diff(x))/2,,S =,,1.9982 0.0000 1.9995,,>> S1=trapz(x1,y) % 得出和上述完全一致的结果,,S1 =,,1.9982 0.0000 1.9995,,例：,,画图,,>> x=[0:0.01:3*pi/2, 3*pi/2]; % 这样赋值能确保 3*pi/2点被包含在内,,>> y=cos(15*x); plot(x

45、,y),,,% 求取理论值,,>> syms x, A=int(cos…,,(15*x),0,3*pi/2),,A =,,1/15,,,随着步距h的减小，计算精度逐渐增加：,,>> h0=[0.1,0.01,0.001,0.0001,0.00001,0.000001]; v=[];,,>> for h=h0,,,x=[0:h:3*pi/2, 3*pi/2]; y=cos(15*x); I=trapz(x,y);,,v=[v; h, I, 1/15-I ];,,end,,>> v,,v =,,0.1000 0.0539 0.0128,,0.0100 0.0665 0.00

46、01,,0.0010 0.0667 0.0000,,0.0001 0.0667 0.0000,,0.0000 0.0667 0.0000,,0.0000 0.0667 0.0000,,>> format long，v,,3.4.2 单变量数值积分问题求解,梯形公式,,,,,,格式：(变步长）,,,y=quad(Fun,a,b),,y=quadl(Fun,a,b),% 求定积分,,,y=quad(Fun,a,b, ),,y=quadl(Fun,a,b, ),%限定精度的定积分求解，默认精度为10,－6,。后面函数算法更精，精度更高。,,

47、例：,第三种：匿名函数(MATLAB 7.0),第二种：inline 函数,,第一种，一般函数方法,,函数定义被积函数：,,>,> y=quad('c3ffun',0,1.5),,y =,,0.9661,,用 inline 函数定义被积函数：,,>> f=inline('2/sqrt(pi)*exp(-x.^2)','x');,,>> y=quad(f,0,1.5),,y =,,0.9661,,运用符号工具箱：,,>>,syms x, y0=vpa(int(2/sqrt(pi)*exp(-x^2),0,1.5),60),,y0 =,,,>> y=quad(f,0,1.5,1e-20) %

48、 设置高精度，但该方法失效,,例：,,,提高求解精度：,,>> y=quadl(f,0,1.5,1e-20),,y =,,0.9661,,>> abs(y-y0),,ans =,,.6402522848913892e-16,,>> format long ％16位精度,,>> y=quadl(f,0,1.5,1e-20),,y =,,,例：,求解,,,,绘制函数:,,>> x=[0:0.01:2, 2+eps:0.01:4,4];,,>> y=exp(x.^2).*(x2);,,>> y(end)=0;,,>> x=[eps, x];,,>> y=[0,y];,,>> fill(x,y,'g

49、'),,％为减少视觉上的误,,差，对端点与间断点,,（有跳跃）进行处理。,,调用quad( ):,,>> f=inline('exp(x.^2).*(x2)./(4-sin(16*pi*x))','x');,,>> I1=quad(f,0,4),,I1 =,,57.76435412500863,,调用quadl( ):,,>> I2=quadl(f,0,4),,I2 =,,57.76445016946768,,,,>> syms x; I=vpa(int(exp(x^2),0,2)+int(80/(4-sin(16*pi*x)),2,4)),,I =,,,3.4.3 Gauss求积公式,为

50、使求积公式得到较高的代数精度,,,,,,,对求积区间[a,b]，通过变换,,有,,,以n=2的高斯公式为例：,,function g=gauss2(fun,a,b),,h=(b-a)/2;,,c=(a+b)/2;,,x=[h*(-0.7745967)+c, c, h*0.7745967+c];,,g=h*(0.55555556*(gaussf(x(1))+gaussf(x(3)))+0.88888889*gaussf(x(2)));,,,function y=gaussf(x),,y=cos(x);,,,>> gauss2('gaussf',0,1),,ans =,,0.8415,,3.4.4

51、 基于样条插值的数值微积分运算,基于样条插值的数值微分运算,,格式：,,,S,d,=fnder(S,k),,该函数可以求取S的k阶导数。,,,格式：,,,S,d,=fnder(S,[k,1,,…,k,n,]),,可以求取多变量函数的偏导数,,,例：,,,,>> syms x; f=(x^2-3*x+5)*exp(-5*x)*sin(x);,,>> ezplot(diff(f),[0,1]), hold on,,>> x=0:.12:1; y=(x.^2-3*x+5).*exp(-5*x).*sin(x);,,>> sp1=csapi(x,y);％建立三次样条函数,,>> dsp1=fnder(

52、sp1,1);,,>> fnplt(dsp1,‘--’)％绘制样条图,,>> sp2=spapi(5,x,y);％5阶次B样条,,>> dsp2=fnder(sp2,1);,,>> fnplt(dsp2,':');,,>> axis([0,1,-0.8,5]),,例：,,,,拟合曲面,,>> x0=-3:.3:3; y0=-2:.2:2; [x,y]=ndgrid(x0,y0);,,>> z=(x.^2-2*x).*exp(-x.^2-y.^2-x.*y);,,>> sp=spapi({5,5},…,,{x0,y0},z); ％B样条,,>>dspxy=fnder(sp,[1,1]);,,>>

53、 fnplt(dspxy),,理论方法：,,>> syms x y; z=(x^2-2*x)*exp(-x^2-y^2-x*y);,,>> ezsurf(diff(diff(z,x),y),[-3 3],[-2 2]),,％对符号变量表达式做三维表面图,,基于样条插值的数值积分运算,,格式：,,,f=fnint(S),,其中S为样条函数。,,,例：考虑,中较稀疏的样本点,用样条积分的方式求出定积分及积分函数。,,,>> x=[0,0.4,1 2,pi]; y=sin(x);,,>> sp1=csapi(x,y); a=fnint(sp1,1); ％建立三次样条函数,,>> xx=fnval(a

54、,[0,pi]); xx(2)-xx(1),,ans =,,2.0191,,>> sp2=spapi(5,x,y); b=fnint(sp2,1);,,>> xx=fnval(b,[0,pi]); xx(2)-xx(1),,ans =,,1.9999,,绘制曲线,,>> ezplot('-cos(t)+2',[0,pi]); hold on,,％不定积分可上下平移,,>> fnplt(a,'--');,,>> fnplt(b,':'),,3.4.5 双重积分问题的数值解,矩形区域上的二重积分的数值计算,,,,格式：,,矩形区域的双重积分：,,,y=dblquad(Fun,x,m,,x,M,,y

55、,m,,y,M,),,,限定精度的双重积分：,,,y=dblquad(Fun,x,m,,x,M,,y,m,,y,M，,),,例：,求解,,,>> f=inline('exp(-x.^2/2).*sin(x.^2+y)','x','y');,,>> y=dblquad(f,-2,2,-1,1),,y =,,1.57449318974494,,任意区域上二元函数的数值积分,(调用工具箱NIT），该函数指定顺序先x后y.,,,,,,,,,例,>> fh=inline('sqrt(1-x.^2/2)','x'); % 内积分上限,,>> fl=inline('-sqrt(1-x.^2/2)','x'

56、); % 内积分下限,,>> f=inline('exp(-x.^2/2).*sin(x.^2+y)','y','x'); % 交换顺序的被积函数,,>> y=quad2dggen(f,fl,fh,-1/2,1,eps),,y =,,,解析解方法：,,>> syms x y,,>> i1=int(exp(-x^2/2)*sin(x^2+y), y, -sqrt(1-x^2/2), sqrt(1-x^2/2));,,>> int(i1, x, -1/2, 1),,Warning: Explicit integral could not be found.,,> In D:\MATLAB6p5\

57、toolbox\symbolic\@sym\int.m at line 58,,ans =,,int(2*exp(-1/2*x^2)*sin(x^2)*sin(1/2*(4-2*x^2)^(1/2)), x = -1/2 .. 1),,>> vpa(ans),,ans =,,,例：,计算单位圆域上的积分：,,,,,先把二重积分转化：,>> syms x y,,i1=int(exp(-x^2/2)*sin(x^2+y), x, -sqrt(1-y.^2), sqrt(1-y.^2));,,Warning: Explicit integral could not be found.,,> In D

58、:\MATLAB6p5\toolbox\symbolic\@sym\int.m at line 58,,对x是不可积的，故调用解析解方法不会得出结果，而数值解求解不受此影响。,,>> fh=inline('sqrt(1-y.^2)','y'); % 内积分上限,,>> fl=inline('-sqrt(1-y.^2)','y'); % 内积分下限,,>> f=inline('exp(-x.^2/2).*sin(x.^2+y)','x','y'); %交换顺序的被积函数,,>> I=quad2dggen(f,fl,fh,-1,1,eps),,Integral did not converge

59、--singularity likely,,I =,,0.53686038269795,,3.4.6 三重定积分的数值求解,,,,格式:,,,,I=triplequad(Fun,x,m,,x,M,,y,m,,y,M,,z,m,,z,M， ,,@quadl),,,其中@quadl为具体求解一元积分的数值函数,也可选用@quad或自编积分函数,但调用格式要与quadl一致。,,例：,,,,>> triplequad(inline('4*x.*z.*exp(-x.*x.*y-z.*z)', …,,'x','y','z'), 0, 1, 0, pi, 0, pi,1e-7,@quadl),,,a

60、ns =,,,1.7328,,3.5 曲线积分与曲面积分的计算,3.5.1 曲线积分及MATLAB求解,,第一类曲线积分,,起源于对不均匀分布的空间曲线总质量的求取.设空间曲线,L,的密度函数为f(x,y,z),则其总质量,,,,其中s为曲线上某点的弧长,又称这类曲线积分为对弧长的曲线积分.,,数学表示,,若,,,,,弧长表示为,,例：,,,,,>> syms t; syms a positive; x=a*cos(t); y=a*sin(t); z=a*t;,,,>> I=int(z^2/(x^2+y^2)*sqrt(diff(x,t)^2+diff(y,t)^2+ diff(z,t)^2)

61、,t,0,2*pi),,I =,,8/3*pi^3*a*2^(1/2),,,,>> pretty(I),,3 1/2,,8/3 pi a 2,,例：,,,>> x=0:.001:1.2; y1=x; y2=x.^2; plot(x,y1,x,y2),,％绘出两条曲线,,>> syms x; y1=x; y2=x^2; I1=int((x^2+y2^2)*sqrt(1+diff(y2,x)^2),x,0,1);,,>> I2=int((x^2+y1^2)*sqrt(1+diff(y1,x)^2),x,1,0); I=I2+I1,,I =,,-2/3*2^(1/2)+349/768*5^

62、(1/2)+7/512*log(-2+5^(1/2)),,3.5.1.2 第二类曲线积分,又称对坐标的曲线积分，起源于变力,,沿曲线 移动时作功的研究,,,,,曲线 亦为向量,若曲线可以由参数方程表示,,,,则两个向量的点乘可由这两个向量直接得出.,,例：求曲线积分,,,,>> syms t; syms a positive; x=a*cos(t); y=a*sin(t);,,>> F=[(x+y)/(x^2+y^2),-(x-y)/(x^2+y^2)]; ds=[diff(x,t);diff(y,t)];,,>> I=int(F*ds,t,2*pi,0) % 正向圆周,,I

63、 =,,2*pi,,例：,,,,>> syms x; y=x^2; F=[x^2-2*x*y,y^2-2*x*y]; ds=[1; diff(y,x)];,,>> I=int(F*ds,x,-1,1),,,,I =,,,,-14/15,,曲面积分与MATLAB语言求解,3.5.2.1 第一类曲面积分,,,其中 为小区域的面积，故又称为对面积的曲面积分。曲面 由 给出，则该积分可转换成x-y平面的二重积分为,,例：,,,,％四个平面，其中三个被积函数的值为0，只须计算一个即可。,,>> syms x y; syms a positive; z=a-x-y

64、;,,>> I=int(int(x*y*z*sqrt(1+diff(z,x)^2+ diff(z,y)^2),y,0,a-x),x,0,a),,,,I =,,1/120*3^(1/2)*a^5,,若曲面由参数方程,,,曲面积分,,例：,,,,,>> syms u v; syms a positive;,,>> x=u*cos(v); y=u*sin(v); z=v;f=x^2*y+z*y^2;,,>> E=simple(diff(x,u)^2+diff(y,u)^2+diff(z,u)^2);,,>> F=diff(x,u)*diff(x,v)+diff(y,u)*diff(y,v)+diff

65、(z,u)* diff(z,v);,,>> G=simple(diff(x,v)^2+diff(y,v)^2+diff(z,v)^2);,,>> I=int(int(f*sqrt(E*G-F^2),u,0,a),v,0,2*pi),,I =,,1/4*a*(a^2+1)^(3/2)*pi^2+1/8*log(-a+(a^2+1)^(1/2)) *pi^2-1/8*(a^2+1)^(1/2)*a*pi^2,,3.5.2.2 第二类曲面积分,又称对坐标的曲面积分,,,,,,可转化成第一类曲面积分,,,若曲面由参数方程给出,,,例：,,,的上半部，且积分沿椭球面的上面。,,％引入参数方程 x=a*sin(u)*cos(v); y=b*sin(u)*sin(v); z=c*cos(u)， u[0,pi/2], v[0,2*pi].,,>> syms u v; syms a b c positive;,,>> x=a*sin(u)*cos(v); y=b*sin(u)*sin(v); z=c*cos(u);,,>> A=diff(y,u)*diff(z,v)-diff(z,u)*diff(y,v);,,>> I=int(int(x^3*A,u,0,pi/2),v,0,2*pi),,I =,,2/5*pi*a^3*c*b,,