(1.8)-2023數(shù)值實(shí)驗(yàn)-練習(xí)8

一、Newton-Cotes公式：梯形、Simpson、Cotes方法clear,clca=0;b=1;h=b-a;formatlongT=(f(a)+f(b))*h/2S=(f(a)+4*f((a+b)/2)+f(b))*h/6C=(7*f(a)+32*f((3*a+b)/4)+12*f((a+b)/2)+32*f((a+3*b)/4)+7*f(b))*h/90Q=integral(@f,0,1)1functiony=f(x)ifx==0y=1;elsey=sin(x)./x;end第六章：數(shù)值積分與數(shù)值微分二、復(fù)化梯形、復(fù)化Simpsonclear,clca=0;b=1;n=8;h=(b-a)/n;T8=(f(a)+f(b))*h/2;fori=1:n-1T8=T8+f(a+i*h)*h;endT821、復(fù)化梯形公式clear,clca=0;b=1;n=4;h=(b-a)/n;S4=(f(a)+f(b))*h/6;fori=1:nS4=S4+4*f(a+(i-0.5)*h)*h/6;endfori=1:n-1S4=S4+2*f(a+i*h)*h/6;endQ=integral(@f,0,1),S43二、復(fù)化梯形、復(fù)化Simpson2、復(fù)化Simpson公式三、自適應(yīng)的復(fù)化梯形方法clear,clca=0;b=1;h=b-a;formatlongT1=(f(a)+f(b))*h/2;T2=T1/2+h*f((a+b)/2)/2;k=1;whileabs(T2-T1)>0.5*10^(-6)&k<20k=k+1;h=h/2;n=(b-a)/h;T1=T2;T2=T1/2;fori=1:n

T2=T2+h*f(a+(i-1/2)*h)/2;endendT=T2,k=k,Q=integral(@f,0,1)四*、Romberg算法function[s,k]=Romberg_2(a,b,eps)ifnargin==2eps=1.0e-6;elseifnargin<2errorreturnendt1=10^4;t2=-10^4;n=2;t(1,1)=0.5*(b-a)*(f(a)+f(b));area=t(1,1)/2;whileabs(t2-t1)>=eps

h=(b-a)/2^(n-1);fori=1:2:(2^(n-1))area=area+h*f(a+i*h);end

t(n,1)=area;forj=2:nfori=1:(n-j+1)t(i,j)=(4^(j-1)*t(i+1,j-1)-t(i,j-1))/(4^(j-1)-1);endendt1=t(i,n);t2=t(i+1,n-1);n=n+1;area=area/2;ends=t1;k=n-1;clear,clc,formatlonga=0;b=1;h=b-a;[s,k]=Romberg_2(a,b,eps)Q=integral(@f,0,1)五、數(shù)值求導(dǎo)（數(shù)值微分）clear,clc,formatshortx=[0,1,2,3,4,5];y=[1,2,4,5,7,6];n=length(x);h=x(2)-x(1);dx=zeros(1,n);dx(1)=(-3*y(1)+4*y(2)-y(3))/2/h;dx(n)=(y(n-2)-4*y(n-1)+3*y(n))/2/h;fori=2:n-1dx(i)=(y(i+1)-y(i-1))/2/h;enddx1、等距節(jié)點(diǎn)求導(dǎo)公式五、數(shù)值求導(dǎo)（數(shù)值微分）clear,clc,formatshortx=[0,1,2,15,18,20];y=[1,2,4,5,7,6];n=length(x);fori=1:nifi==1p=polyfit(x(i:i+2),y(i:i+2),2);elseifi==np=polyfit(x(i-2:i),y(i-2:i),2);elsep=polyfit(x(i-1:i+1),y(i-1:i+1),2);enddx(i)=2*p(1)*x(i)+p(2);ddx(i)=2*p(1);enddx,ddx2、非等距節(jié)點(diǎn)求導(dǎo)公式六、線性方程組的迭代解法clear,clcA=[3,2,1;2,5,-2;1,-1,-4];b=[5;16;7];n=length(b);D=diag(diag(A));L=-tril(A,-1);U=-triu(A,1);X0=zeros(n,1);X=D\((L+U)*X0+b);k=1;whilenorm(X-X0)>10^(-12)&k<10^3k=k+1;X0=X;X=D\((L+U)*X0+b);endX,k1、Jacobi迭代法六、線性方程組的迭代解法clear,clcA=[3,2,1;2,5,-2;1,-1,-4];b=[5;16;7];n=length(b);D=diag(diag(A));L=-tril(A,-1);U=-triu(A,1);X0=zeros(n,1);X=(D-L)\(U*X0+b);k=1;whilenorm(X-X0)>10^(-12)&k<10^3k=k+1;X0=X;