matlab第五章函数、导数与积分等ppt课件

1、第五章 函数、导数与积分等 的数值计算函数值、多项式、导数、定积分、二重积分、非线性方程组）、级数的计算1.函数值的计算解：x=pi/6;y=sin(x)+sin(2.*x)+sin(3.*x);x,yx = 0.5236y = 2.3660处的函数值。在计算例63sin2sinsin1xxxxy解：x=1:2:5;y=2:2:6;z=log(x.2+y.2)+2.*x.*y.*tan(y./x); x,y,zx = 1 3 5y = 2 4 6z = -7.1307 102.3804 158.4400处的函数值。在计算例6 , 4 , 2, 5 , 3 , 1tan2)ln(222yxxyx

2、yyxz2.多项式的计算多项式的表示法：p=1,0,-12,5多项式在给定点的值：p=4,-3,2,1; y=polyval(p,5) y = 436多项式方程求根:rootsp1=1,0,-12,5;p2=1,0,-12,20;x1i=roots(p1)x1i = -3.6562 3.2332 0.4230 x2i=roots(p2)x2i = -4.1072 2.0536 + 0.8075i 2.0536 - 0.8075i多项式相乘：convp1=5,-4,3,-2,1,-6;p2=1,0,-2,3;p3=conv(p1,p2)p3 = 5 -4 -7 21 -17 7 -8 15 -1

3、8多项式相除：q,r=deconv(p1,p2)p1=5,-4,-7,21,-17,7,-8,15,-18;p2=1,0,-2,3;q,r=deconv(p1,p2)q = 5 -4 3 -2 1 -6r = 0 0 0 0 0 0 0 0 0p2=1,0,-2,5;q,r=deconv(p1,p2)q = 5 -4 3 -12 9 -32r = 0 0 0 0 0 0 70 -94 142=poly2sym(q) =5*x5-4*x4+3*x3-12*x2+9*x-32 rr=poly2sym(r) rr =70*x2-94*x+142解： p=1,0,0,0;q=1,1,-2;a,b,r=

4、residue(p,q)a = 2.6667 0.3333b = -2 1r = 1 -1分解为最简分式之和。将有理分式例2323 xxxw有理分式的分解与合并：a,b,r=residue(p,q) 1(13333. 026667. 2223xxxxxx解：p=1,1;q=1,-6,11,-6; a,b,r=residue(p,q) a = 2.0000 -3.0000 1.0000 b = 3.0000 2.0000 1.0000 r = 为最简分式之和。分解例61161423xxxx解：p=1;q=1,0,0,0,-1; a,b,r=residue(p,q)a = -0.2500 0.25

5、00 -0.0000 + 0.2500 -0.0000 - 0.2500ib = -1.0000 1.0000 0.0000 + 1.0000 0.0000 - 1.0000ir = 为最简分式之和。分解例1154x)121111(411124xxxx将简单分式之和并为有理分式例6a=2.6667,0.3333;b=-2,1;r=1,-1;p,q=residue(a,b,r)p = 1.0000 0 0 -0.0001q = 1 1 -23.导数的数值计算fx,fy=gradient(f, x, y)yx=diff(y)./diff(x)例7 x=0:0.5:3;f=x.2;fx=gradie

6、nt(f,0.5)fx = 0.5000 1.0000 2.0000 3.0000 4.0000 5.0000 5.5000例8x,y=meshgrid(0:0.2:1,0:0.3:1.5);f=x.2+x.*y;fx,fy=gradient(f,0.2,0.3);fxfx = 0.2000 0.4000 0.8000 1.2000 1.6000 1.8000 0.5000 0.7000 1.1000 1.5000 1.9000 2.1000 0.8000 1.0000 1.4000 1.8000 2.2000 2.4000 1.1000 1.3000 1.7000 2.1000 2.5000

7、 2.7000 1.4000 1.6000 2.0000 2.4000 2.8000 3.0000 1.7000 1.9000 2.3000 2.7000 3.1000 3.3000例9x,y=meshgrid(-2:0.2:2);f=x.*exp(-x.2-y.2);fx,fy=gradient(f,0.2,0.2);contour(f)hold onquiver(fx,fy)hold off246810121416182024681012141618204.定积分的数值计算wSimpson法：S,nf=quad(F,a,b,tol)解：function y=f(x) y=sin(x)./x

8、;quad(simp11,1,10)S1 = 0.7123nf = 65。计算例dxxxS1011sin10例11 计算解： function y=f(x)y=exp(-x.2/2); S2,nf=quad(simp12,-8,8)S2 = 2.5066nf = 105dxeSx222Newton Cotes法：S,nf=quad8(F,a,b,tol)例12：同11。解：function y=f(x)y=exp(-x.2/2); format long;S3,nf=quad8(Newton,-8,8)S3 = 2.50662827463083nf = 168例例13 分别用分别用quad函数

9、和函数和quad8函数函数求定积分的近似值，并在相同的积求定积分的近似值，并在相同的积分精度下，比较函数的调用次数。分精度下，比较函数的调用次数。 解：解： 调用函数调用函数quad求定积分：求定积分： format long; fx=inline(exp(-x); I,n=quad(fx,1,2.5,1e-10) I =0.28579444254766 n =65调用函数quad8求定积分： format long; fx=inline(exp(-x); I,n=quad8(fx,1,2.5,1e-10) I = 0.28579444254754 n =335.二重积分的数值计算w矩形域上的

10、二重积分：S=dblquad(F,c,d,a,b,tol) f=inline(sin(x+y),x,y); S=dblquad(f,-2,2,-1,1) S = 3.3307e-016解：function z=f(x,y)global nf;nf=nf+1;z=sin(x+y);S=db1quad(simp21,-2,2,-1,1),nfS = 3.3307e-016nf = 3122 , 2,1 , 1,)sin(S14yxdxdyyxD计算例w一般有界域上的二重积分w例15 计算 ，D为 与 所为成区域。Dyxdxdye)(22解：syms x y;f=exp(-x*x-y*y);y1=x

11、*x;y2=2*x;S1=int(f,y,y1,y2);S2=int(S1,x,0,2);SS=double(S2)SS = 0.2652xy22xy 例16 计算 ，D为 与 所围区域。解：syms x y;f=sin(x+y)/(x+y);x1=y*y;x2=y+2;S1=int(f,x,x1,x2);S2=int(S1,y,-1,2);SS=double(S2)SS = 1.9712Ddxdyyxyx)/()sin(2 xyxy 26.非线性方程组的数值解wx=fsolve(fun,x0)解： function f=fun2(t) x=t(1);y=t(2);f(1)=x3-y2;f(2

12、)=exp(-x)-y;t0=1,1;t=fsolve(fun2,t0) t = 0.6488 0.5226f=fun2(t) f = 1.0e-006 * 0.2196 0.0321.001723的实根求方程组例yeyxx解：function f=fun3(t) x=t(1);y=t(2);z=t(3);f(1)=x+y+z-6;f(2)=x+y*z+z*x-8;f(3)=exp(-x)+log(y)+z-2;t0=1,1,1;t=fsolve(fun3,t0) t = 2.6132 2.2877 1.0992f=fun3(t) f = 1.0e-012 * 0.0036 -0.2019 -0.0868的一个实根。求方程组例2ln, 8, 618zyezxyzxzyxx7.级数的数值和wS=symsum(fn,n,n1,n2)解：syms n;fn=1/(n*n);S10=symsum(fn,n,1,10);Sinf=symsum(fn,n,1,inf);S10,SinfS10 =1968329/1270080Sinf =1/6*pi211012,/191nnnnnffnf