用Romberg方法求解积分

1、1.用Romberg方法求解积分：、仅心，要求误差不超过玄=102解：Romberg.m 文件：function I, step = Rombergf, % b,EPS)%Romberg.m是用龙贝格公式求积分%f为被积函数% EPS为积分结果精度%a,b为积分区间的上下限%1为积分结果；step为积分的于区间数m = 1k = 0E r = 0.1H =b-aS = zcros(l, 1)S(l, 1) = (H/2) * (subs(sym(f)findsym(sym(O),a)+subs(syTm(f),findsym(sym(0),b) while Er EPSk = k + 1fl

2、=0H = H/2for i = l:mx = a +H*(2*i-1)fl = fl + subs(sym(f)findsym(sym(0)endS(k+l,l) = S(k, l)/2 + H*flm = 2* mf()r n = l:kS(k+l,n+l) = S(k+l,n) + (S(k+1, n)-S(k, n)/(4An-l)endEr = abs(S(k+l, n+l)-S(k,n)endI = S(k+hk+l)step = k命令：clearclc format shorta = 0; b = 0.& EPS = lc-2;P. step = Mmbcrgf/(1/2)：

3、a, b, EPS)计算结果:1Er =0.1000H =0.8000S =0.3578fl =00.40000.4000 fl =0.6325S =0.35780.4319S =0.357800.43190.4566Er =0.0988 fl =0.200000.2000 fl =0.4472 x =0.6000 fl =1.22180.357800.43190.45660.460300.357800.43190.45660.46030.46980.3578000.43190.456600.46030.46980.4707Er =0.0141 fl =0H =0.1000 x =0.1000

4、 fl =0.3162 x =0.3000 fl =0.864()0.50001.5711 x =0.70002.40770.3578000.43190.456600.46030.46980.47070.47090080.3578000.43190.456600.46030.46980.47070.47090.474500.3578000.43190.456600.46030.46980.47070.47090.47450.47480.35780000.43190.4566000.46030.46980.470700.47090.47450.47480.4748Er =0.00420.4748

5、 step =3I =0.4748 step =35xt +2x2 +x3 =一12,2设方程组-召+4龙2+2 =20,试用Jacobi迭代法求解此方程，x（0 =（0,0,0）r ,当 2xt 3x2 +10x3 = 3.maxx；n-xru)10-5 时终止迭代。解：Jacobi.m 文件:function Jacobi(A, b, max, ups) %max为呈大迭代次数，ups为容许误差 n = lcngth(A); x = zcr()s(n, 1); xl = zcros(n, 1); k = 0;while 1Xl(l) = (b(1)- A(l,2:n) * x(2:n,l)

6、 )/A(l,l)f()r i = 2：n-lxl = (b(i)* x(l:i-l,l) - A(i,i+l：n) * x(i+l：n,l)/A(i3)endxl(n) = (b(n)- A(n,l:n-1) * x(l:n-13) )/A(n.n)k = k + 1if sum(abs(xl - x) = maxfprintffThc Method is disec)nvcrgrntn,)breakendX = xlendif k maxf()r i = l：nfprintf( %d = %fni,xl(i)endend命令：clearclcformat shortA = 5 2 1;-1

7、4 2; 2-3 10； b = -12 20 31;max = 100;cps = lc-5Jacobi(A, b, max, cps)计算结果：i =1A =521-1422-310b =-120010203-2-2-2D0 =0.20000.2500000.1000-0.40(H)0.2500-0.2000-2.40005.00000.3000-2.40005.00000.30000.3000-0.2000-0.50000-2.40005.00000.3000-4.46004.25002.28002xO4.46004.25002.2800i =3x =-4.55602.74502.467

8、0 xO =-4.55602.74502.4670i =4x =-3.99142.62752.0347 xO =-3.99142.62752.0347i =5x =3.85792.98481.8865 xO =-3.85792.98481.8865i =6x =-3.97123.09221.9670 xO = 3.97123.09221.9670x =-4.03033.02372.0219 x0 =-4.03033.02372.0219i =8x =-4.01392.98152.0132 xO =-4.01392.98152.0132i =9x =-3.99522.99001.9972 xO

9、=-3.99522.99001.9972i =10x =-3.99543.00261.9960 xO =-3.99543.00261.9960i =11-4.0002xi =3.00311.9999 x0 =-4.00023.00311.9999i =12x =-4.00123.00002.0010 xO =-4.00123.00002.0010i =13x =-4.00022.99922.0002 xO =-4.00022.99922.0002i =14x =-3.99972.99981.9998 xO =-3.99972.99981.9998i =15x =-3.99993.00021.9999 xO =3.99993.00021.9999i =16x =-4.00003.00012.0000 x0 =-4.00003.00012.0000i =17x =-4.00003.00002.0000 xO =-4.00003.00002.00